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}{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 65 "The Taylor series method for solv ing first order DE's numerically" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Pe ter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Versi on: 10.10.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 7 "de solve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 270 7 "DEsol.m" }{TEXT -1 32 " is required by this worksheet . " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple sessio n by a command similar to the one that follows, where the file path gi ves its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \" K:\\\\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "load root-finding procedures including: \+ " }{TEXT 0 13 "bisect,secant" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 270 7 "roots.m" }{TEXT -1 32 " is required by this wo rksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file \+ path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " read \"K:\\\\Maple/procdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 69 "The Taylor series method for the \+ numerical solution of 1st order DE's" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "Euler's method uses " }{TEXT 265 20 "linear approximation" }{TEXT -1 73 " to provide the \+ step for the next point an an approximate solution curve." }}{PARA 0 " " 0 "" {TEXT -1 75 "\nOne way of looking at this is to observe that we are essentially usng the " }{TEXT 265 15 "first two terms" }{TEXT -1 32 " in the Taylor series expansion " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(x[0]+h)=y(x[0])+`y '`(x[0])*h+`y ''`(x[0])" "6#/- %\"yG6#,&&%\"xG6#\"\"!\"\"\"%\"hGF,,(-F%6#&F)6#F+F,*&-%$y~'G6#&F)6#F+F ,F-F,F,-%%y~''G6#&F)6#F+F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "h^2/2!+`y \+ '''`(x[0])" "6#,&*&%\"hG\"\"#-%*factorialG6#F&!\"\"\"\"\"-%&y~'''G6#&% \"xG6#\"\"!F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "h^3/3!+` . . . `" "6#,& *&%\"hG\"\"$-%*factorialG6#F&!\"\"\"\"\"%(~.~.~.~GF+" }{TEXT -1 14 " - ------ (i) " }}{PARA 0 "" 0 "" {TEXT -1 17 "of the solution " } {XPPEDIT 18 0 "y = y(x)" "6#/%\"yG-F$6#%\"xG" }{TEXT -1 7 " about " } {XPPEDIT 18 0 "x = x[0];" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 72 "To get Euler's formula for the first step of the numerical solution of " }{XPPEDIT 18 0 "dy/dx = f(x,y);" "6#/ *&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$%\"xG%\"yG" }{TEXT -1 28 " with the ini tial condition " }{XPPEDIT 18 0 "y(0) = y[0];" "6#/-%\"yG6#\"\"!&F%6#F '" }{TEXT -1 15 " and step size " }{TEXT 268 1 "h" }{TEXT -1 59 ", tak e the first 2 terms on the right side, and substitute " }{XPPEDIT 18 0 "f(x[0],y[0]);" "6#-%\"fG6$&%\"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 5 " fo r " }{XPPEDIT 18 0 "`y '`(x[0])" "6#-%$y~'G6#&%\"xG6#\"\"!" }{TEXT -1 25 ". \nThis gives the formula" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y[1] = y[0]+f(x[0],y[0])*h;" "6#/&%\"yG6#\"\"\",&&F%6# \"\"!F'*&-%\"fG6$&%\"xG6#F+&F%6#F+F'%\"hGF'F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "To improv e on Euler's method, we can incorporate the next term of the Taylor se ries expansion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "In order to do this we need to find an expression for " } {XPPEDIT 18 0 "`y ''`(x[0])" "6#-%%y~''G6#&%\"xG6#\"\"!" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Con sider the differential equation " }{XPPEDIT 18 0 "dy/dx = -x/y;" "6#/ *&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yGF(F(" }{TEXT -1 34 " subject to the initial condition " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 77 "This differential e quation can be solved by separating the variables to give " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(y,y) = -Int(x,x);" "6#/ -%$IntG6$%\"yGF',$-F%6$%\"xGF+!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2/2=-x^2/2+c" "6#/*&%\"yG\"\"#F&!\"\",&*&%\"xGF&F&F'F'%\"cG\" \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "The intial condi tion implies that " }{XPPEDIT 18 0 "c=1/2" "6#/%\"cG*&\"\"\"F&\"\"#!\" \"" }{TEXT -1 13 " which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y^2/2=1/2-x^2/2" "6#/*&%\"yG\"\"#F&!\"\",&*&\"\"\"F*F&F 'F**&%\"xGF&F&F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2=1-x^2" "6#/*$ %\"yG\"\"#,&\"\"\"F(*$%\"xGF&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "This is equation is equivalent to " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 280 1 "y" }{TEXT -1 3 " = " }{TEXT 281 1 "+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1-x^2)" "6#-%%sqrtG6#,&\"\"\"F'*$% \"xG\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Because \+ " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 6 " when " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 28 ", we take the + sign giving " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1- x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 279 6 "______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "W e now consider the problem of constructing a numerical solution using \+ the " }{TEXT 265 17 "first three terms" }{TEXT -1 27 " of the Taylor s eries (i). " }}{PARA 0 "" 0 "" {TEXT -1 69 "We need to find an appropr iate expression for the second derivative. " }}{PARA 0 "" 0 "" {TEXT -1 16 "Differentiating " }{XPPEDIT 18 0 "-x/y;" "6#,$*&%\"xG\"\"\"%\"y G!\"\"F(" }{TEXT -1 2 " " }{TEXT 265 10 "implicitly" }{TEXT -1 17 " w ith respect to " }{TEXT 269 1 "x" }{TEXT -1 32 ", using the quotient r ule gives:" }}{PARA 256 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 "[-x/y];" "6#7#,$*& %\"xG\"\"\"%\"yG!\"\"F)" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "(-y+x*``( dy/dx))/(y^2);" "6#*&,&%\"yG!\"\"*&%\"xG\"\"\"-%!G6#*&%#dyGF)%#dxGF&F) F)F)*$F%\"\"#F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "We can now substitute " }{XPPEDIT 18 0 "d y/dx=-x/y" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yGF(F(" }{TEXT -1 62 " to obtain a formula for the 2nd derivative at a general point" } {XPPEDIT 18 0 " ``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 9 ", namely:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = - (y^2+x^2)/(y^3);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\",$* &,&*$F'F&F(*$F+F&F(F(*$F'\"\"$F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "We can perform this calcu lation using Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "diff(-x/y(x),x);\nexp1 := normal(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%-%\"yG6#%\"xG!\"\"F**(F)F% F&!\"#-%%diffG6$F&F)F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exp1G*& ,&-%\"yG6#%\"xG!\"\"*&F*\"\"\"-%%diffG6$F'F*F-F-F-F'!\"#" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Now we can make th e sustitution " }{XPPEDIT 18 0 "diff(y(x),x) = -x/y(x);" "6#/-%%diffG 6$-%\"yG6#%\"xGF*,$*&F*\"\"\"-F(6#F*!\"\"F0" }{TEXT -1 68 ", which is \+ just the statement of the original differential equation." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "subs( diff(y(x),x)=-x/y(x),exp1);\nexp2 := normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%\"yG6#%\"xG!\"\"*&F(\"\"#F%F)F)\"\"\"F%!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exp2G,$*&,&*$)-%\"yG6#%\"xG\"\"#\" \"\"F/*$)F-F.F/F/F/F*!\"$!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG " }{TEXT -1 62 " still appears in the expression, so we make the subst itution " }{XPPEDIT 18 0 "y(x)=y" "6#/-%\"yG6#%\"xGF%" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "exp3 := subs(y(x)=y,exp2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%exp3G,$*&,&*$)%\"yG\"\"#\"\"\"F,*$)%\"xGF+F,F,F,F*!\"$!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "All of this can be achieved mor e readily by using " }{TEXT 265 45 "the chain rule for functions of tw o variables" }{TEXT -1 27 ", which gives the formula: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f( x,y) = diff(f(x,y),x)+diff(f(x,y),y)*``(dy/dx);" "6#/-%\"fG6$%\"xG%\"y G,&-%%diffG6$-F%6$F'F(F'\"\"\"*&-F+6$-F%6$F'F(F(F/-%!G6#*&%#dyGF/%#dxG !\"\"F/F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "`` = diff(f(x,y),x)+diff (f(x,y),y)*f(x,y);" "6#/%!G,&-%%diffG6$-%\"fG6$%\"xG%\"yGF,\"\"\"*&-F' 6$-F*6$F,F-F-F.-F*6$F,F-F.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= ``" "6# /%!GF$" }{TEXT -1 1 "f" }{XPPEDIT 18 0 "``[x](x,y)+``" "6#,&-&%!G6#%\" xG6$F(%\"yG\"\"\"F&F+" }{TEXT -1 1 "f" }{XPPEDIT 18 0 "``[y](x,y)*f(x, y)" "6#*&-&%!G6#%\"yG6$%\"xGF(\"\"\"-%\"fG6$F*F(F+" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "This for mula involves the partial derivatves " }{XPPEDIT 18 0 "diff(f(x,y),x) = ``;" "6#/-%%diffG6$-%\"fG6$%\"xG%\"yGF*%!G" }{TEXT -1 1 "f" } {XPPEDIT 18 0 "``[x](x,y)" "6#-&%!G6#%\"xG6$F'%\"yG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "diff(f(x,y),y) = ``;" "6#/-%%diffG6$-%\"fG6$%\"xG% \"yGF+%!G" }{TEXT -1 1 "f" }{XPPEDIT 18 0 "``[y](x,y)" "6#-&%!G6#%\"yG 6$%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "For the curr ent example, where " }{XPPEDIT 18 0 "f(x,y) = -x/y;" "6#/-%\"fG6$%\"x G%\"yG,$*&F'\"\"\"F(!\"\"F," }{TEXT -1 22 " , the derivative of " } {XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 17 " with resp ect to " }{TEXT 274 1 "x" }{TEXT -1 51 " can be obtained in simplified form as a function " }{XPPEDIT 18 0 "f[2](x,y);" "6#-&%\"fG6#\"\"#6$ %\"xG%\"yG" }{TEXT -1 22 " of the two variables " }{TEXT 272 1 "x" } {TEXT -1 5 " and " }{TEXT 273 1 "y" }{TEXT -1 13 " as follows. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "f := (x,y) -> -x/y;\ndiff(f(x,y),x) + diff(f(x,y),y)*f(x,y);\nf2 : = unapply(normal(%),x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6 $%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&9$\"\"\"9%!\"\"F2F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%%\"yG!\"\"F'*&%\"xG\"\"#F& !\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2Gf*6$%\"xG%\"yG6\"6$%)o peratorG%&arrowGF),$*&,&*$)9%\"\"#\"\"\"F4*$)9$F3F4F4F4F2!\"$!\"\"F)F) F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The Taylor series estimat e for the " }{TEXT 271 1 "y" }{TEXT -1 56 " coordinate of the next poi nt on the solution curve for " }{XPPEDIT 18 0 "x = x[0]+h;" "6#/%\"xG, &&F$6#\"\"!\"\"\"%\"hGF)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x[1];" "6# &%\"xG6#\"\"\"" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y[1] = y[0]+f(x[0],y[0])*h+f[2](x[0],y[0])*``(h^2/2);" "6#/&%\"yG6#\"\"\",(&F%6#\"\"!F'*&-%\"fG6$&%\"xG6#F+&F%6#F+F'%\"hGF'F' *&-&F.6#\"\"#6$&F16#F+&F%6#F+F'-%!G6#*&F5F:F:!\"\"F'F'" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We say that this is an " }{TEXT 265 15 "order 2 formula" }{TEXT 257 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "The fi rst omitted term of the Taylor series " }{XPPEDIT 18 0 "`y '''`(x[0]) " "6#-%&y~'''G6#&%\"xG6#\"\"!" }{TEXT -1 1 " " }{XPPEDIT 18 0 "h^3/3!; " "6#*&%\"hG\"\"$-%*factorialG6#F%!\"\"" }{TEXT -1 22 " is proportio nal to " }{XPPEDIT 18 0 "h^3;" "6#*$%\"hG\"\"$" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 166 "It is often reasonable to suppose that t his this term gives the substantial part of the truncation error, in w hich case the error is approximately is proportional to " }{XPPEDIT 18 0 "h^3;" "6#*$%\"hG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "The following loop uses the or der 2 Taylor series formula (i) to construct a numerical solution curv e for the differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = -x/y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&% \"yGF(F(" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 18 "over the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\" !%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 28 " with the \+ initial condition " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "20 steps of width " } {XPPEDIT 18 0 "h=1/20" "6#/%\"hG*&\"\"\"F&\"#?!\"\"" }{TEXT -1 74 " ar e performed, and the solution is compared with the analytical solution " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " y = sqrt(1-x^2);" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#!\"\"" } {TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 905 "f := (x, y) -> -x/y;\nf2 := unapply(simplify(diff(f(x,y),x)+diff(f(x,y),y)*f(x, y)),x,y);\nh := 0.05: # width of step\nn := 20: # number of \+ steps\nxk := 0: # initial value of x\nyk := 1.0: # initial v alue of y\nsoln := [xk,yk]:\nprintf(\" xk yk \+ 'exact' y absolute relative\\n\");\nprintf(\" \+ error error\\n\\n\");\nfor k from 1 to \+ n do\n yk := evalf(yk+h*f(xk,yk)+f2(xk,yk)*h^2/2); # Taylor formula \n xk := xk + h;\n soln := soln,[xk,yk];\n zk := sqrt(1-xk^2); \+ # \"exact\" value\n abserr := abs(yk-zk);\n if zk <> 0 then relerr := abserr/abs(zk) else relerr := infinity end if;\n printf( \"%12.10f %12.10f %12.10f %4.2e %4.2e\\n\",xk,yk,zk,abserr,rel err);\nend do:\nsoln := [soln];\nplot([sqrt(1-x^2),soln$2],x=0..1,colo r=[red,green,blue],\n thickness=[1,2],style=[line$2,point], scaling=constrained);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\" xG%\"yG6\"6$%)operatorG%&arrowGF),$*&9$\"\"\"9%!\"\"F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2Gf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowG F),$*&,&*$)9%\"\"#\"\"\"F4*$)9$F3F4F4F4F2!\"$!\"\"F)F)F)" }}{PARA 6 " " 1 "" {TEXT -1 64 " xk yk 'exact' y absol ute relative" }}{PARA 6 "" 1 "" {TEXT -1 62 " \+ error error" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 " .0500000000 .9987500000 .9987 492178 7.82e-07 7.83e-07" }}{PARA 6 "" 1 "" {TEXT -1 64 " .1000000 000 .9949921699 .9949874371 4.73e-06 4.76e-06" }}{PARA 6 "" 1 "" {TEXT -1 64 " .1500000000 .9886980237 .9886859967 1.20e-0 5 1.22e-05" }}{PARA 6 "" 1 "" {TEXT -1 64 " .2000000000 .97981890 04 .9797958971 2.30e-05 2.35e-05" }}{PARA 6 "" 1 "" {TEXT -1 64 " .2500000000 .9682840333 .9682458366 3.82e-05 3.94e-05" }}{PARA 6 "" 1 "" {TEXT -1 64 " .3000000000 .9539975985 .9539392 014 5.84e-05 6.12e-05" }}{PARA 6 "" 1 "" {TEXT -1 64 " .3500000000 .9368344408 .9367496998 8.47e-05 9.05e-05" }}{PARA 6 "" 1 " " {TEXT -1 64 " .4000000000 .9166339987 .9165151390 1.19e-04 \+ 1.30e-04" }}{PARA 6 "" 1 "" {TEXT -1 64 " .4500000000 .8931916722 \+ .8930285550 1.63e-04 1.83e-04" }}{PARA 6 "" 1 "" {TEXT -1 64 " \+ .5000000000 .8662464118 .8660254038 2.21e-04 2.55e-04" }} {PARA 6 "" 1 "" {TEXT -1 64 " .5500000000 .8354624989 .835164654 4 2.98e-04 3.57e-04" }}{PARA 6 "" 1 "" {TEXT -1 64 " .6000000000 \+ .8004020046 .8000000000 4.02e-04 5.03e-04" }}{PARA 6 "" 1 "" {TEXT -1 64 " .6500000000 .7604815413 .7599342077 5.47e-04 7 .20e-04" }}{PARA 6 "" 1 "" {TEXT -1 64 " .7000000000 .7149009659 \+ .7141428429 7.58e-04 1.06e-03" }}{PARA 6 "" 1 "" {TEXT -1 64 " .7 500000000 .6625182774 .6614378278 1.08e-03 1.63e-03" }} {PARA 6 "" 1 "" {TEXT -1 64 " .8000000000 .6016114216 .600000000 0 1.61e-03 2.69e-03" }}{PARA 6 "" 1 "" {TEXT -1 64 " .8500000000 \+ .5293715470 .5267826876 2.59e-03 4.91e-03" }}{PARA 6 "" 1 "" {TEXT -1 64 " .9000000000 .4406384999 .4358898944 4.75e-03 1 .09e-02" }}{PARA 6 "" 1 "" {TEXT -1 64 " .9500000000 .3238427367 \+ .3122498999 1.16e-02 3.71e-02" }}{PARA 6 "" 1 "" {TEXT -1 60 "1.0 000000000 .1400901398 0.0000000000 1.40e-01 Inf" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solnG777$\"\"!$\"#5!\"\"7$$\"\"&!\"#$\"+++]( )**!#57$$F)F.$\"+*p@*\\**F17$$\"#:F.$\"+P-)p))*F17$$\"#?F.$\"+/!*=)z*F 17$$\"#DF.$\"+L.%Go*F17$$\"#IF.$\"+&)f(*R&*F17$$\"#NF.$\"+3WMo$*F17$$ \"#SF.$\"+()*Rj;*F17$$\"#XF.$\"+An\">$*)F17$$\"#]F.$\"+=TYi')F17$$\"#b F.$\"+*)\\ia$)F17$$\"#gF.$\"+Y+-/!)F17$$\"#lF.$\"+8a\"[g(F17$$\"#qF.$ \"+f'4!\\rF17$$\"#vF.$\"+uF=DmF17$$\"#!)F.$\"+;U6;gF17$$\"#&)F.$\"+qar $H&F17$$\"#!*F.$\"+**\\Q1WF17$$\"#&*F.$\"+ntUQKF17$$\"$+\"F.$\"+)R,4S \"F1" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVE SG6&7Y7$$\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#>$\"3e(=:\"QTi(***!#=7$$\"3[ LL$e9ui2%F/$\"3!e2E0a)o\"***F27$$\"3nmmm\"z_\"4iF/$\"3)f\\s^f/2)**F27$ $\"3[mmmT&phN)F/$\"3)37fk0E]'**F27$$\"3CLLe*=)H\\5F2$\"3R\\N+$H'zW**F2 7$$\"3gmm\"z/3uC\"F2$\"3^l4$yi$*=#**F27$$\"3%)***\\7LRDX\"F2$\"3%p#zm \"3WR*)*F27$$\"3]mm\"zR'ok;F2$\"3iaa\"Q\\n/')*F27$$\"3w***\\i5`h(=F2$ \"3\"Rj1r%eUA)*F27$$\"3WLLL3En$4#F2$\"3_.Ayu2Py(*F27$$\"3qmm;/RE&G#F2$ \"3wIBk#=x`t*F27$$\"3\")*****\\K]4]#F2$\"31q9\"H%H@#o*F27$$\"3$****** \\PAvr#F2$\"3B_^A([sOi*F27$$\"3)******\\nHi#HF2$\"3m:#e7,zAc*F27$$\"3j mm\"z*ev:JF2$\"3)>#\\;*p8A]*F27$$\"3?LLL347TLF2$\"3pW!GHZL`U*F27$$\"3, LLLLY.KNF2$\"3#=1W(>`Yb$*F27$$\"3w***\\7o7Tv$F2$\"3!QZNcQ$eo#*F27$$\"3 'GLLLQ*o]RF2$\"3=`?'yv9l=*F27$$\"3A++D\"=lj;%F2$\"3]kxU3;t!4*F27$$\"31 ++vV&R5j')F27$$\" 3&em;zRQb@&F2$\"3e0&=5t$=K&)F27$$\"3\\***\\(=>Y2aF2$\"3ld&yZ.e=T)F27$$ \"39mm;zXu9cF2$\"3]/X@$oS\\F)F27$$\"3l******\\y))GeF2$\"3k!yShX>b7)F27 $$\"3'*)***\\i_QQgF2$\"35nMlUm1rzF27$$\"3@***\\7y%3TiF2$\"3bX$z*QqP8yF 27$$\"35****\\P![hY'F2$\"3QXhIao;GwF27$$\"3kKLL$Qx$omF2$\"3q\"p\\+YH?X (F27$$\"3!)*****\\P+V)oF2$\"378IrHv-`sF27$$\"3?mm\"zpe*zqF2$\"3QynnKd; iqF27$$\"3%)*****\\#\\'QH(F2$\"3G\"QTZ_=5%oF27$$\"3GKLe9S8&\\(F2$\"3a% Rp()p\"*)>mF27$$\"3R***\\i?=bq(F2$\"3=&F27$$\"3=LLe9tOc()F2$\"3 PnBuRIqH[F27$$\"3u******\\Qk\\*)F2$\"3+@I:/fPhWF27$$\"3CLL$3dg6<*F2$\" 3au)*4GH?')RF27$$\"3ImmmmxGp$*F2$\"3s!RL6R._\\$F27$$\"3A++D\"oK0e*F2$ \"3!yTAE(=!f'GF27$$\"3C+++]oi\"o*F2$\"3?UJ1O#=K]#F27$$\"3A++v=5s#y*F2$ \"37vO>U4Dt?F27$$\"3;+D1k2/P)*F2$\"3mLW(eL^zz\"F27$$\"35+]P40O\"*)*F2$ \"31H.f%oH+Z\"F27$$\"3k]7.#Q?&=**F2$\"3G#>z^*=&RF\"F27$$\"31+voa-oX**F 2$\"3(pV\\8H')3/\"F27$$\"3[\\PMF,%G(**F2$\"3?&=hJ*p=ltF/7$F*F(-%'COLOU RG6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG-%*THICKNESSG6#F+-F$6&77 F'7$$\"3G+++++++]F/$\"3E+++++]()**F27$$\"3/+++++++5F2$\"3Y******)p@*\\ **F27$$\"3%**************\\\"F2$\"3S+++P-)p))*F27$$\"35+++++++?F2$\"3q *****R+*=)z*F27$$\"3++++++++DF2$\"39+++L.%Go*F27$$\"3))**************H F2$\"3))*****\\)f(*R&*F27$$\"3w*************\\$F2$\"3++++3WMo$*F27$$\" 3A+++++++SF2$\"3_*****p)*Rj;*F27$$\"35+++++++XF2$\"3`*****>s;>$*)F27$$ \"3++++++++]F2$\"3]*****z6kCm)F27$$\"3U+++++++bF2$\"3;+++*)\\ia$)F27$$ \"3w**************fF2$\"3%******f/?S+)F27$$\"3A+++++++lF2$\"33+++8a\"[ g(F27$$\"3a**************pF2$\"3i******e'4!\\rF27$$\"3++++++++vF2$\"3A +++uF=DmF27$$\"3U+++++++!)F2$\"3H+++;U6;gF27$$\"3w*************\\)F2$ \"3'*******par$H&F27$$\"3A+++++++!*F2$\"3!)******)*\\Q1WF27$$\"3a***** ********\\*F2$\"3A+++ntUQKF27$F*$\"38+++)R,4S\"F2-Fc\\l6&Fe\\lF(Ff\\lF (Fi\\l-F^]l6#\"\"#-F$6&Fb]l-Fc\\l6&Fe\\lF(F(Ff\\l-Fj\\l6#%&POINTGF]]l- %(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$Q\"x6\"Q!Fidl-%%VIEWG6$;F(F* %(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "C urve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 77 " \nThe er rors are considerably smaller than those obtained by Euler's method. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 823 "f := (x,y) -> -x/y;\nh := 0.05: # width of step\nn := 20: # number of steps\nxk := 0: # initial value of x\nyk := 1 .0: # initial value of y\nsoln := [xk,yk]:\nprintf(\" xk \+ yk 'exact' y absolute relative\\n\");\nprintf(\" \+ error error\\n\\n\") ;\nfor k from 1 to n do\n yk := evalf(yk+h*f(xk,yk)); # Euler's form ula\n xk := xk + h;\n soln := soln,[xk,yk];\n zk := sqrt(1-xk^2) ; # \"exact\" value\n abserr := abs(yk-zk);\n if zk <> 0 t hen relerr := abserr/abs(zk) else relerr := infinity end if;\n print f(\"%12.10f %12.10f %12.10f %4.2e %4.2e\\n\",xk,yk,zk,abserr,r elerr);\nend do:\nsoln := [soln];\nplot([sqrt(1-x^2),soln$2],x=0..1,co lor=[red,green,blue],\n thickness=[1,2],style=[line$2,point ],scaling=constrained);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$% \"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&9$\"\"\"9%!\"\"F2F)F)F)" }} {PARA 6 "" 1 "" {TEXT -1 64 " xk yk 'exact' y absolute relative" }}{PARA 6 "" 1 "" {TEXT -1 62 " \+ error error" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 " .0500000000 1.00000000 00 .9987492178 1.25e-03 1.25e-03" }}{PARA 6 "" 1 "" {TEXT -1 64 " .1000000000 .9975000000 .9949874371 2.51e-03 2.53e-03" }}{PARA 6 "" 1 "" {TEXT -1 64 " .1500000000 .9924874687 .9886859 967 3.80e-03 3.84e-03" }}{PARA 6 "" 1 "" {TEXT -1 64 " .2000000000 .9849306982 .9797958971 5.13e-03 5.24e-03" }}{PARA 6 "" 1 " " {TEXT -1 64 " .2500000000 .9747776996 .9682458366 6.53e-03 \+ 6.75e-03" }}{PARA 6 "" 1 "" {TEXT -1 64 " .3000000000 .9619542630 \+ .9539392014 8.02e-03 8.40e-03" }}{PARA 6 "" 1 "" {TEXT -1 64 " \+ .3500000000 .9463610060 .9367496998 9.61e-03 1.03e-02" }} {PARA 6 "" 1 "" {TEXT -1 64 " .4000000000 .9278691198 .916515139 0 1.14e-02 1.24e-02" }}{PARA 6 "" 1 "" {TEXT -1 64 " .4500000000 \+ .9063143557 .8930285550 1.33e-02 1.49e-02" }}{PARA 6 "" 1 "" {TEXT -1 64 " .5000000000 .8814885325 .8660254038 1.55e-02 1 .79e-02" }}{PARA 6 "" 1 "" {TEXT -1 64 " .5500000000 .8531274148 \+ .8351646544 1.80e-02 2.15e-02" }}{PARA 6 "" 1 "" {TEXT -1 64 " .6 000000000 .8208930738 .8000000000 2.09e-02 2.61e-02" }} {PARA 6 "" 1 "" {TEXT -1 64 " .6500000000 .7843475102 .759934207 7 2.44e-02 3.21e-02" }}{PARA 6 "" 1 "" {TEXT -1 64 " .7000000000 \+ .7429117951 .7141428429 2.88e-02 4.03e-02" }}{PARA 6 "" 1 "" {TEXT -1 64 " .7500000000 .6957998765 .6614378278 3.44e-02 5 .20e-02" }}{PARA 6 "" 1 "" {TEXT -1 64 " .8000000000 .6419050696 \+ .6000000000 4.19e-02 6.98e-02" }}{PARA 6 "" 1 "" {TEXT -1 64 " .8 500000000 .5795905594 .5267826876 5.28e-02 1.00e-01" }} {PARA 6 "" 1 "" {TEXT -1 64 " .9000000000 .5062629330 .435889894 4 7.04e-02 1.61e-01" }}{PARA 6 "" 1 "" {TEXT -1 64 " .9500000000 \+ .4173763148 .3122498999 1.05e-01 3.37e-01" }}{PARA 6 "" 1 "" {TEXT -1 60 "1.0000000000 .3035701444 0.0000000000 3.04e-01 \+ Inf" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solnG777$\"\"!$\"#5!\"\"7$$ \"\"&!\"#F(7$$F)F.$\"++++v**!#57$$\"#:F.$\"+(ou[#**F37$$\"#?F.$\"+#)pI \\)*F37$$\"#DF.$\"+'*pxZ(*F37$$\"#IF.$\"+IEa>'*F37$$\"#NF.$\"+g+hj%*F3 7$$\"#SF.$\"+)>\"py#*F37$$\"#XF.$\"+dN9j!*F37$$\"#]F.$\"+D`)[\"))F37$$ \"#bF.$\"+[TFJ&)F37$$\"#gF.$\"+Q2$*3#)F37$$\"#lF.$\"+-^ZVyF37$$\"#qF.$ \"+^z6HuF37$$\"#vF.$\"+l()*z&pF37$$\"#!)F.$\"+'p]!>kF37$$\"#&)F.$\"+%f 0fz&F37$$\"#!*F.$\"+I$HE1&F37$$\"#&*F.$\"+[JwtTF37$$\"$+\"F.$\"+W9qNIF 3" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6 &7Y7$$\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#>$\"3e(=:\"QTi(***!#=7$$\"3[LL$ e9ui2%F/$\"3!e2E0a)o\"***F27$$\"3nmmm\"z_\"4iF/$\"3)f\\s^f/2)**F27$$\" 3[mmmT&phN)F/$\"3)37fk0E]'**F27$$\"3CLLe*=)H\\5F2$\"3R\\N+$H'zW**F27$$ \"3gmm\"z/3uC\"F2$\"3^l4$yi$*=#**F27$$\"3%)***\\7LRDX\"F2$\"3%p#zm\"3W R*)*F27$$\"3]mm\"zR'ok;F2$\"3iaa\"Q\\n/')*F27$$\"3w***\\i5`h(=F2$\"3\" Rj1r%eUA)*F27$$\"3WLLL3En$4#F2$\"3_.Ayu2Py(*F27$$\"3qmm;/RE&G#F2$\"3wI Bk#=x`t*F27$$\"3\")*****\\K]4]#F2$\"31q9\"H%H@#o*F27$$\"3$******\\PAvr #F2$\"3B_^A([sOi*F27$$\"3)******\\nHi#HF2$\"3m:#e7,zAc*F27$$\"3jmm\"z* ev:JF2$\"3)>#\\;*p8A]*F27$$\"3?LLL347TLF2$\"3pW!GHZL`U*F27$$\"3,LLLLY. 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[x[n], y[n]]];" "6#7$7$&%\"xG6#\"\"\"&%\"yG6#F(;7$&F&6#\"\" #&F*6#F07$&F&6#%\"nG&F*6#F6" }{TEXT 302 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 15 " f or fx - " }{TEXT -1 87 " a function of one variable or an expression fx defining a f unction of one variable." }}{PARA 0 "" 0 "" {TEXT -1 6 " " }} {PARA 0 "" 0 "" {TEXT 23 17 " x - " }{TEXT 299 65 "the ind ependent variable is required when the 2nd argument is an " }{TEXT -1 10 "expression" }{TEXT 303 4 " fx." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 23 3 " " }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "Descr iption:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 14 "c omparewithfcn" }{TEXT -1 78 " tabulates the list of points vertically \+ as the first two columns of a matrix." }}{PARA 0 "" 0 "" {TEXT -1 263 "The values of the 2nd components are compared with the values obtaine d by applying the given function to the corresponding 1st components, \+ and listing these \"exact values\" in the 3rd column of the matrix. Th e absolute or relative error is given in the 4th column." }}{PARA 0 " " 0 "" {TEXT -1 68 "Alternatively, the same information can be printed out line by line." }}{PARA 0 "" 0 "" {TEXT -1 128 "The maximum of the absolute or relative errors is given. When the absolute error is tabu lated, the mean absolute error is given." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 22 "mode=linebyline/matrix" }}{PARA 0 "" 0 "" {TEXT -1 106 "W ith the option \"mode=linebyline\" the information is printed out line by line. This is the default option." }}{PARA 0 "" 0 "" {TEXT -1 112 "With the option \"mode=matrix\" the tabulated information is given in a matrix as a return value of the procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "errtype=relative/absolute " }}{PARA 0 "" 0 "" {TEXT -1 152 "This option determines whether the a bsolute or relative error is given. The default is \"errtype=relative \" in which case the relative error is tabulated." }}{PARA 0 "" 0 "" {TEXT -1 73 "\"errtype=RELATIVE\" and \"errtype=REL\" are equivalent t o \"errtype=relative\"" }}{PARA 0 "" 0 "" {TEXT -1 75 "\"errtype=ABSOL UTE\" and \"errtype=ABS\" are equivalent to \"errtype=absolute\"." }} {PARA 0 "" 0 "" {TEXT -1 43 "\"errtype\" can also be typed as \"errort ype\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 4 "Note" }{TEXT -1 77 ": If the true value of the function is zero, the \+ relative error is infinite. " }}{PARA 0 "" 0 "" {TEXT -1 113 "The maxi mum relative error is computed for the remaining relative errors, igno ring any infinite relative errors. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 15 "How to activate" }{TEXT -1 1 ":" }{TEXT 267 1 "\n" }{TEXT -1 154 "To make th e procedure active open the subsection, place the cursor anywhere afte r the prompt [ > and press [Enter].\nYou can then close up the subsec tion." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "comparewithfcn: implemen tation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "compa rewithfcn" {MPLTEXT 1 0 6095 "comparewithfcn := proc(pts,f)\n local \+ xx,y1,y2,r,i,j,n,rows,fn,x,prec,prcsn,saveDigits,\n startoptions,Opt ions,md,t,maxerr,format1,format2,g,e,zero,\n proctype,ertyp,xmax,pre c1,prec2,vars,averr;\n\n proctype := false;\n if nargs<2 then\n \+ error \"invalid arguments; the basic syntax is 'comparewithfcn([[x1 ,y1],..,[xn,yn]],f(x),x)' or 'comparewithfcn([[x1,y1],..,[xn,yn]],f)' \"\n end if;\n if not type(pts,listlist) then\n error \"the 1 st argument, %1, is invalid .. it should be a list of points\",pts;\n \+ end if;\n if nargs>2 and not type(args[3],equation) then\n x \+ := args[3];\n if not type(x,name) then\n error \"the 3rd \+ optional argument must be the name of the independent variable\"\n \+ end if;\n startoptions := 4;\n vars := indets(f,name) minu s indets(f,realcons);\n if vars<>\{x\} then \n if not typ e(f,algebraic) or not has(indets(f),\{Int,Sum\}) then\n err or \"the 2nd argument, %1, is invalid .. it should be an expression wh ich depends only on the single variable %2\",f,x;\n end if;\n \+ end if;\n else\n if type(f,procedure) or (type(f,`@`) and \+ type(\{op(f)\},set(procedure))) then\n proctype := true;\n \+ startoptions := 3;\n else\n error \"the 2nd argument , %1, is invalid .. it should be a function of one variable or an expr ession in the one variable given as a 3rd argument\",f;\n end if; \n end if; \n\n # Get the options.\n md := 0;\n ertyp := 1; \n if nargs>=startoptions then\n Options :=[args[startoptions.. nargs]];\n if not type(Options,list(equation)) then\n err or \"each optional argument must be an equation\"\n end if;\n \+ if hasoption(Options,'mode','md','Options') then\n if not (m d='linebyline' or md='matrix') then\n error \"\\\"mode\\\" \+ must be 'matrix' or 'linebyline'\"\n end if;\n if md=' linebyline' then md := 0 else md := 1 end if;\n end if;\n if hasoption(Options,'errtype','ertyp','Options') or \n hasoptio n(Options,'errortype','ertyp','Options') then\n if not member( ertyp,\{'absolute','ABSOLUTE','ABS','relative','RELATIVE','REL'\}) the n\n error \"\\\"errtype\\\" option must be 'absolute' <-> ' ABSOLUTE' <-> 'ABS' or 'relative' <-> 'RELATIVE' <-> 'REL'\"\n \+ end if;\n if member(ertyp,\{'absolute','ABSOLUTE','ABS'\})\n \+ then ertyp := 0 else ertyp := 1 end if;\n end if;\n \+ if nops(Options)>0 then\n error \"%1 is not a valid option for %2\",op(1,Options), procname;\n end if;\n end if;\n n := nop s(pts);\n prcsn := 0;\n\n # Check the data and find its maximum pr ecision.\n for i to n do\n if nops(pts[i])<>2 then\n er ror \"the 1st argument must be a list of points, where each point is i tself a list with two members\"\n end if;\n t := pts[i,1];\n if type(t,float) and type(t,numeric) then\n prec1 := len gth(convert(op(1,t),string));\n elif type(t,realcons) then\n \+ prec1 := Digits;\n else\n error \"the 1st argument mu st be a list of points, where each point is itself a list of two real \+ numbers\"\n end if;\n t := pts[i,2];\n if type(t,float) and type(t,numeric) then\n prec2 := length(op(1,t));\n e lif type(t,realcons) then\n prec2 := Digits;\n else\n \+ error \"the 1st argument must be a list of points, where each poi nt is itself a list of two real numbers\"\n end if;\n prec : = max(prec1,prec2);\n if prec>prcsn then prcsn := prec end if;\n \+ end do;\n saveDigits := Digits;\n Digits := prcsn;\n prec := t runc(prcsn/2);\n \n if proctype then\n fn := f;\n else\n \+ fn := unapply(evalf(f),x);\n end if;\n rows := NULL;\n maxerr := 0;\n averr := 0;\n zero := false;\n format1 := cat(\"%\",con vert(prcsn+5,string),\".\",convert(prcsn,string),g);\n format2 := ca t(\"%\",convert(prec+4,string),\".\",convert(prec-1,string),e);\n\n \+ for i from 1 to n do\n xx := evalf(pts[i,1]);\n y1 := evalf( pts[i,2]);\n y2 := traperror(evalf(fn(xx)));\n if y2=lasterr or or not type(y2,numeric) then\n error \"function failed to e valuate to a real floating point number at %1\",xx;\n end if;\n \+ \n if ertyp=0 then\n r := evalf(abs(y1-y2));\n \+ averr := averr+r;\n if r>maxerr then\n maxerr := \+ r;\n xmax := xx;\n end if;\n else\n if y2<>0 then\n r := evalf(abs(y1-y2)/abs(y2));\n \+ if r>maxerr then\n maxerr := r;\n xmax := \+ xx;\n end if;\n else\n zero := true;\n \+ r := infinity;\n end if;\n end if;\n\n if \+ md=0 then\n printf(format1,xx);\n printf(` `);\n \+ printf(format1,y1);\n printf(` function val: `);\n \+ printf(format1,y2);\n if ertyp=0 then\n printf(` \+ abs err: `);\n printf(format2,r);\n else\n \+ printf(` rel err: `);\n if y2<>0 then\n \+ printf(format2,r)\n else\n printf(infin ity);\n end if;\n end if;\n printf(`\\n`); \n else\n rows := rows,[xx,y1,y2,r];\n end if;\n e nd do;\n \n print(``);\n if ertyp=0 then\n printf(` \+ Maximum absolute error: `);\n else\n printf(` \+ Maximum relative error: `); \n end if;\n printf(format2,max err);\n\n if maxerr<>0 then\n printf(`\\n obtained for the input value: `);\n printf(format1,xmax);\n end if;\n \+ \n if ertyp=1 and zero then\n printf(`\\n excludi ng any cases where the function value is zero.`);\n end if;\n\n if ertyp=0 then\n averr := averr/n;\n print(``);\n printf (` Mean absolute error: `);\n printf(format2,averr); \n end if;\n\n Digits := saveDigits;\n if md=1 then\n p rint(``);\n if ertyp=0 then\n return array([[x,\"discrete value\",\"function value\",\"absolute err\"],rows]);\n else\n \+ return array([[x,\"discrete value\",\"function value\",\"relativ e err\"],rows]);\n end if;\n else\n return NULL;\n end i f;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 34 "Examples appear in a later section" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolveTA" } {TEXT -1 12 ": basic idea" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 106 "The procedure in the next section \+ constructs a numerical solution for a differential equation of the for m " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "dy/dx = f(x,y);" "6#/*&%#dyG\" \"\"%#dxG!\"\"-%\"fG6$%\"xG%\"yG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 54 "by using the Taylor series method of prescribed order " } {TEXT 275 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "The numerical solution can be in the form of a " }{TEXT 265 18 "sequence of points" }{TEXT -1 96 " along the so lution curve, which are equally spaced horizontally, or it can be in t he form of a " }{TEXT 265 27 "function of a real variable" }{TEXT -1 99 " which can interpolate between the pre-computed points by using an approximate local Taylor series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 197 "Computer algebra systems such as Maple a nd Mathematica, which allow automatic computation of derivatives, and \+ and the high speed of current processors now make this a reasonable me thod to use when " }{TEXT 265 13 "high accuracy" }{TEXT 257 1 " " } {TEXT -1 11 "is desired." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 21 "The step-size can be " }{TEXT 265 5 "fixed" } {TEXT -1 4 " or " }{TEXT 265 8 "variable" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 62 "When a variable step-size is used, the error in us ing an order" }{XPPEDIT 18 0 "``(n-1);" "6#-%!G6#,&%\"nG\"\"\"F(!\"\" " }{TEXT -1 91 " Taylor series formula for a particular step is approx imated by the last term of the order " }{TEXT 276 1 "n" }{TEXT -1 212 " Taylor series formula. \nIf the error is too large for a particular \+ step, a smaller step is taken instead, and if the error is smaller tha n required, the step-size is increased for the next step. This is cal led " }{TEXT 265 26 "adaptive step-size control" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 53 "The actual value returned is that given b y the order " }{TEXT 277 1 "n" }{TEXT -1 70 " Taylor series, even thou gh the error estimates used are for the order" }{XPPEDIT 18 0 "``(n-1) " "6#-%!G6#,&%\"nG\"\"\"F(!\"\"" }{TEXT -1 15 " Taylor series." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 " A procedure implementing the Taylor series method: " }{TEXT 0 9 "desol veTA" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "desolveTA: \+ usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 258 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 259 2 " \+ " }{TEXT -1 28 " desolveTA( \{de,ic\}, rng )" }{TEXT 260 1 "\n" } {TEXT -1 36 " desolveTA( \{de,ic\}, y(x), rng )" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " de - \+ " }{TEXT -1 92 " a first order differential equation with the deri vative given in the form diff(y(x),x)," }}{PARA 0 "" 0 "" {TEXT -1 94 " (if x and y are the independent and dependent variables respectively)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 10 " ic - " }{TEXT -1 50 " an initial condition in the form y( x0) = y0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 23 8 "rng - " }{TEXT -1 83 " a range of value s containing the initial value x0 of the dependent variable. " }} {PARA 0 "" 0 "" {TEXT -1 9 " " }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "desolveTA" }{TEXT -1 132 " us es the Taylor series method of prescribed order to find a numerical so lution to a first order differential equation of the form " } {XPPEDIT 18 0 "dy/dx = f(x,y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$%\" xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The numerical solution can be given in two ways:" }} {PARA 15 "" 0 "" {TEXT -1 13 "As a list of " }{TEXT 265 6 "points" } {TEXT -1 26 " along the solution curve." }}{PARA 15 "" 0 "" {TEXT -1 5 "As a " }{TEXT 265 19 "numerical procedure" }{TEXT -1 175 " of a rea l variable, defined throughout the specified range, which uses an appr oximate local Taylor series to interpolate between pre-computed points along the solution curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 180 "In both cases there is a choice of using a fixed or variable step-size. In the case of variable step-size a method of \+ adaptive step-size is used in an attempt to control the error." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 8 "Options: " }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 264 46 "Options for both fi xed and variable step-size:" }{TEXT -1 25 "\n\nstepsize=fixed/variable " }}{PARA 0 "" 0 "" {TEXT -1 22 "OR adaptive=false/true" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The default option i s \"stepsize=variable\"." }}{PARA 0 "" 0 "" {TEXT -1 143 "With the opt ion \"stepsize=fixed\", or equivalently \"adaptive=false\", equally sp aced x values are used in construction of the numerical solution." }} {PARA 0 "" 0 "" {TEXT -1 136 "With the option \"stepsize=variable\", \+ or equivalently \"adaptive=true\", the step-size is controlled by esti mating the error at each step." }}{PARA 0 "" 0 "" {TEXT -1 75 "The def ault option is \"stepsize=variable\", or equivalently \"adaptive=true \"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "ord er=n" }}{PARA 0 "" 0 "" {TEXT -1 38 "The order of the Taylor series me thod." }}{PARA 0 "" 0 "" {TEXT -1 119 "The default is \"order =order = min(Digits,15+trunc(abs(Digits-15)^.9))\", which is equal to, or a bit less than,\"Digits\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "output=points/localtaylor" }}{PARA 15 "" 0 "" {TEXT -1 152 "\"output=points\" gives the numerical solution as a set of poi nts along the approximate solution curve.\n\"output=discrete\" is the \+ sames as \"output=points\"." }}{PARA 15 "" 0 "" {TEXT -1 278 "\"output =localtaylor\" gives a function as output.\nThis function uses an appr oximate local Taylor series to interpolate the solution between pre-co mputed equally spaced data points.\n\"output=procedure\", \"output=con tinuous\", \"output=loctaylor\" are the sames as \"output=localtaylor \"." }}{PARA 0 "" 0 "" {TEXT -1 43 "The default option is \"output=loc altaylor\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "info=0, 1, 2, 3 , 4, true or false" }}{PARA 0 "" 0 "" {TEXT -1 116 "With the option \"info=false\" or \"info=0\" no information is pr inted during the computation of the numerical solution." }}{PARA 0 "" 0 "" {TEXT -1 665 "With the option \"info=1\" the formulas for the var ious derivatives to be used is printed, and, in the case of variable s tep-size, the total number of steps is printed.\nWith the option \"inf o=2\", information concerning intermediate steps is printed as the val ues are computed, and, in the case of variable step-size, the total nu mber of steps is printed.\nWith the option \"info=3\", the information of both the previous two cases is printed, including the formulas for the derivatives and values for intermediate steps.\nWith the option \+ \"info=4\", the total number of steps used for the case of variable st ep-size is the only printout.\n\"info=true\" is the same as \"info=2\" ." }}{PARA 0 "" 0 "" {TEXT -1 28 "The default is \"info=false\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 33 "Options \+ for fixed step-size only:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 99 "steps=n\nThe number of steps to be used in covering the interva l. The default number of steps is 20." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 47 "Options for adaptive \+ (variable) step-size only:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 8 "hstart=h" }}{PARA 0 "" 0 "" {TEXT -1 67 "The step-size for the f irst trial step. The default is \"hstart=0.1*" }{XPPEDIT 18 0 "10^ceil (-Digits/10);" "6#)\"#5-%%ceilG6#,$*&%'DigitsG\"\"\"F$!\"\"F," }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "hmax=h" }}{PARA 0 "" 0 "" {TEXT -1 56 "The maximum step-size. The d efault is \"hmax=order*0.05\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 60 "hmin=h\nThe minimum step-size. The defaul t is \"hmin=min(0.5*" }{XPPEDIT 18 0 "10^(-5);" "6#)\"#5,$\"\"&!\"\" " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "hstart/2000;" "6#*&%'hstartG\"\"\" \"%+?!\"\"" }{TEXT -1 3 ")\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "Note: hstart must lie between hmin and hm ax." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "ma xsteps=n" }}{PARA 0 "" 0 "" {TEXT -1 188 "The maximum number of steps \+ to be used. An error message results if the maximum number of steps is reached before reaching the end of the solution interval. The default is \"maxsteps=2000\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "tolerance=" }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 166 "When the error control option is set to \"relativeerror\", the step size is chosen adaptivel y to try to ensure that the relative error for each step is no greater than " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 34 ". Other error control options use " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 106 " in an appropriate way. See \"errorcontrol\" for more i nformation. The default is \"tolerance=10^(-Digits)\".\n" }}{PARA 0 " " 0 "" {TEXT -1 48 "errorcontrol=auto/absolute/relative/accumulative" }}{PARA 0 "" 0 "" {TEXT -1 37 "For the current setting of tolerance=" }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 72 ", the step size \+ is chosen to try to ensure that when \"errorcontrol\" is:" }}{PARA 15 "" 0 "" {TEXT -1 12 "auto - " }{XPPEDIT 18 0 "abs(y-y[k]) <= e psilon*max(abs(y[k]),abs(f(x[k],y[k])*h),tiny);" "6#1-%$absG6#,&%\"yG \"\"\"&F(6#%\"kG!\"\"*&%(epsilonGF)-%$maxG6%-F%6#&F(6#F,-F%6#*&-%\"fG6 $&%\"xG6#F,&F(6#F,F)%\"hGF)%%tinyGF)" }{TEXT -1 3 " \n" }}{PARA 15 " " 0 "" {TEXT -1 16 "absolute - " }{XPPEDIT 18 0 "abs(y-y[k]) <= e psilon*abs(max(y[max],tiny));" "6#1-%$absG6#,&%\"yG\"\"\"&F(6#%\"kG!\" \"*&%(epsilonGF)-F%6#-%$maxG6$&F(6#F3%%tinyGF)" }{TEXT -1 8 ", where \+ " }{XPPEDIT 18 0 "y[max];" "6#&%\"yG6#%$maxG" }{TEXT -1 188 " is updat ed as the computation progresses, or is provided by the option \"maxim um value\".\nThis error control option would be an appropriate choice \+ if the solution is oscillatory in nature.\n" }}{PARA 15 "" 0 "" {TEXT -1 17 "relative - " }{XPPEDIT 18 0 "abs(y-y[k]) <= epsilon*max(a bs(y[k]),tiny);" "6#1-%$absG6#,&%\"yG\"\"\"&F(6#%\"kG!\"\"*&%(epsilonG F)-%$maxG6$-F%6#&F(6#F,%%tinyGF)" }{TEXT -1 169 " \nThis is a general \+ purpose choice except that problems would arise if y is close to 0. If this is likely to happen it would be better to use the default option \"auto\".\n" }}{PARA 15 "" 0 "" {TEXT -1 20 "accumulative - " } {XPPEDIT 18 0 "abs(y-y[k]) <= epsilon*max(abs(f(x[k],y[k])*h),tiny);" "6#1-%$absG6#,&%\"yG\"\"\"&F(6#%\"kG!\"\"*&%(epsilonGF)-%$maxG6$-F%6#* &-%\"fG6$&%\"xG6#F,&F(6#F,F)%\"hGF)%%tinyGF)" }{TEXT -1 2 " " }} {PARA 15 "" 0 "" {TEXT -1 92 "This is the most stringent error control option, which attempts to control the global error." }}{PARA 0 "" 0 " " {TEXT -1 10 "\"tiny\" is " }{XPPEDIT 18 0 "10^(-3*Digits);" "6#)\"#5 ,$*&\"\"$\"\"\"%'DigitsGF(!\"\"" }{TEXT -1 60 ", and is included to av oid the possibility of division by 0." }}{PARA 0 "" 0 "" {TEXT -1 46 " The default value of \"errorcontrol\" is \"auto\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "maxvalue=a" }}{PARA 0 "" 0 "" {TEXT -1 202 "This is only used in the case where the \"errorcont rol\" option has been set to \"absolute\". See the option \"error cont rol. If no value is provided the maximum value is updated as the compu tation procedes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 16 "How to activate:" } {TEXT -1 158 "\nTo make the procedures active open the subsections, pl ace the cursor anywhere after the prompt [ > and press [Enter].\nYou \+ can then close up the subsections." }}{PARA 0 "" 0 "" {TEXT 265 4 "Not e" }{TEXT -1 40 ": Both procedures must be activated for " }{TEXT 0 9 "desolveTA" }{TEXT -1 10 " to work. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "desolveTA: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "desolveTA" {MPLTEXT 1 0 8166 "desolveTA := proc () \n local ff,vars,derivs,x,yx,y,df,dff,Options,mthd,rg,dblrk,\n \+ de,ic,lsic,rsic,x0,y0,tt,rghtprt,lftprt,outpt,reverse,\n loc taylor,stps,stpsz,adptv,lftstps,rghtstps,drv,\n lft,rght,rng,star topts,xx,yy,ee,z,arg2OK,nvars;\n\nreverse := proc(lst::list)\n local n,i;\n n := nops(lst);\n [seq(lst[n-i],i=0..n-2)];\nend proc:\n \+ \ndblrk := proc(lft::realcons,rght::realcons,x0::realcons,\n lf tprc::procedure,rghtprc::procedure)\n proc(x::realcons)\n local left,right,leftproc,rightproc,xx0,xx,saveDigits;\n options `Copy right 2002 by Peter Stone`;\n \n leftproc := lftprc;\n rig htproc := rghtprc;\n \n saveDigits := Digits;\n Digits := \+ max(trunc(evalhf(Digits)),Digits+5);\n\n xx0 := evalf(x0);\n \+ left := evalf(lft);\n right := evalf(rght);\n xx := evalf(x );\n\n if xx<=right and xx>=xx0 then\n Digits := saveDigi ts;\n return rightproc(xx);\n elif xx>=left and xx<=xx0 t hen\n Digits := saveDigits;\n return leftproc(xx);\n \+ else\n error \"argument must be between %1 and %2\",left,r ight; \n end if;\n end proc;\nend proc: # of dblrk\n \n # \+ start of main procedure desolveTA\n if nargs>0 then \n ff := ar gs[1]\n else\n error \"at least one argument must be supplied\" \n end if;\n if type(ff,\{set(equation),list(equation)\}) and nops (ff)=2 then\n ff := map(_u -> if has(_u,D) then convert(_u,diff) \+ else _u end if,ff);\n de := op(1,ff);\n ic := op(2,ff);\n \+ if not has(de,diff) then\n de := op(2,ff);\n ic := \+ op(1,ff);\n end if;\n else\n error \"the 1st argument, %1, is invalid .. it should be a set (or list) of 2 equations\",ff;\n e nd if;\n\n startopts := 3;\n if nargs>1 then\n ee := args[2]; \n if type(ee,range) or type(ee,name=range) then\n rng := ee;\n else\n arg2OK := true;\n if type(ee,functi on) and nops(ee)=1 then\n yy := op(0,ee);\n xx : = op(1,ee);\n if type(xx,name) and type(yy,name) then\n \+ startopts := 4;\n if nargs>2 then\n \+ rng := args[3];\n else\n error \+ \"expecting a 3rd argument\"\n end if;\n else \n arg2OK := false;\n end if;\n else \n arg2OK := false;\n end if;\n if not arg2 OK then\n error \"the 2nd argument, %1, has incorrect form \+ for the dependent variable\",ee; \n end if;\n end if;\n \+ else\n error \"expecting a 2nd argument\"\n end if;\n\n out pt := 'localtaylor';\n adptv := true;\n stps := 20;\n Options := [];\n if nargs>=startopts then\n Options:=[args[startopts..narg s]];\n if not type(Options,list(equation)) then\n error \+ \"each optional argument must be an equation\"\n end if;\n i f hasoption(Options,'adaptive','adptv') then\n if not member(a dptv,\{true,false\}) then\n error \"\\\"adaptive\\\" must b e 'true' or 'false'\"\n end if;\n elif hasoption(Options, 'stepsize','stpsz') then\n if not member(stpsz,\{'fixed','vari able'\}) then\n error \"\\\"stepsize\\\" must be 'fixed' or 'variable'\"\n end if;\n if stpsz='fixed' then adptv \+ := false end if;\n end if; \n if hasoption(Options,'step s','stps') then \n if not type(stps,'posint') then\n \+ error \"\\\"steps\\\" must be a positive integer\"\n end if; \n adptv := false;\n end if;\n if hasoption(Options, 'output','outpt') then\n if not member(outpt,\{'points','discr ete','loctaylor',\n 'localtaylor','continuous','pro cedure'\}) then\n error \"\\\"output\\\" must be one of 'po ints','discrete','continuous','localtaylor','loctaylor' or 'procedure' \"\n end if;\n if outpt='discrete' then outpt := 'poin ts' end if;\n if member(outpt,\{'continuous','procedure','loct aylor'\}) then \n outpt := 'localtaylor'\n end if; \n end if;\n end if;\n\n derivs := indets(de,'specfunc(anythi ng,diff)');\n if derivs=\{\} then\n error \"the 1st argument, % 1, is invalid .. it should be a set (or list) containing a differentia l equation and an initial condition\",ff;\n end if;\n nvars := nop s(indets(derivs,'name'));\n if nvars<>1 then\n if nvars=0 then \n error \"there is a problem with the independent variable oc curring in the derivative(s)\";\n else\n error \"there sh ould only be one independent variable in the differential equation\"\n end if;\n end if;\n nvars := nops(indets(derivs,'anyfunc(nam e)'));\n if nvars<>1 then\n if nvars=0 then\n error \"t here is a problem with the dependent variable occurring in the derivat ive(s)\"\n else\n error \"there should only be one depend ent variable in the differential equation\"\n end if;\n end if; \n\n if nops(derivs)<>1 then\n error \"there are too many deriv atives in the differential equation .. note that the differential equa tion must be of order 1\"\n end if;\n df := op(1,derivs);\n if t ype(df,function) and op(0,df)=diff and nops(df)=2 then\n yx := op (1,df);\n if not type(yx,anyfunc(name)) then\n error \"th e 1st argument %1, in the derivative, %2, is invalid .. it should be t he 'unknown' dependent variable\",yx,df;\n end if; \n x := o p(2,df);\n if not type(x,name) then\n error \"the 2nd arg ument %1, in the derivative, %2, is invalid .. it should be the indepe ndent variable\",x,df;\n end if; \n else\n error \"the der ivative, %1, does not make sense\",df;\n end if;\n\n y := op(0,yx) ;\n vars := indets(de,name);\n if member(y,vars) then\n error \"%1 and %2 cannot both appear in the differential equation\",yx,y;\n end if;\n if op(1,yx)<>x then\n error \"the derivative, %1, \+ does not make sense\",df;\n end if;\n\n if startopts=4 then \n \+ if x<>xx or y<>yy then\n error \"cannot solve the ODE for %1 \",ee;\n end if;\n end if;\n \n lsic := lhs(ic);\n if type( lsic,function) and op(0,lsic)=y and nops(lsic)=1 \n \+ and type(op(1,lsic),algebraic) then\n x0 := op(1,lsic); \n else\n error \"the initial condition is not decipherable\"\n end if;\n rsic := rhs(ic);\n if type(rsic,realcons) then\n \+ y0 := rsic;\n else\n error \"the initial condition is not deci pherable\"\n end if;\n\n if type(rng,equation) then\n z := op (1,rng);\n if not type(z,name) or z<>x then\n error \"lef t side, %1, of equation for solution range must be the independent var iable\",x;\n end if;\n rg := op(2,rng);\n else\n rg : = rng;\n end if;\n\n if not type(rg,realcons..realcons) then\n \+ error \"the range for the solution must have real end values\"\n e nd if;\n \n lft := op(1,rg);\n rght := op(2,rg);\n \n if sig num(rght-lft)=0 then\n error \"the range for the solution must ha ve distinct end points\"\n end if;\n if signum(rght-lft)<0 then # swap over\n tt := lft; lft := rght; rght := tt; \n end if;\n \+ if signum(x0-rght)>0 or signum(x0-lft)<0 then\n error \"the ran ge for the solution must contain the initial value of the independent \+ variable\"\n end if; \n\n drv := rhs(isolate(de,df));\n if nop s([drv])<>1 then\n error \"cannot obtain a unique expression for \+ the derivative\"\n end if;\n drv := subs(yx=y,drv);\n\n if signu m(x0-lft)=0 then\n return taylor1(drv,x=x0..rght,y=y0,op(Options) );\n elif signum(x0-rght)=0 then\n return taylor1(drv,x=x0..lft ,y=y0,op(Options));\n else\n if not adptv then\n lftstp s := trunc(evalf((x0-lft)/(rght-lft)*stps,Digits+5));\n rghtst ps := stps-lftstps;\n Options := convert(\{op(Options)\} minus \{steps=stps\},list);\n rghtprt := taylor1(drv,x=x0..rght,y=y 0,steps=rghtstps,op(Options));\n lftprt := taylor1(drv,x=x0..l ft,y=y0,steps=lftstps,op(Options));\n else\n rghtprt := t aylor1(drv,x=x0..rght,y=y0,op(Options));\n lftprt := taylor1(d rv,x=x0..lft,y=y0,op(Options));\n end if;\n if outpt=localta ylor then\n return dblrk(lft,rght,x0,lftprt,rghtprt);\n e lse\n return [op(reverse(lftprt)),op(rghtprt)];\n end if; \n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "taylor1: implementation" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13703 "taylor1 := proc(fxy::algebraic,xrng::(name=realcons..realcons),\n \+ iy::(name=realcons)) \n local soln,xk,yk,k,xe nd,xstart,h,safety,finished,\n yout,yerr,sum,term,dk,dn,derivs,fact, i,f1,stps,outpt,\n maxstps,tiny,yscale,err,htemp,xnew,hnext,saveDigi ts,eps,\n Options,j,prntflg,t,tt,errcntl,ymax,ymaxtemp,x,y,errc,x0,y 0,xn,f,\n hmx,hmn,hstrt,sgn,loctaylor,maxstepsize,laststep,xrg,\n \+ ordr,minstepsize,pgrow,pshrink,errcontrol,inc,stpsz,adptv;\n \nloctayl or := proc(x_taylor::realcons)\n local xk,yk,jF,jM,jS,n,h,j,data,xx, saveDigits,\n sum,term,fact,dk;\n options `Copyright 2002 by Peter Stone`;\n \n data := _SOLN;\n\n saveDigits := Digits;\n Digit s :=\n min(max(trunc(evalhf(Digits)),trunc(Digits*4/3)),Digits+10); \n\n xx := evalf(x_taylor);\n n := nops(data);\n if (data[1,1]data[n,1] or xxdata[1,1])) then\n error \"independ ent variable is outside of the interpolation interval: %1\",evalf(data [1,1])..evalf(data[n,1]);\n end if;\n \n # Peform a binary searc h for the interval containing x.\n n := nops(data);\n jF := 0;\n \+ jS := n+1;\n\n if data[1,1]1 do \n jM := trunc((jF+jS)/2);\n if xx>=data[jM,1] then jF : = jM else jS := jM end if;\n end do;\n if jM = n then jF := \+ n-1; jS := n end if;\n else\n while jS-jF>1 do\n jM := t runc((jF+jS)/2);\n if xx<=data[jM,1] then jF := jM else jS := j M end if;\n end do;\n if jM = n then jF := n-1; jS := n end \+ if;\n end if;\n \n # Get the data needed from the list.\n xk : = data[jF,1];\n yk := data[jF,2];\n h := xx-xk;\n sum := yk;\n \+ fact := evalf(1);\n \n for j from 1 to nops(data[jF])-2 do\n \+ fact := fact*h/j;\n dk := data[jF,j+2];\n term := dk*fact; \n sum := sum + term;\n end do;\n Digits := saveDigits;\n e valf(sum);\nend proc: # of loctaylor\n \n # start of main procedure \n x := op(1,xrng);\n y := op(1,iy);\n if not type(indets(fxy,na me) minus \{x,y\},set(realcons)) then\n error \"the 1st argument, %1, must depend only on the variables %2 and %3\",fxy,x,y;\n end if ;\n xrg := op(2,xrng);\n x0 := op(1,xrg);\n xn := op(2,xrg);\n \+ y0 := op(2,iy);\n \n # Get the options.\n # Set the default val ues to start with.\n maxstps := 2000;\n stps := 20;\n adptv := t rue;\n outpt := 'localtaylor';\n t := Float(1,-Digits);\n ordr : = min(Digits,15+trunc(abs((Digits-15))^0.9));\n hmx := ordr*0.05;\n \+ hstrt := evalf(0.1*10^(-ceil(Digits/10)));\n hmn := min(0.000005,h strt/2000);\n errcntl := 0;\n ymaxtemp := 0;\n prntflg := 0;\n \+ if nargs>3 then\n Options:=[args[4..nargs]];\n if not type( Options,list(equation)) then\n error \"each optional argument \+ must be an equation\"\n end if;\n if hasoption(Options,'orde r','ordr','Options') then\n if not type(ordr,posint) or ordr<2 or ordr>100 then\n error \"\\\"order\\\" must be a positiv e integer between 2 and 100 inclusive\"\n end if;\n hm x := ordr*0.05;\n end if;\n if hasoption(Options,'info','prn tflg','Options') then\n if not member(prntflg,\{0,1,2,3,4,true ,false\}) then\n error \"\\\"info\\\" must be 0 <-> false, \+ 1, 2 <-> true, 3 or 4\"\n end if;\n if prntflg=false t hen prntflg := 0 end if; \n if prntflg=true then prntflg := 2 \+ end if;\n end if;\n if hasoption(Options,'adaptive','adptv', 'Options') then\n if not member(adptv,\{true,false\}) then\n \+ error \"\\\"adaptive\\\" must be 'true' or 'false'\"\n \+ end if;\n elif hasoption(Options,'stepsize','stpsz','Options') then\n if not member(stpsz,\{'fixed','variable'\}) then\n \+ error \"\\\"stepsize\\\" must be 'fixed' or 'variable'\"\n \+ end if;\n if stpsz='fixed' then adptv := false end if;\n \+ end if;\n if hasoption(Options,'steps','stps','Options') the n \n if not type(stps,'posint') then\n error \"\\\" steps\\\" must be a positive integer\"\n end if;\n adp tv := false;\n end if;\n if hasoption(Options,'output','outp t','Options') then\n if not member(outpt,\{'points','discrete' ,'loctaylor','localtaylor',\n \+ 'continuous','procedure'\}) then\n error \"\\\"output\\ \" must be one of 'points','discrete','continuous','localtaylor','loct aylor' or 'procedure'\"\n end if;\n if outpt='discrete ' then outpt := 'points' end if;\n if member(outpt,\{'continuo us','procedure','loctaylor'\}) then \n outpt := 'localtaylo r'\n end if;\n else\n if not adptv then outpt := \+ 'points' end if; \n end if;\n\n if adptv then\n \+ if hasoption(Options,'maxsteps','maxstps','Options') then\n \+ if not type(maxstps,posint) then\n error \"\\\"maxsteps \\\" must be a positive integer\"\n end if;\n end i f;\n if hasoption(Options,'tolerance','t','Options') then \n \+ tt := evalf(t); \n if not type(tt,float) or \n \+ tt>Float(1,-iquo(Digits,2)) or tthmx) or evalf(hstrt0 then\n error \"%1 is not a val id option for %2\",op(1,Options), procname;\n end if;\n end if; \n\n # Convert any real constants in the expression to floats.\n f := unapply(evalf(fxy),x,y);\n saveDigits := Digits;\n Digits :=\n min(max(trunc(evalhf(Digits)),trunc(Digits*4/3)),Digits+10);\n\n \+ xk := evalf(x0);\n yk := evalf(y0);\n soln := NULL;\n\n dn := ar ray(1..ordr);\n dn[1] := f;\n\n # Use implicit differentiation to \+ obtain the derivatives.\n for i from 2 to ordr do\n dn[i] := un apply(simplify(diff(dn[i-1](x,y),x)\n + diff(dn[i-1](x ,y),y)*f(x,y)),x,y);\n if prntflg=1 or prntflg=3 then print(`deri vative`,i,` `,dn[i]) end if;\n end do;\n\n if not adptv then\n \+ h := evalf((xn-x0)/stps);\n \n # Use the Taylor series to \+ calculate the new yk at each step.\n for k from 1 to stps do\n \+ sum := yk;\n term := sum;\n fact := evalf(1);\n \+ derivs := yk;\n for j from 1 to ordr do\n f act := evalf(fact*h/j);\n dk := traperror(dn[j](xk,yk));\n \+ if dk=lasterror or not type(dk,numeric) then\n \+ if j=1 then\n error \"evaluation of gradient field \+ failed at %1\",[xk,yk];\n else\n error \+ \"evaluation of %-1 derivative of gradient field failed at %2\",j-1,[x k,yk];\n end if;\n end if;\n term \+ := evalf(dk*fact);\n sum := sum + term;\n derivs := derivs,dk;\n end do;\n if outpt='points' then\n \+ soln := soln,[xk,yk];\n else\n soln := sol n,[xk,derivs];\n end if;\n yk := sum;\n xk := \+ xk + h;\n if prntflg=2 or prntflg=3 then\n print(`s tep`,k,` `,[xk,yk])\n end if;\n end do;\n soln := \+ soln,[xk,yk];\n Digits := saveDigits;\n if outpt='points' th en\n return evalf([soln]);\n else\n return subs(_ SOLN=[soln],eval(loctaylor));\n end if;\n else # stepsize is v ariable\n xstart := evalf(x0);\n xend:= evalf(xn);\n\n \+ sgn := sign(xend-xstart);\n h := sgn*hstrt;\n\n safety := 0. 9;\n eps := evalf(t);\n tiny := Float(1,-3*saveDigits);\n \+ pgrow := -evalf(1/ordr);\n pshrink := -evalf(1/(ordr-1));\n \+ errcontrol := evalf((5/safety)^(-ordr));\n \n xk := evalf(x0) ;\n yk := evalf(y0);\n \n if ymaxtemp<>0 then\n ym ax := abs(evalf(ymaxtemp))\n else\n ymax := max(abs(yk),t iny)\n end if;\n \n soln := NULL;\n finished := false ;\n f1 := evalf(f(xk,yk));\n if f1=lasterror or not type(f1, numeric) then\n error \"evaluation of gradient field failed at %1\",[xk,yk];\n end if;\n for k from 1 to maxstps do\n \+ if errcntl=0 then yscale := max(abs(yk),abs(f1*h),tiny)\n \+ elif errcntl=1 then yscale := max(abs(yk),tiny)\n elif errcntl =2 then yscale := abs(ymax)\n else yscale := max(abs(f1*h),tin y) end if;\n\n if abs(h)>=hmx then\n h := sgn*hmx; \n maxstepsize := true;\n else\n maxstep size := false;\n end if;\n if abs(h)<=hmn then\n \+ h := sgn*hmn;\n minstepsize := true;\n else\n minstepsize := false;\n end if;\n if (xk+h -xend)*(xk+h-xstart)>0 then \n h := xend-xk;\n l aststep := true;\n else\n laststep := false;\n \+ end if;\n\n # Do step.\n do\n sum := yk ;\n term := sum;\n fact := evalf(1);\n \+ derivs := yk;\n for j from 1 to ordr\n do\n \+ fact := evalf(fact*h/j);\n dk := dn[j](xk,yk );\n if dk=lasterror or not type(dk,numeric) then\n \+ if j=1 then\n error \"evaluation of g radient field failed at %2\",j-1,[xk,yk];\n else\n \+ error \"evaluation of %-1 derivative of gradient fie ld failed at %2\",j-1,[xk,yk];\n end if;\n \+ end if;\n term := evalf(dk*fact);\n su m := sum + term;\n derivs := derivs,dk;\n end do;\n\n yout := sum;\n\n # error estimate - las t term in Taylor series\n yerr := evalf(dk*fact);\n \+ err := abs(yerr/yscale)/eps;\n if err<=1.0 or minsteps ize then break end if;\n\n # Shrink, but not too much.\n \+ if prntflg=2 or prntflg=3 then\n print(`reducin g step-size and repeating step`);\n end if;\n ht emp := safety*h*err^pshrink;\n if h>=0 then\n \+ h := max(htemp,0.1*h)\n else\n h := min(htem p,0.1*h)\n end if;\n xnew := xk + h;\n \+ if xnew=xk then\n error \"stepsize underflow\"\n \+ end if;\n end do;\n\n if outpt='points' then\n \+ soln := soln,[xk,yk];\n else\n soln := so ln,[xk,derivs];\n end if;\n\n if err>errcontrol then\n hnext := safety*h*err^pgrow;\n inc := false;\n \+ else\n if abs(h) `,evalf(abs(yerr),5),`abs err bound -> `,evalf(abs(yscale)*eps,5) );\n print(`step`,k,` `,h,` `,[xk,yk]);\n if laststep then\n print(`last step`);\n elif i nc then\n print(`increasing step-size by a factor of 5`) \n elif maxstepsize then\n print(`used maximu m step-size`)\n elif not minstepsize then\n p rint(`using error to adjust step-size`)\n else \n \+ print(`used minimum step-size`)\n end if;\n \+ print(``);\n end if;\n \n if (xk-xend)*(xend-xstart)> =0 then\n if prntflg>0 then\n print(`the tota l number of steps is`,k);\n end if;\n finished : = true;\n break;\n end if;\n\n if abs(yk)>y max then ymax := abs(yk) end if;\n h := hnext;\n end do; \n\n if not finished and k>=maxstps then\n error \"reache d maximum number of steps before reaching end of interval\"\n end if;\n \n soln := soln,[xk,yk];\n Digits := saveDigits; \n if outpt=points then\n return evalf([soln]);\n el se\n return subs(_SOLN=[soln],eval(loctaylor));\n end if; \n end if;\nend proc: # of taylor1" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolve TA" }{TEXT -1 10 ": examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 4 "Note" }{TEXT -1 22 ": The utility routine " } {TEXT 0 14 "comparewithfcn" }{TEXT -1 5 " . . " }{HYPERLNK 17 "compare withfcn" 1 "" "comparewithfcn" }{TEXT -1 30 " is needed for these exam ples." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }{TEXT 288 79 " .. a discrete numeric al solution obtained by the order 2 Taylor series method " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Consider the diffe rential equation " }{XPPEDIT 18 0 "dy/dx = -x/y;" "6#/*&%#dyG\"\"\"%# dxG!\"\",$*&%\"xGF&%\"yGF(F(" }{TEXT -1 20 " over the interval " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 28 " with the initial condition " } {XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 27 "The analytical solution is " }{XPPEDIT 18 0 "y(x)=sqrt(1-x^2)" "6#/-%\"yG6#%\"xG-%%sqrtG6#,&\"\"\"F,*$F'\"\"# !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "de := diff(y(x),x)=-x/y(x);\nic := y(0)=1 ;\ndsolve(\{de,ic\},y(x));\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&F,\"\"\"F)!\"\"F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*$)F'\"\"#\"\"\"!\" \"F-F-#F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)o peratorG%&arrowGF(*$,&*$)9$\"\"#\"\"\"!\"\"F2F2#F2F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "To obtain a discrete numerical solutio n with 20 steps by an order 2 Taylor series method can be obtained by \+ using the procedure " }{TEXT 0 9 "desolveTA" }{TEXT -1 19 " with the o ptions \"" }{TEXT 270 14 "stepsize=fixed" }{TEXT -1 4 "\", \"" }{TEXT 270 13 "output=points" }{TEXT -1 5 "\" , \"" }{TEXT 270 8 "steps=20" } {TEXT -1 8 "\" and \"" }{TEXT 270 7 "order=2" }{TEXT -1 4 "\" . " }} {PARA 0 "" 0 "" {TEXT 296 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 " " {TEXT -1 12 "The option \"" }{TEXT 270 14 "stepsize=fixed" }{TEXT -1 22 "\" can be replaced by \"" }{TEXT 270 14 "adaptive=false" } {TEXT -1 2 "\"." }}{PARA 15 "" 0 "" {TEXT -1 12 "The option \"" } {TEXT 270 14 "stepsize=fixed" }{TEXT -1 68 "\", could be omitted becau se, when the number of steps is specified, " }{TEXT 0 9 "desolveTA" } {TEXT -1 59 " will assume that you want a solution with fixed step-siz e." }}{PARA 15 "" 0 "" {TEXT -1 58 "Since the default number of steps \+ used is 20, the option \"" }{TEXT 270 8 "steps=20" }{TEXT -1 85 "\" ca n be omitted, but if this is done, it is necessary to include either t he option \"" }{TEXT 270 14 "stepsize=fixed" }{TEXT -1 6 "\" or \"" } {TEXT 270 14 "adaptive=false" }{TEXT -1 48 "\" if you want a solution \+ with fixed step-size. " }}{PARA 15 "" 0 "" {TEXT -1 12 "The option \" " }{TEXT 270 13 "output=points" }{TEXT -1 43 "\" can be omitted since \+ this is the default." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 98 "The discrete solution can be compared with the analytic al solution by using the special procedure " }{TEXT 0 14 "comparewithf cn" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 152 "de := diff(y(x),x)=-x/y(x);\nic := y(0)=1; \nsoln := desolveTA(\{de,ic\},y(x),x=0..1,stepsize=fixed,order=2,steps =20,info=true);\ncomparewithfcn(soln,g(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&F,\"\"\"F)!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"\"\"%$~~~G7$$\"/++++++]!#:$\"/++++] ()**!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"\"#%$~~~G7$$\"/+++ +++5!#9$\"/=&))p@*\\**F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\" \"$%$~~~G7$$\"/++++++:!#9$\"/*zsB!)p))*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"\"%%$~~~G7$$\"/++++++?!#9$\"/S3/!*=)z*F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"\"&%$~~~G7$$\"/++++++D!#9$\"/ ,$*)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"#5%$~~~G7$$\" /++++++]!#9$\"/8Q;TYi')F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\" #6%$~~~G7$$\"/++++++b!#9$\"/@T()\\ia$)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"#7%$~~~G7$$\"/++++++g!#9$\"/jqV+-/!)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"#8%$~~~G7$$\"/++++++l!#9$\"/f;5a\"[g(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"#9%$~~~G7$$\"/++++++q!#9$\" /$Rbl4!\\rF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"#:%$~~~G7$$\" /++++++v!#9$\"/$p3x#=DmF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\" #;%$~~~G7$$\"/++++++!)!#9$\"/#=B@9h,'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"#<%$~~~G7$$\"/++++++&)!#9$\"/!)Qlar$H&F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&%%stepG\"#=%$~~~G7$$\"/++++++!*!#9$\"/fB$*\\Q1WF )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"#>%$~~~G7$$\"/++++++&*!# 9$\"/beftUQKF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%%stepG\"#?%$~~~G7$$ \"/++++++5!#8$\"/s*\\Q,4S\"!#9" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%s olnG777$$\"\"!F($\"\"\"F(7$$\"+++++]!#6$\"+++]()**!#57$$\"+++++5F1$\"+ *p@*\\**F17$$\"+++++:F1$\"+P-)p))*F17$$\"+++++?F1$\"+/!*=)z*F17$$\"+++ ++DF1$\"+K.%Go*F17$$\"+++++IF1$\"+%)f(*R&*F17$$\"+++++NF1$\"+2WMo$*F17 $$\"+++++SF1$\"+')*Rj;*F17$$\"+++++XF1$\"+@n\">$*)F17$$F-F1$\"+;TYi')F 17$$\"+++++bF1$\"+()\\ia$)F17$$\"+++++gF1$\"+W+-/!)F17$$\"+++++lF1$\"+ 5a\"[g(F17$$\"+++++qF1$\"+c'4!\\rF17$$\"+++++vF1$\"+rF=DmF17$$\"+++++! )F1$\"+7U6;gF17$$\"+++++&)F1$\"+lar$H&F17$$\"+++++!*F1$\"+$*\\Q1WF17$$ \"+++++&*F1$\"+gtUQKF17$$F4!\"*$\"+&Q,4S\"F1" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0 1 function val: 1 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0. 05 0.99875 function val: 0.9987492178 rel err: 7.8318 e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.1 0.9949921699 \+ function val: 0.9949874371 rel err: 4.7566e-06" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.15 0.9886980237 function val: 0 .9886859967 rel err: 1.2165e-05" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.2 0.9798189004 function val: 0.9797958971 rel \+ err: 2.3478e-05" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.25 0.9 682840332 function val: 0.9682458366 rel err: 3.9449e-05" }} {PARA 6 "" 1 "" {TEXT -1 87 " 0.3 0.9539975984 functio n val: 0.9539392014 rel err: 6.1217e-05" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.35 0.9368344407 function val: 0.9367 496998 rel err: 9.0463e-05" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.4 0.9166339986 function val: 0.916515139 rel err: \+ 1.2969e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.45 0.893191 6721 function val: 0.893028555 rel err: 1.8266e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.5 0.8662464116 function val: 0.8660254038 rel err: 2.5520e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.55 0.8354624987 function val: 0.8351646544 \+ rel err: 3.5663e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.6 \+ 0.8004020044 function val: 0.8 rel err: 5.0251e-0 4" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.65 0.760481541 fu nction val: 0.7599342077 rel err: 7.2024e-04" }}{PARA 6 "" 1 " " {TEXT -1 87 " 0.7 0.7149009656 function val: 0.71 41428429 rel err: 1.0616e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.75 0.6625182771 function val: 0.6614378278 rel err : 1.6335e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.8 0.6016 114212 function val: 0.6 rel err: 2.6857e-03" }} {PARA 6 "" 1 "" {TEXT -1 87 " 0.85 0.5293715465 functio n val: 0.5267826876 rel err: 4.9145e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.9 0.4406384993 function val: 0.4358 898944 rel err: 1.0894e-02" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.95 0.323842736 function val: 0.3122498999 rel err: \+ 3.7127e-02" }}{PARA 6 "" 1 "" {TEXT -1 85 " 1 0.140090 1385 function val: 0 rel err: infinity" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " \+ Maximum relative error: 3.7127e-02" }}{PARA 6 "" 1 "" {TEXT -1 59 " obtained for the input value: 0.95" }} {PARA 6 "" 1 "" {TEXT -1 67 " excluding any cases where t he function value is zero." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "A discrete numerical solution with 20 steps by an order 8 Taylor series method can be obtained by using the procedur e " }{TEXT 0 9 "desolveTA" }{TEXT -1 19 " with the options \"" }{TEXT 270 8 "steps=20" }{TEXT -1 8 "\" and \"" }{TEXT 270 7 "order=8" } {TEXT -1 4 "\" . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "de := diff(y(x),x)=-x/y(x);\nic := y(0)=1;\n soln2 := desolveTA(\{de,ic\},x=0..1,order=8,steps=20):\ncomparewithfcn (soln2,sqrt(1-x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%di ffG6$-%\"yG6#%\"xGF,,$*&F,\"\"\"F)!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0 1 function val: 1 \+ rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.05 \+ 0.9987492178 function val: 0.9987492178 rel err: 0.0000e+0 0" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.1 0.9949874371 fu nction val: 0.9949874371 rel err: 0.0000e+00" }}{PARA 6 "" 1 " " {TEXT -1 87 " 0.15 0.9886859967 function val: 0.98 86859967 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.2 0.9797958971 function val: 0.9797958971 rel err : 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.25 0.9682 458366 function val: 0.9682458366 rel err: 0.0000e+00" }} {PARA 6 "" 1 "" {TEXT -1 87 " 0.3 0.9539392014 functio n val: 0.9539392014 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.35 0.9367496998 function val: 0.9367 496998 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.4 0.916515139 function val: 0.916515139 rel err: \+ 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.45 0.89302 8555 function val: 0.893028555 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.5 0.8660254038 function val: 0.8660254038 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.55 0.8351646544 function val: 0.8351646544 \+ rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.6 \+ 0.8000000001 function val: 0.8 rel err: 1.2500e-1 0" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.65 0.7599342078 fu nction val: 0.7599342077 rel err: 1.3159e-10" }}{PARA 6 "" 1 " " {TEXT -1 87 " 0.7 0.7141428433 function val: 0.71 41428429 rel err: 5.6011e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.75 0.6614378292 function val: 0.6614378278 rel err : 2.1166e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.8 0.6000 000064 function val: 0.6 rel err: 1.0667e-08" }} {PARA 6 "" 1 "" {TEXT -1 87 " 0.85 0.5267827287 functio n val: 0.5267826876 rel err: 7.8021e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.9 0.4358903717 function val: 0.4358 898944 rel err: 1.0950e-06" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.95 0.3122673505 function val: 0.3122498999 rel err: \+ 5.5887e-05" }}{PARA 6 "" 1 "" {TEXT -1 85 " 1 0.0622102 3914 function val: 0 rel err: infinity" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " \+ Maximum relative error: 5.5887e-05" }}{PARA 6 "" 1 "" {TEXT -1 59 " obtained for the input value: 0.95" }} {PARA 6 "" 1 "" {TEXT -1 67 " excluding any cases where t he function value is zero." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "The points of the discrete solution can be plo tted along with the graph of the analytical solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "plot([ sqrt(1-x^2),soln$3],x=0..1,style=[line,point$3],color=[red,black$3],\n symbol=[circle,diamond,cross],labels=[`x`,`y(x)`],scaling=co nstrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6)- %'CURVESG6&7Y7$$\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#>$\"3e(=:\"QTi(***!#= 7$$\"3[LL$e9ui2%F/$\"3!e2E0a)o\"***F27$$\"3nmmm\"z_\"4iF/$\"3)f\\s^f/2 )**F27$$\"3[mmmT&phN)F/$\"3)37fk0E]'**F27$$\"3CLLe*=)H\\5F2$\"3R\\N+$H 'zW**F27$$\"3gmm\"z/3uC\"F2$\"3^l4$yi$*=#**F27$$\"3%)***\\7LRDX\"F2$\" 3%p#zm\"3WR*)*F27$$\"3]mm\"zR'ok;F2$\"3iaa\"Q\\n/')*F27$$\"3w***\\i5`h (=F2$\"3\"Rj1r%eUA)*F27$$\"3WLLL3En$4#F2$\"3_.Ayu2Py(*F27$$\"3qmm;/RE& G#F2$\"3wIBk#=x`t*F27$$\"3\")*****\\K]4]#F2$\"31q9\"H%H@#o*F27$$\"3$** ****\\PAvr#F2$\"3B_^A([sOi*F27$$\"3)******\\nHi#HF2$\"3m:#e7,zAc*F27$$ \"3jmm\"z*ev:JF2$\"3)>#\\;*p8A]*F27$$\"3?LLL347TLF2$\"3pW!GHZL`U*F27$$ \"3,LLLLY.KNF2$\"3#=1W(>`Yb$*F27$$\"3w***\\7o7Tv$F2$\"3!QZNcQ$eo#*F27$ $\"3'GLLLQ*o]RF2$\"3=`?'yv9l=*F27$$\"3A++D\"=lj;%F2$\"3]kxU3;t!4*F27$$ \"31++vV&R5j')F27 $$\"3&em;zRQb@&F2$\"3e0&=5t$=K&)F27$$\"3\\***\\(=>Y2aF2$\"3ld&yZ.e=T)F 27$$\"39mm;zXu9cF2$\"3]/X@$oS\\F)F27$$\"3l******\\y))GeF2$\"3k!yShX>b7 )F27$$\"3'*)***\\i_QQgF2$\"35nMlUm1rzF27$$\"3@***\\7y%3TiF2$\"3bX$z*Qq P8yF27$$\"35****\\P![hY'F2$\"3QXhIao;GwF27$$\"3kKLL$Qx$omF2$\"3q\"p\\+ YH?X(F27$$\"3!)*****\\P+V)oF2$\"378IrHv-`sF27$$\"3?mm\"zpe*zqF2$\"3Qyn nKd;iqF27$$\"3%)*****\\#\\'QH(F2$\"3G\"QTZ_=5%oF27$$\"3GKLe9S8&\\(F2$ \"3a%Rp()p\"*)>mF27$$\"3R***\\i?=bq(F2$\"3=&F27$$\"3=LLe9tOc()F 2$\"3PnBuRIqH[F27$$\"3u******\\Qk\\*)F2$\"3+@I:/fPhWF27$$\"3CLL$3dg6<* F2$\"3au)*4GH?')RF27$$\"3ImmmmxGp$*F2$\"3s!RL6R._\\$F27$$\"3A++D\"oK0e *F2$\"3!yTAE(=!f'GF27$$\"3C+++]oi\"o*F2$\"3?UJ1O#=K]#F27$$\"3A++v=5s#y *F2$\"37vO>U4Dt?F27$$\"3;+D1k2/P)*F2$\"3mLW(eL^zz\"F27$$\"35+]P40O\"*) *F2$\"31H.f%oH+Z\"F27$$\"3k]7.#Q?&=**F2$\"3G#>z^*=&RF\"F27$$\"31+voa-o X**F2$\"3(pV\\8H')3/\"F27$$\"3[\\PMF,%G(**F2$\"3?&=hJ*p=ltF/7$F*F(-%'C OLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG-%'SYMBOLG6#%'CIRCLE G-F$6&77F'7$$\"3G+++++++]F/$\"3E+++++]()**F27$$\"3/+++++++5F2$\"3Y**** **)p@*\\**F27$$\"3%**************\\\"F2$\"3S+++P-)p))*F27$$\"35+++++++ ?F2$\"3q*****R+*=)z*F27$$\"3++++++++DF2$\"31+++K.%Go*F27$$\"3))******* *******HF2$\"3!)*****R)f(*R&*F27$$\"3w*************\\$F2$\"3#******pSW $o$*F27$$\"3A+++++++SF2$\"3c+++')*Rj;*F27$$\"35+++++++XF2$\"3X*****4s; >$*)F27$$\"3++++++++]F2$\"3W+++;TYi')F27$$\"3U+++++++bF2$\"3++++()\\ia $)F27$$\"3w**************fF2$\"3y*****R/?S+)F27$$\"3A+++++++lF2$\"3$)* *****4a\"[g(F27$$\"3a**************pF2$\"3]+++c'4!\\rF27$$\"3++++++++v F2$\"3(******4x#=DmF27$$\"3U+++++++!)F2$\"3'******>@9h,'F27$$\"3w***** ********\\)F2$\"3c*****\\Y:PH&F27$$\"3A+++++++!*F2$\"3')*****H*\\Q1WF2 7$$\"3a*************\\*F2$\"3?+++gtUQKF27$F*$\"3*)*****\\Q,4S\"F2-Fc\\ l6&Fe\\lF)F)F)-Fj\\l6#%&POINTG-F^]l6#%(DIAMONDG-F$6&Fc]lFfclFhcl-F^]l6 #%&CROSSG-F$6&Fc]lFfclFhclF]]l-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELS G6$%\"xG%%y(x)G-%%VIEWG6$;F(F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }{TEXT 287 115 " .. constructing progressively more accurate numerical solutions by increasing the order of the Taylor series used " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 37 "Consider the differential equation " }{XPPEDIT 18 0 "dy/dx = \+ 2*x/(y^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"#F&%\"xGF&*$%\"yGF*F(" } {TEXT -1 34 " subject to the initial condition " }{XPPEDIT 18 0 "y(0)= 1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 77 "This differential equation can be solved by separating the vari ables to give " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (y^2,y) = Int(2*x,x);" "6#/-%$IntG6$*$%\"yG\"\"#F(-F%6$*&F)\"\"\"%\"xG F-F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^3/3 = x^2+c;" "6#/* &%\"yG\"\"$F&!\"\",&*$%\"xG\"\"#\"\"\"%\"cGF," }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 34 "The intial condition implies that " } {XPPEDIT 18 0 "c = 1/3;" "6#/%\"cG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 13 " which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^3/ 3=x^2+1/3" "6#/*&%\"yG\"\"$F&!\"\",&*$%\"xG\"\"#\"\"\"*&F,F,F&F'F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^3 = 3*x^2+1;" "6#/*$%\"yG\"\"$,&*&F& \"\"\"*$%\"xG\"\"#F)F)F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=(3*x^2+1)^(1/3)" "6#/%\"yG),&*&\"\"$\"\"\"*$%\"xG\"\"#F)F)F)F)*&F )F)F(!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 282 10 "__________" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "We can use Maple's " }{TEXT 0 6 "dsolve" }{TEXT -1 72 " to obtain this result and set up the analyti cal solution as a function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "de := diff(y(x),x)=2*x/y(x)^2;\nic := y(0)=1;\nd solve(\{de,ic\},y(x));\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"#\"\"\"F,F0F)! \"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$),&*&\"\"$\"\"\")F'\" \"#F-F-F-F-#F-F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(*$),&*&\"\"$\"\"\")9$\"\"#F1F1F1F1#F1F0F1F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "desolveTA" }{TEXT -1 82 " can be used to obtain a (discrete) soluti on by the order 2 Taylor series method. " }}{PARA 0 "" 0 "" {TEXT -1 52 "We construct a numerical solution over the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 3;" "6#1%!G\"\"$" } {TEXT -1 25 " using 30 steps of width " }{XPPEDIT 18 0 "h=1/10" "6#/% \"hG*&\"\"\"F&\"#5!\"\"" }{TEXT -1 131 ", and compare the values given by the numerical solution (in terms of absolute error) with those giv en by the analytical solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "de := diff(y(x),x)=2*x/y(x) ^2;\nic := y(0)=1;\nsoln := desolveTA(\{de,ic\},y(x),x=0..3,order=2,st eps=30):\ncomparewithfcn(soln,g(x),x,errtype=absolute);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"#\"\"\"F ,F0F)!\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\" \"\"" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0 1 \+ function val: 1 abs err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.1 1.01 function val: \+ 1.009901634 abs err: 9.8366e-05" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.2 1.039028295 function val: 1.03849882 abs \+ err: 5.2948e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.3 1. 084021367 function val: 1.082932133 abs err: 1.0892e-03" }} {PARA 6 "" 1 "" {TEXT -1 87 " 0.4 1.141185657 functio n val: 1.139603843 abs err: 1.5818e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.5 1.20698718 function val: 1.205 071132 abs err: 1.9160e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.6 1.278590415 function val: 1.27650086 abs err: \+ 2.0896e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.7 1.3538 9706 function val: 1.351758112 abs err: 2.1389e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.8 1.43141994 function val: 1.4293139 abs err: 2.1060e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.9 1.510128786 function val: 1.508104283 \+ abs err: 2.0245e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1 \+ 1.589318767 function val: 1.587401052 abs err: 1.9177e-0 3" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.1 1.668511708 fu nction val: 1.66671111 abs err: 1.8006e-03" }}{PARA 6 "" 1 " " {TEXT -1 87 " 1.2 1.747385868 function val: 1.7 45703747 abs err: 1.6821e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 1.3 1.825727228 function val: 1.824159869 abs err : 1.5674e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.4 1.903 396049 function val: 1.901937083 abs err: 1.4590e-03" }} {PARA 6 "" 1 "" {TEXT -1 87 " 1.5 1.980303931 functio n val: 1.978945805 abs err: 1.3581e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.6 2.056398049 function val: 2.055 132884 abs err: 1.2652e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 1.7 2.131650263 function val: 2.130470335 abs err: \+ 1.1799e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.8 2.20604 9545 function val: 2.204947543 abs err: 1.1020e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.9 2.279596691 function val: 2.278565837 abs err: 1.0309e-03" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2 2.352300599 function val: 2.351334688 \+ abs err: 9.6591e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.1 \+ 2.424175639 function val: 2.423269038 abs err: 9.0660e-0 4" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.2 2.495239792 fu nction val: 2.494387409 abs err: 8.5238e-04" }}{PARA 6 "" 1 " " {TEXT -1 87 " 2.3 2.565513319 function val: 2.5 64710565 abs err: 8.0275e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 2.4 2.63501782 function val: 2.634260565 abs err : 7.5726e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.5 2.703 775561 function val: 2.703060088 abs err: 7.1547e-04" }} {PARA 6 "" 1 "" {TEXT -1 87 " 2.6 2.771809001 functio n val: 2.771131964 abs err: 6.7704e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.7 2.839140458 function val: 2.838 498843 abs err: 6.4162e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 2.8 2.905791875 function val: 2.905182963 abs err: \+ 6.0891e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.9 2.97178 4667 function val: 2.971206001 abs err: 5.7867e-04" }}{PARA 6 "" 1 "" {TEXT -1 87 " 3 3.037139614 function val: 3.036588972 abs err: 5.5064e-04" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum absolu te error: 2.1389e-03" }}{PARA 6 "" 1 "" {TEXT -1 59 " obt ained for the input value: 0.7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 45 " Mean \+ absolute error: 1.1611e-03" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The maximum absolute error occurs when " } {XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 4 ".7. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "plot ([g(x),soln$3],x=0..3,y=0..3.1,style=[line,point$3],color=[red,black$3 ],\n symbol=[circle,diamond,cross],labels=[`x`,`y(x)`],scaling=cons trained);" }}{PARA 13 "" 1 "" {GLPLOT2D 555 400 400 {PLOTDATA 2 "6)-%' CURVESG6&7S7$$\"\"!F)$\"\"\"F)7$$\"3s******\\i9Rl!#>$\"3*=;E!))yD/5!#< 7$$\"3/++vVA)GA\"!#=$\"3wem!z=OZ,\"F27$$\"3+++]Peui=F6$\"3a.jP*QfN.\"F 27$$\"3A++]i3&o]#F6$\"3!HH?**oh#f5F27$$\"3%)***\\(oX*y9$F6$\"3)[^_@RI1 4\"F27$$\"3z***\\P9CAu$F6$\"3k1\\BPW-C6F27$$\"3!)***\\P*zhdVF6$\"31y'f \">#o@;\"F27$$\"31++v$>fS*\\F6$\"3WT*Q_9iY?\"F27$$\"3$)***\\(=$f%GcF6$ \"3mYM(4a:%\\7F27$$\"3Q+++Dy,\"G'F6$\"3G=%yc%)QtH\"F27$$\"33++]7-1\\\"F27$$\"3))*** \\PpnsM*F6$\"3muD_5LhN:F27$$\"3,++]siL-5F2$\"3#[$)Q=Nb#*e\"F27$$\"3-++ +!R5'f5F2$\"3-$e'*)Q1qM;F27$$\"3)***\\P/QBE6F2$\"31A\"fL;vuo\"F27$$\"3 !******\\\"o?&=\"F2$\"3+p1QWx/MF27$$\"3)****\\7B>&)\\\"F2$\"3+NSA\" Q6y(>F27$$\"3)***\\P>:mk:F2$\"33764\"f0$G?F27$$\"3'***\\iv&QAi\"F2$\"3 q!*)z\\#4'>2#F27$$\"31++vtLU%o\"F2$\"3,T')QdF2$\"3W!z>6AhwI#F27$$\"3z**** *\\@80+#F2$\"3'>/99)fq^BF27$$\"31++]7,Hl?F2$\"35z#3eh#F27$$\"3')**\\PfL'zV#F2$\"3=;v'4Jm/m#F27$ $\"3>+++!*>=+DF2$\"3P@n)ei%=.FF27$$\"3-++DE&4Qc#F2$\"3->#eE,zlu#F27$$ \"3=+]P%>5pi#F2$\"3oOI,?*G$*y#F27$$\"39+++bJ*[o#F2$\"3>xsBbhOGGF27$$\" 33++Dr\"[8v#F2$\"3GPxD@Q#G(GF27$$\"3++++Ijy5GF2$\"3hl4!R&eL7HF27$$\"31 +]P/)fT(GF2$\"3A<&[\"zw?aHF27$$\"31+]i0j\"[$HF2$\"3)G?nc^TS*HF27$$\"\" $F)$\"3Bic(=(*)eOIF2-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%L INEG-%'SYMBOLG6#%'CIRCLEG-F$6&7AF'7$$\"3/+++++++5F6$\"3,++++++55F27$$ \"35+++++++?F6$\"30+++&HG!R5F27$$\"3))**************HF6$\"31+++n8-%3\" F27$$\"3A+++++++SF6$\"33+++dc=T6F27$$\"3++++++++]F6$\"30+++!=()p?\"F27 $$\"3w**************fF6$\"3/+++:/fy7F27$$\"3a**************pF6$\"3)*** ****fq*QN\"F27$$\"3U+++++++!)F6$\"31+++S*>9V\"F27$$\"3A+++++++!*F6$\"3 6+++'yG,^\"F27$F*$\"3'******pw=$*e\"F27$$\"33+++++++6F2$\"3/+++3<^o;F2 7$$\"3%**************>\"F2$\"31+++oeQZF27$$\"3++++++++:F2$\"3# ******4$RI!)>F27$$\"33+++++++;F2$\"3y******[!)Rc?F27$$\"3%************ **p\"F2$\"30+++j-lJ@F27$$\"3/+++++++=F2$\"3&******\\a\\g?#F27$$\"3!*** ************=F2$\"31+++\"p'fzAF27$$\"\"#F)$\"3))******)f+BN#F27$$\"33+ ++++++@F2$\"3********Qcy,NEF27$$\"3++++++++DF2$\"30+++hbx.FF27$$\"33+++++++EF2$\"3%)***** 4+4=x#F27$$\"3;+++++++FF2$\"39+++e/9RGF27$$\"3#)*************z#F2$\"3! )*****\\(=z0HF27$$\"3!***************GF2$\"35+++nYyrHF27$Fez$\"3****** *RhRr.$F2-Fjz6&F\\[lF)F)F)-Fa[l6#%&POINTG-Fe[l6#%(DIAMONDG-F$6&Fj[lF]e lF_el-Fe[l6#%&CROSSG-F$6&Fj[lF]elF_elFd[l-%(SCALINGG6#%,CONSTRAINEDG-% +AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fez;F($\"#J!\"\"" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Suppose that we are aked to find a disrete numerical solu tion with 30 steps over the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"! %\"xG" }{XPPEDIT 18 0 "``<=3" "6#1%!G\"\"$" }{TEXT -1 60 " which exhib its a maximum relative error which is less than " }{XPPEDIT 18 0 "10^( -6)" "6#)\"#5,$\"\"'!\"\"" }{TEXT -1 133 ". Then, by gradually inctrea sing the order of the Taylor series method, we see that this can be ac hieved by using an order 6 method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "de := diff(y(x),x)=2*x/y(x )^2;\nic := y(0)=1;\nsoln := desolveTA(\{de,ic\},y(x),x=0..3,order=6,s teps=30):\ncomparewithfcn(soln,g(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"#\"\"\"F,F0F)!\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0 1 function val: \+ 1 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.1 1.009901667 function val: 1.009901634 r el err: 3.2676e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.2 \+ 1.038499046 function val: 1.03849882 rel err: 2.1762e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.3 1.082932449 funct ion val: 1.082932133 rel err: 2.9180e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.4 1.139604115 function val: 1.139 603843 rel err: 2.3868e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.5 1.205071327 function val: 1.205071132 rel err: \+ 1.6182e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.6 1.27 6501 function val: 1.27650086 rel err: 1.0967e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.7 1.351758222 function val: 1.351758112 rel err: 8.1376e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.8 1.429313994 function val: 1.4293139 \+ rel err: 6.5766e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.9 \+ 1.508104368 function val: 1.508104283 rel err: 5.6362e-0 8" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1 1.58740113 fu nction val: 1.587401052 rel err: 4.9137e-08" }}{PARA 6 "" 1 " " {TEXT -1 87 " 1.1 1.666711182 function val: 1. 66671111 rel err: 4.3199e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 1.2 1.745703814 function val: 1.745703747 rel err : 3.8380e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.3 1.824 159931 function val: 1.824159869 rel err: 3.3988e-08" }} {PARA 6 "" 1 "" {TEXT -1 87 " 1.4 1.90193714 functio n val: 1.901937083 rel err: 2.9969e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.5 1.978945858 function val: 1.978 945805 rel err: 2.6782e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 1.6 2.055132934 function val: 2.055132884 rel err: \+ 2.4329e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.7 2.13047 0382 function val: 2.130470335 rel err: 2.2061e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.8 2.204947587 function val: 2.204947543 rel err: 1.9955e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.9 2.278565878 function val: 2.278565837 \+ rel err: 1.7994e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2 \+ 2.351334726 function val: 2.351334688 rel err: 1.6161e-0 8" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.1 2.423269074 fu nction val: 2.423269038 rel err: 1.4856e-08" }}{PARA 6 "" 1 " " {TEXT -1 87 " 2.2 2.494387443 function val: 2.4 94387409 rel err: 1.3631e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 2.3 2.564710598 function val: 2.564710565 rel err : 1.2867e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.4 2.634 260595 function val: 2.634260565 rel err: 1.1388e-08" }} {PARA 6 "" 1 "" {TEXT -1 87 " 2.5 2.703060117 functio n val: 2.703060088 rel err: 1.0729e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.6 2.771131992 function val: 2.771 131964 rel err: 1.0104e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 2.7 2.83849887 function val: 2.838498843 rel err: \+ 9.5121e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.8 2.90518 2988 function val: 2.905182963 rel err: 8.6053e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.9 2.971206025 function val: 2.971206001 rel err: 8.0775e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 3 3.036588995 function val: 3.036588972 \+ rel err: 7.5743e-09" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum relative error: 2.9180e-0 7" }}{PARA 6 "" 1 "" {TEXT -1 59 " obtained for the input value: 0.3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }{TEXT 295 45 " .. an accurate discrete nuneric al solution " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 36 "Consider the differential equation " } {XPPEDIT 18 0 "dy/dx+``(2/x)*y = sin(x)/x;" "6#/,&*&%#dyG\"\"\"%#dxG! \"\"F'*&-%!G6#*&\"\"#F'%\"xGF)F'%\"yGF'F'*&-%$sinG6#F0F'F0F)" }{TEXT -1 34 " subject to the initial condition " }{XPPEDIT 18 0 "y(1) = 0;" "6#/-%\"yG6#\"\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "This is a linear differential equation which can be solved by the \+ integrating factor method." }}{PARA 0 "" 0 "" {TEXT -1 42 "The integra ting factor to be used here is " }{XPPEDIT 18 0 "exp(Int(2/x,x))=exp(2 *ln(x))" "6#/-%$expG6#-%$IntG6$*&\"\"#\"\"\"%\"xG!\"\"F--F%6#*&F+F,-%# lnG6#F-F," }{XPPEDIT 18 0 " ``= x^2" "6#/%!G*$%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "Multiplying both sides of the diff erential equation by " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 8 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2 " "6#*$%\"xG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+ 2*x*y = x*si n(x)" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'%\"xGF'%\"yGF'F'*&F,F'-% $sinG6#F,F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\" \"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ x^2*y] = x*sin(x)" " 6#/7#*&%\"xG\"\"#%\"yG\"\"\"*&F&F)-%$sinG6#F&F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 12 "which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*y=Int(x*sin(x),x)" "6#/*&%\"xG\"\"#%\"yG \"\"\"-%$IntG6$*&F%F(-%$sinG6#F%F(F%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= sin(x)-x*cos(x)+c" "6#/%!G,(-% $sinG6#%\"xG\"\"\"*&F)F*-%$cosG6#F)F*!\"\"%\"cGF*" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 53 "using integration by parts to evaluate th e integral. " }}{PARA 0 "" 0 "" {TEXT -1 48 "Taking the initial condit ion into account gives " }{XPPEDIT 18 0 "c=cos(1)-sin(1)" "6#/%\"cG,&- %$cosG6#\"\"\"F)-%$sinG6#F)!\"\"" }{TEXT -1 11 ", so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*y=sin(x)-x*cos(x)+cos(1 )-sin(1)" "6#/*&%\"xG\"\"#%\"yG\"\"\",*-%$sinG6#F%F(*&F%F(-%$cosG6#F%F (!\"\"-F/6#F(F(-F+6#F(F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=(sin(x)-x *cos(x)+cos(1)-sin(1))/x^2" "6#/%\"yG*&,*-%$sinG6#%\"xG\"\"\"*&F*F+-%$ cosG6#F*F+!\"\"-F.6#F+F+-F(6#F+F0F+*$F*\"\"#F0" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " We can u se Maple's " }{TEXT 0 6 "dsolve" }{TEXT -1 72 " to obtain this result \+ and set up the analytical solution as a function " }{XPPEDIT 18 0 "g(x )" "6#-%\"gG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "de := diff(y(x),x)+2/x*y(x )=sin(x)/x;\nic := y(1)=0;\ndsolve(\{de,ic\},y(x));\ng := unapply(rhs( %),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#% \"xGF-\"\"\"*(\"\"#F.F-!\"\"F*F.F.*&-%$sinGF,F.F-F1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,*-%$sinGF&\"\"\"*&F'F,-%$cosGF&F,!\"\" -F+6#F,F0-F/F2F,F,F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6# %\"xG6\"6$%)operatorG%&arrowGF(*&,*-%$sinG6#9$\"\"\"*&F1F2-%$cosGF0F2! \"\"-F/6#F2F6-F5F8F2F2F1!\"#F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 15 "\n The procedure " }{TEXT 0 9 "desolveTA" }{TEXT -1 74 " can be used to o btain a (discrete) solution by the Taylor series method. " }}{PARA 0 " " 0 "" {TEXT -1 52 "We construct a numerical solution over the interva l " }{XPPEDIT 18 0 "1 <= x;" "6#1\"\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 16 ;" "6#1%!G\"#;" }{TEXT -1 64 " using the order 12 Taylor series method with 75 steps of width " }{XPPEDIT 18 0 "h = 1/5;" "6#/%\"hG*&\"\"\"F &\"\"&!\"\"" }{TEXT -1 131 ", and compare the values given by the nume rical solution (in terms of absolute error) with those given by the an alytical solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "de := diff(y(x),x)+2/x*y(x)=sin(x)/x;\nic \+ := y(1)=0;\nsoln := desolveTA(\{de,ic\},y(x),x=1..16,steps=75,order=8) :\ncomparewithfcn(soln,g(x),x,errtype=absolute);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(\"\"#F.F-!\" \"F*F.F.*&-%$sinGF,F.F-F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-% \"yG6#\"\"\"\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1 \+ 0 function val: 0 abs err: 0.0000e+00" } }{PARA 6 "" 1 "" {TEXT -1 87 " 1.2 0.1361383904 functi on val: 0.136139654 abs err: 1.2636e-06" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.4 0.2277167795 function val: 0.2277 178832 abs err: 1.1037e-06" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 1.6 0.291063247 function val: 0.291064125 abs err: \+ 8.7800e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.8 0.333839 6451 function val: 0.3338403463 abs err: 7.0120e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2 0.360105035 function val: 0.3601056052 abs err: 5.7020e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.2 0.3723198138 function val: 0.3723202857 \+ abs err: 4.7190e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.4 \+ 0.372228669 function val: 0.3722290658 abs err: 3.9680e-0 7" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.6 0.3612782789 fu nction val: 0.3612786172 abs err: 3.3830e-07" }}{PARA 6 "" 1 " " {TEXT -1 87 " 2.8 0.3408213952 function val: 0.34 08216869 abs err: 2.9170e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 3 0.3122140591 function val: 0.3122143132 abs err : 2.5410e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 3.2 0.2768 552904 function val: 0.2768555138 abs err: 2.2340e-07" }} {PARA 6 "" 1 "" {TEXT -1 87 " 3.4 0.2361939262 functio n val: 0.236194124 abs err: 1.9780e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 3.6 0.1917159637 function val: 0.1917 161402 abs err: 1.7650e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 3.8 0.1449202527 function val: 0.144920411 abs err: \+ 1.5830e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 4 0.097287 5639 function val: 0.09728770688 abs err: 1.4298e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 4.2 0.05024652559 function val: 0.05024665522 abs err: 1.2963e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 4.4 0.005139028315 function val: 0.005139146446 \+ abs err: 1.1813e-07" }}{PARA 6 "" 1 "" {TEXT -1 87 " 4.6 \+ -0.03681287077 function val: -0.0368127627 abs err: 1.0807e-0 7" }}{PARA 6 "" 1 "" {TEXT -1 87 " 4.8 -0.07453692253 fu nction val: -0.07453682326 abs err: 9.9270e-08" }}{PARA 6 "" 1 " " {TEXT -1 87 " 5 -0.1071362467 function val: -0.10 71361553 abs err: 9.1400e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 5.2 -0.133909479 function val: -0.1339093945 abs err : 8.4500e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 5.4 -0.15 436478 function val: -0.1543647015 abs err: 7.8500e-08" }} {PARA 6 "" 1 "" {TEXT -1 87 " 5.6 -0.1682272488 functio n val: -0.1682271759 abs err: 7.2900e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 5.8 -0.1754395465 function val: -0.1754 394785 abs err: 6.8000e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 6 -0.1761557829 function val: -0.1761557194 abs err: \+ 6.3500e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.2 -0.170728 9638 function val: -0.1707289043 abs err: 5.9500e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.4 -0.1596925108 function val: -0.1596924549 abs err: 5.5900e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.6 -0.1437365636 function val: -0.1437365111 \+ abs err: 5.2500e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.8 \+ -0.1236799427 function val: -0.1236798932 abs err: 4.9500e-0 8" }}{PARA 6 "" 1 "" {TEXT -1 87 " 7 -0.1004387785 fu nction val: -0.1004387318 abs err: 4.6700e-08" }}{PARA 6 "" 1 " " {TEXT -1 87 " 7.2 -0.07499291217 function val: -0.074 99286804 abs err: 4.4130e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 7.4 -0.04835122524 function val: -0.04835118347 abs err : 4.1770e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 7.6 -0.02151 707232 function val: -0.02151703274 abs err: 3.9580e-08" }} {PARA 6 "" 1 "" {TEXT -1 87 " 7.8 0.004545037789 functio n val: 0.00454507538 abs err: 3.7591e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 8 0.02894043049 function val: 0.02894 046622 abs err: 3.5730e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 8.2 0.05087194305 function val: 0.05087197707 abs err: \+ 3.4020e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 8.4 0.069663 4444 function val: 0.0696634768 abs err: 3.2400e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 8.6 0.08477850918 function val: 0.0847785401 abs err: 3.0920e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 8.8 0.09583373903 function val: 0.09583376857 \+ abs err: 2.9540e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 9 \+ 0.1026064182 function val: 0.1026064465 abs err: 2.8300e-0 8" }}{PARA 6 "" 1 "" {TEXT -1 87 " 9.2 0.1050363925 fu nction val: 0.1050364196 abs err: 2.7100e-08" }}{PARA 6 "" 1 " " {TEXT -1 87 " 9.4 0.1032222618 function val: 0.10 32222877 abs err: 2.5900e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 9.6 0.09741217088 function val: 0.0974121957 abs err : 2.4820e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 9.8 0.08798 966443 function val: 0.08798968824 abs err: 2.3810e-08" }} {PARA 6 "" 1 "" {TEXT -1 87 " 10 0.07545523214 functio n val: 0.07545525501 abs err: 2.2870e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 10.2 0.06040430601 function val: 0.06040 432798 abs err: 2.1970e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 10.4 0.04350257805 function val: 0.0435025992 abs err: \+ 2.1150e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 10.6 0.0254595 8023 function val: 0.02545960058 abs err: 2.0350e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 10.8 0.007001507149 function val: 0.007001526758 abs err: 1.9609e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 11 -0.0111557343 function val: -0.0111557154 \+ abs err: 1.8900e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 11.2 \+ -0.02833229568 function val: -0.02833227745 abs err: 1.8230e-0 8" }}{PARA 6 "" 1 "" {TEXT -1 87 " 11.4 -0.04390808993 fu nction val: -0.04390807234 abs err: 1.7590e-08" }}{PARA 6 "" 1 " " {TEXT -1 87 " 11.6 -0.05734363307 function val: -0.057 34361608 abs err: 1.6990e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 11.8 -0.06819735207 function val: -0.06819733565 abs err : 1.6420e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 12 -0.07613 882909 function val: -0.07613881319 abs err: 1.5900e-08" }} {PARA 6 "" 1 "" {TEXT -1 87 " 12.2 -0.08095761022 functio n val: -0.08095759487 abs err: 1.5350e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 12.4 -0.08256737571 function val: -0.08256 736089 abs err: 1.4820e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 12.6 -0.08100544026 function val: -0.08100542586 abs err: \+ 1.4400e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 12.8 -0.0764277 2206 function val: -0.07642770807 abs err: 1.3990e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 13 -0.06909947981 function val: -0.06909946627 abs err: 1.3540e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 13.2 -0.05938226407 function val: -0.05938225098 \+ abs err: 1.3090e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 13.4 \+ -0.04771765647 function val: -0.04771764373 abs err: 1.2740e-0 8" }}{PARA 6 "" 1 "" {TEXT -1 87 " 13.6 -0.03460847453 fu nction val: -0.03460846216 abs err: 1.2370e-08" }}{PARA 6 "" 1 " " {TEXT -1 87 " 13.8 -0.02059819687 function val: -0.020 59818486 abs err: 1.2010e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 14 -0.006249411557 function val: -0.006249399888 abs err : 1.1669e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 14.2 0.007877 892047 function val: 0.007877903387 abs err: 1.1340e-08" }} {PARA 6 "" 1 "" {TEXT -1 87 " 14.4 0.02124738011 functio n val: 0.02124739114 abs err: 1.1030e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 14.6 0.03336601436 function val: 0.03336 602508 abs err: 1.0720e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 14.8 0.04380157893 function val: 0.04380158937 abs err: \+ 1.0440e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 15 0.0521974 9141 function val: 0.05219750156 abs err: 1.0150e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 15.2 0.05828440904 function val: 0.05828441893 abs err: 9.8900e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 15.4 0.06188827083 function val: 0.06188828049 \+ abs err: 9.6600e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 15.6 \+ 0.06293455637 function val: 0.06293456575 abs err: 9.3800e-0 9" }}{PARA 6 "" 1 "" {TEXT -1 87 " 15.8 0.06144868773 fu nction val: 0.06144869688 abs err: 9.1500e-09" }}{PARA 6 "" 1 " " {TEXT -1 87 " 16 0.0575526461 function val: 0.0 57552655 abs err: 8.9000e-09" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum absolute error : 1.2636e-06" }}{PARA 6 "" 1 "" {TEXT -1 59 " obtained fo r the input value: 1.2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 6 "" 1 "" {TEXT -1 45 " Mean absolute error: 1.2167e-07" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The maximum absolute error occurs when " }{XPPEDIT 18 0 "x = 1. 2;" "6#/%\"xG-%&FloatG6$\"#7!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "plot([g(x), soln],x=1..16,y=-0.3..0.5,color=[red,blue],style=[line,point],\n \+ symbol=circle,labels=[`x`,`y(x)`]);" }} {PARA 13 "" 1 "" {GLPLOT2D 563 326 326 {PLOTDATA 2 "6'-%'CURVESG6%7]q7 $$\"\"\"\"\"!$F*F*7$$\"3#**\\iSmp3/\"!#<$\"3_A;dA6.\"G$!#>7$$\"31+]7G$ R<3\"F/$\"3%*eCZ-hytiF27$$\"3@+v=#**3E7\"F/$\"3c%e<+ggU,*F27$$\"37++Dc 'yM;\"F/$\"3)e_&GvI@`6!#=7$$\"3)***\\P%)z@X7F/$\"3q&y&\\\\8X*f\"FB7$$ \"3/++]7t&pK\"F/$\"3I14T2'o8)>FB7$$\"39+v=<#*>p9F/$\"3=JlYJbHBDFB7$$\" 3****\\(=7T9h\"F/$\"3#*eP1#*4;SHFB7$$\"33+DJ?qSrF/$\"3'[zz1d)zFNFB7$$\"3&***\\(o/')=,#F/$\"3,*e#o:q/7 OFB7$$\"3;+++v\"*R#4#F/$\"3=V%\\=&zytOFB7$$\"3$)*\\i!RdlK@F/$\"3%3^PG$ Qf'p$FB7$$\"3%***\\7.B\"H<#F/$\"3H#eY$*f2Ur$FB7$$\"30+v=n)oJ@#F/$\"3/k ,!oifns$FB7$$\"3;++DJaU`AF/$\"3'3u70HvVt$FB7$$\"3ID1*Gm!\\$H#F/$\"3\\# >L'\\P`t$FB7$$\"3qu=\\7:LFB7$$\"3!***\\ PM0'\\-$F/$\"3%RWZj[493$FB7$$\"3'***\\(o**3)yJF/$\"3Z$o\\g#yy3GFB7$$\" 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\"\"%#dxG!\"\",&*&F&F&,&F&F&*$%\"xG\"\"#F&F(F&*&F.F&*$%\"yGF.F&F(" } {TEXT -1 21 " over the interval " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\" !" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 28 " with the initial condition " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6# \"\"!F'" }{TEXT -1 30 ".\nThe analytical solution is " }{XPPEDIT 18 0 "y = x/(1+x^2);" "6#/%\"yG*&%\"xG\"\"\",&F'F'*$F&\"\"#F'!\"\"" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "de := diff(y(x),x)=1/(1+x^2)-2*y(x)^2;\nic := y (0)=0;\nsoln := dsolve(\{de,ic\},y(x));\ng := unapply(rhs(soln),x);\np lot1 := DEtools[DEplot](de,y(x),x=0..2,y=0..1,arrows=medium,color=blue ):\nplot2 := plot(g(x),x=0..2,y=0..1,thickness=2,color=coral):\nplots[ display]([plot1,plot2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%% diffG6$-%\"yG6#%\"xGF,,&*&\"\"\"F/,&F/F/*$)F,\"\"#F/F/!\"\"F/*&F3F/)F) F3F/F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }} {PARA 11 "" 1 "" 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "de := diff(y(x),x)=1/(1+x^2)-2*y(x)^2;\nic := y(0)=0 ;\nsoln := desolveTA(\{de,ic\},x=0..2,stepsize=fixed,order=9):\ng := x -> x/(1+x^2);\ncomparewithfcn(soln,g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*&\"\"\"F/,&F/F/*$)F,\"\"#F/F/ !\"\"F/*&F3F/)F)F3F/F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"y G6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)op eratorG%&arrowGF(*&,&*$)9$\"\"#\"\"\"F2F2F2!\"\"F0F2F(F(F(" }}{PARA 6 "" 1 "" {TEXT -1 85 " 0 0 function val: \+ 0 rel err: infinity" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.1 0.099009901 function val: 0.09900990099 rel err: 1.0100e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.2 0. 1923076924 function val: 0.1923076923 rel err: 5.2000e-10" }} {PARA 6 "" 1 "" {TEXT -1 87 " 0.3 0.2752293579 functio n val: 0.2752293578 rel err: 3.6333e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.4 0.3448275863 function val: 0.3448 275862 rel err: 2.9000e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.5 0.4000000001 function val: 0.4 rel err: \+ 2.5000e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.6 0.441176 4706 function val: 0.4411764706 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.7 0.4697986577 function val: 0.4697986577 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.8 0.4878048781 function val: 0.4878048781 \+ rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.9 \+ 0.4972375691 function val: 0.497237569 rel err: 2.0111e-1 0" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1 0.5 fu nction val: 0.5 rel err: 0.0000e+00" }}{PARA 6 "" 1 " " {TEXT -1 87 " 1.1 0.4977375566 function val: 0.49 77375566 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 1.2 0.4918032787 function val: 0.4918032787 rel err : 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.3 0.4832 713755 function val: 0.4832713755 rel err: 0.0000e+00" }} {PARA 6 "" 1 "" {TEXT -1 87 " 1.4 0.472972973 functio n val: 0.4729729729 rel err: 2.1143e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.5 0.4615384615 function val: 0.4615 384616 rel err: 2.1667e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 1.6 0.4494382023 function val: 0.4494382022 rel err: \+ 2.2250e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.7 0.437017 9949 function val: 0.4370179948 rel err: 2.2882e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.8 0.4245283019 function val: 0.4245283019 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.9 0.4121475054 function val: 0.4121475054 \+ rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2 \+ 0.4 function val: 0.4 rel err: 0.0000e+0 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum relative error: 5.2000e-10" }}{PARA 6 "" 1 " " {TEXT -1 59 " obtained for the input value: \+ 0.2" }}{PARA 6 "" 1 "" {TEXT -1 67 " excluding any cases where the function value is zero." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 22 "A continuous solution " }{TEXT 270 2 "gn " }{TEXT -1 114 " (numerical function) based on 20 equally spaced data points can be constructed as follows by using the procedure " }{TEXT 0 9 "desolveTA" }{TEXT -1 18 " with the option \"" }{TEXT 270 16 "outp ut=procedure" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "de := diff(y(x),x)=1/(1+x^2 )-2*y(x)^2;\nic := y(0)=0;\ngn := desolveTA(\{de,ic\},x=0..2,stepsize= fixed,order=9,output=procedure);\ng := x -> x/(1+x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*&\"\"\"F/,&F/F/ *$)F,\"\"#F/F/!\"\"F/*&F3F/)F)F3F/F4" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gnGf*6# '%)x_taylorG%)realconsG61%#xkG%#ykG%#jFG%#jMG%#jSG%\"nG%\"hG%\"jG%%dat aG%#xxG%+saveDigitsG%$sumG%%termG%%factG%#dkG6#%>Copyright~2002~by~Pet er~StoneG6\"C4>8,777-$\"\"!FCFB$\"\"\"FCFB$!/++++++g!#8FB$\"/++++++7!# 6FB$!/+++++S]!#5FB$\"/++++!)GO!\")7-$\"/++++++5!#9$\"/+++,*4!**!#:$\"/ $*3*)3$\\q*FU$!/g(\\3HT!eFU$!/=-Jd]?aFH$\"/5'Q[Y*=6!#7$\"/s-4p!ei*F[o$ !/tFWOEnWFK$!/YDIhe$Q$FN$\"/Mn)H7H,$!\"*$\"/hmKn5v=FQ7-$\"/++++++?FU$ \"/y.Cp2B>FU$\"/8BW'Rd())FU$!/1&*)orD0\"FH$!/7=IK61RFH$\"/R,#HYa\"=F[o $\"/GkTL^?SF[o$!/?:v]nkhFK$\"/&GiMO^g$F[o$\"/Qz(pZ$3LFdo$!/E_;f:q6FQ7- $\"/++++++IFU$\"/7YzNH_FFU$\"/o4PzGfwFU$!/1YcBB[8FH$!/.qXIo*)>FH$\"/F2 _yQA>F[o$!/N&=;\"=S;F[o$!/9%)HXE\\ZFK$\"/Lf[#))>Y#FN$\"/'eX!3\"eN\"Fdo $!/%3Pb`zH#FQ7-$\"/++++++SFU$\"/LGjeF[MFU$\"/?Jp$oDC'FU$!/#eVBubX\"FH$ !/s#[uTt'Hq>FK$\"/CiydJqFFN$!/9B 8NssdFN$!/%=&o$*>h8FQ7-$\"/++++++]FU$\"/7m++++SFU$\"/?%*)******z%FU$!/ y&******zS\"FH$\"/6r++?v5FH$\"/Zl**fh25F[o$!/`$***Ry]dF[o$\"/]o5#z4U$F [o$\"/x3)e)oS-=!*FN7-$\"/++++++gFU$\"/(*=1Zw 6WFU$\"/W\">h2-Y$FU$!/!p+!QTf7FH$\"/ILI-=2=FH$\"/))=S:>MZFH$!/Hb+>BYZF [o$\"/B')[SSo9FK$\"/8M\\k#ze&FK$!/g-$G)y!***FN$\"/pB\"GZb(\\Fdo7-$\"/+ +++++qFU$\"/1Rxl)zp%FU$\"/[q>Q>(H#FU$!//Na'*Gi5FH$\"/#G7.H$p?FH$\"/\"G o5*3!y(FU$!/Se-/8VJF[o$\"/`iV+K:;FK$!/JBr>Uk\"G/Utr#FH$!/')H&eN$ziFH$\"/'\\Grn 1D)F[o$!/J$RMvkG%FK$\"/!R&[vTv&)FK$\"/VEUq$fP*FN7-$FTFH$\"/LB++++]FU$! %mYFU$!/q1********\\FU$\"/)f)*******\\\"FH$!/*R,++++$FH$\"/++++].G!#A$ \"/'=$)******\\%F[o$!/bT******\\JFK$\"/#e++++E\"FN$!/+++D&)RJ!#>7-$\"/ ++++++6FH$\"/h\"ecvt(\\FU$!.&3oim*H%FU$!/ShB,P[OFU$\"/rH>:H17FH$!/S'[H F[o$!/t2))\\dh>FK$\"/zC6\"R_1\"FN$!/TN y'>J:$FN7-$FJFH$\"/i0(yK!=\\FU$!.V!yK%FH$!/\"y$=hCcFFH$!/mr^@M#*[F[o$\"/m* oG%)H(H(HZFU$!/;4&G!p&4\"FU$ !/(fWwNG7\"FU$\"/I0R'>uS&FU$!/&Rf)p%3f\"FH$\"/HlmVhqQFH$!/%Q%HE!\\%eFH $!/#R(*e9=i\"F[o$\"/xHQ:6LBFK$!/n@?A!=f\"FN7-$\"/++++++:FH$\"/aZ:YQ:YF U$!/.$G&>V$=\"FU$!/!)o\\BRalFX$\"/]4[*f)**RFU$!/<`KI'[B\"FH$\"/+umb\"z B$FH$!/tdY^LMlFH$\"/.W?$f-y$FU$\"/to[/+)4\"FK$!/htkrf0#*FK7-$\"/++++++ ;FH$\"/haA?Q%\\%FU$!/B())=04B\"FU$!/Sf&f/27$FX$\"/E!\\i(4;HFU$!/)Ga\\A HV*FU$\"/dlN'e7g#FH$!/[3n!o,3'FH$\"/o<,4BovFH$\"/D,uf)>7%F[o$!/'[!zPHl [FK7-$\"/++++++Txlc\"FH$!/#[\")yI0@%FH$\"/-!**** oo]*FH$!/r+^_FH$\"/GFa]Z@TFU$!/bjFR6G 7FU$\"//lZx(fO#FX$\"/1+DQL85FU$!/ue(z;q&RFU$\"/s%HQ]7>\"FH$!/)))GK&e;L FH$\"/p%yxsQG)FH$!/=^\"=gQV\"F[o$!/@bY;i+BF[o7$$FioFH$\"/i.++++SFU>8.% 'DigitsG>F\\jl-%$minG6$-%$maxG6$-%&truncG6#-%'evalhfG6#F\\jl-Fejl6#,$* (\"\"%FE\"\"$!\"\"F\\jlFEFE,&F\\jlFE\"#5FE>8--%&evalfG6#9$>8)-%%nopsG6 #F?@$532&F?6$FEFE&F?6$Fj[mFE52Fd\\mFd[m2Fd[mFb\\m32Fd\\mFb\\m52Fd[mFd \\m2Fb\\mFd[mY6$Q\\oindependent~variable~is~outside~of~the~interpolati on~interval:~%1F<;-Ff[m6#Fb\\m-Ff[m6#Fd\\m>Fj[mF[\\m>8&FC>8(,&Fj[mFEFE FE@%Fa\\mC$?(F8'-Fejl6#,&*&#FE\"\"#FEFh]m FEFE*&Fh^mFEFj]mFEFE@%1&F?6$Fc^mFEFd[m>Fh]mFc^m>Fj]mFc^m@$/Fc^mFj[mC$> Fh]m,&Fj[mFEFEF`[m>Fj]mFj[mC$?(FFc^mFd^m@%1Fd[mF]_m>Fh]m Fc^m>Fj]mFc^m@$Fb_mC$>Fh]mFe_m>Fj]mFj[m>8$&F?6$Fh]mFE>8%&F?6$Fh]mFi^m> 8*,&Fd[mFEFd`mF`[m>8/Fh`m>81-Ff[m6#FE?(8+FEFE,&-F\\\\m6#&F?6#Fh]mFEFi^ mF`[m%%trueGC&>Faam*(FaamFEF\\amFEFeamF`[m>82&F?6$Fh]m,&FeamFEFi^mFE>8 0*&F`bmFEFaamFE>F_am,&F_amFEFebmFE>F\\jlF[jl-Ff[m6#F_amF%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&,&*$)9$\"\"#\"\"\"F2F2F2!\"\"F0F2F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The code for the numerica l function " }{TEXT 270 2 "gn" }{TEXT -1 79 " can be viewed as follows . You need to re-execute the commands to see the code." }}{PARA 0 "" 0 "" {TEXT -1 215 "It contains a data consisting of numerical values \+ for the solution and appropriate derivatives to be used in local Taylo r series. This allows the function to interpolate between the points o f the discrete solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "interface(verboseproc=2);\neval(gn) ;\ninterface(verboseproc=0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "The values of the numerical function " } {TEXT 270 2 "gn" }{TEXT -1 53 " are close to those given by the analyt ical solution " }{TEXT 270 1 "g" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "xx := sqrt(2 );\nevalf(g(xx));\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG*$ \"\"##\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1_/9Z!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3_/9Z!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The numerical function " }{TEXT 270 2 "gn" }{TEXT -1 51 " cannot be evaluated outside the origi nal interval " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to \+ " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 22 " specified when u sing " }{TEXT 0 9 "desolveTA" }{TEXT -1 16 " to constuct it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "xx := 1.234567891;\nevalf(g(xx));\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+\"*ycM7!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1!4 5*[!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1!45*[!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "It is better to allow the use of " } {TEXT 265 25 "adaptive stepsize control" }{TEXT -1 18 " with the optio n \"" }{TEXT 270 13 "adaptive=true" }{TEXT -1 6 "\" or \"" }{TEXT 270 17 "stepsize=variable" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 65 "If the \"steps\" option is omitted then this is the default option ." }}{PARA 0 "" 0 "" {TEXT -1 18 "With this option, " }{TEXT 0 9 "deso lveTA" }{TEXT -1 137 " controls the stepsize used in the construction \+ of the underlying discrete solution by estimating the error after each step is performed." }}{PARA 0 "" 0 "" {TEXT -1 274 "If the error is t oo large, the step is repeated. If the error is within the required bo unds, the the step is accepted, and the next step is performed with a \+ step-size which is chosen on the basis of a numerical comparison of th e error estimate with the required error bound." }}{PARA 0 "" 0 "" {TEXT 265 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 40 "I f you want to see how this works, give " }{TEXT 0 9 "desolveTA" } {TEXT -1 13 " the option \"" }{TEXT 270 9 "info=true" }{TEXT -1 53 "\" . \nYou can get varying amounts of information with \"" }{TEXT 270 6 " info=n" }{TEXT -1 27 "\", where n=0, 1, 2, 3 or 4." }}{PARA 15 "" 0 " " {TEXT -1 50 "When plotting the graph of the numerical function " } {TEXT 270 2 "gn" }{TEXT -1 17 ", the expression " }{TEXT 270 5 "gn(x) " }{TEXT -1 31 " must be enclosed in quotes as " }{TEXT 0 7 "'gn(x)'" }{TEXT -1 27 ", to delay its evaluation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "de := diff(y(x),x)=1 /(1+x^2)-2*y(x)^2;\nic := y(0)=0;\ngn := desolveTA(\{de,ic\},x=0..2,in fo=true);\nplot('gn(x)',x=0..2,thickness=2,color=brown,labels=[`x`,`y( x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\" xGF,,&*&\"\"\"F/,&F/F/*$)F,\"\"#F/F/!\"\"F/*&F3F/)F)F3F/F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"\"!F%%2abs~err~bound~->~G$\"& ++\"!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"\"%$~~~G$\"+++++ 5!#6F%7$F&$\"/+***4+!****!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Fincr easing~step-size~by~a~factor~of~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&92\" !#=%2abs~err~bound~->~G$\"&&)*\\!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 (%%stepG\"\"#%$~~~G$\"+++++]!#6F%7$$\"+++++gF($\"/(o5[x%yf!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Freducing~step-size~and~repeating~stepG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&I#G!#;%2abs~err~bound~->~G$ \"&-B)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"$%$~~~G$\"/j(RT (zlt!#:F%7$$\"/wRT(zlL\"!#9$\"//&>a@JJ\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&&4V!#;%2abs~err~bound~->~G$\"&JJ\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"%%$~~~G$\"/d$o-))yP(!#:F%7$$\"/73W&oV2#!#9$ \"/8A!y!z))>F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adj ust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&O8$!#;%2abs~err~bound~- >~G$\"&)))>!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"&%$~~~G$ \"/wVy?qAu!#:F%7$$\"/]#>vQm\"G!#9$\"/!40!og4EF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&1&*)!#<%2abs~err~bound~->~G$\"&'4E!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"'%$~~~G$\"/#eN&GNO!)!#:F%7$$\"/3GPSF?O!#9$\" /zqi*o2?$F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust ~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%Freducing~step-size~and~repeating~stepG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&+5\"!#:%2abs~err~bou nd~->~G$\"&3?$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"(%$~~~G $\"/8jKGx*)*)!#:F%7$$\"/Ra?8D>X!#9$\"/qf'Q%z_PF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&tD\"!#:%2abs~err~bound~->~G$\"&Gv$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\")%$~~~G$\"/L1mcv-!*!#:F%7$$\"/-:()o_>a!#9$\" /-a6q7*=%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust ~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&sX'!#;%2abs~err~bound~->~G $\"&\"*=%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"*%$~~~G$\"/ YT,5zQ!*!#:F%7$$\"/~G$\"&R/\"!#;%2abs~err~bound~->~G$\"&s^%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#5%$~~~G$\"/(>L9jv!)*!#:F%7$$\"/Pi,B;/t!#9$\"/ g?V>/jZF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~s tep-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Freducing~step-size~and~repeating~stepG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&ee\"!#:%2abs~err~boun d~->~G$\"&Iw%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#6%$~~~G$ \"/EY?Bk06!#9F%7$$\"/j3AY!)4%)F($\"/)H\"*fTf#\\F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&()[\"!#:%2abs~err~bound~->~G$\"&f#\\F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#7%$~~~G$\"/B#[3f26\"!#9F%7$$\"/'3pqj0_*F($\"/ f_<9(R*\\F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust ~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&ZD)!#;%2abs~err~bound~->~G $\"&S*\\!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#8%$~~~G$\"/)f 6#fvE6!#9F%7$$\"/o!G'>tk5!#8$\"/+7K1=!*\\F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&Wf%!#;%2abs~err~bound~->~G$\"&-*\\!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#9%$~~~G$\"/KS]%yS@\"!#9F%7$$\"/r%y!)Rh=\"!#8$ \"/F;nV-G\\F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adju st~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&+7$!#;%2abs~err~bound~->~ G$\"&!G\\!#:" 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"6(%%stepG\"#=%$~~~G$\"/*fk%Gn)e\"!#9F%7$$\"/]Zn96+=!#8$\" /~G$\"&(=d!#;%2abs~err~bound~->~G$ \"&^C%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#>%$~~~G$\"/1\"4 J;'3!#8$\"/pUkU'\\.%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&&>I!#C%2abs~err~bound~->~G$\"&].%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#?%$~~~G$\"-RV,p-H!#8F%7$$\"/++++++?F($\"/2,++ ++S!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*last~stepG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=the~total ~number~of~steps~isG\"#?" }}{PARA 13 "" 1 "" {GLPLOT2D 501 277 277 {PLOTDATA 2 "6'-%'CURVESG6#7S7$$\"\"!F)F(7$$\"+M3VfV!#6$\"+4;;^VF-7$$ \"+#H[D:)F-$\"+=4s)4)F-7$$\"+e0$=C\"!#5$\"+,1(HA\"F87$$\"+3RBr;F8$\"+m U#ei\"F87$$\"+zjf)4#F8$\"+>225?F87$$\"+'4;[\\#F8$\"+gXj[BF87$$\"+j'y]! HF8$\"+6h)*yEF87$$\"+'zs$HLF8$\"+7t9(*HF87$$\"+8iI_PF8$\"+(=%>*G$F87$$ \"+<_M(=%F8$\"+-9niNF87$$\"+4y_qXF8$\"+d8u!y$F87$$\"+]1!>+&F8$\"+e?\"4 +%F87$$\"+]Z/NaF8$\"+Y\"fc>%F87$$\"+]$fC&eF8$\"+RALfVF87$$\"+'z6:B'F8$ \"+Ca`)[%F87$$\"+<=C#o'F8$\"+W\"=&>YF87$$\"+n#pS1(F8$\"+4u[7ZF87$$\"+j `A3vF8$\"+X2Z,[F87$$\"+n(y8!zF8$\"+.*GW'[F87$$\"+j.tK$)F8$\"+.'oz\"\\F 87$$\"+)3zMu)F8$\"+X/Eb\\F87$$\"+#H_?<*F8$\"+b^Q\")\\F87$$\"+!G;cc*F8$ \"+:M2&*\\F87$$\"+4#G,***F8$\"+hv****\\F87$$\"+!o2J/\"!\"*$\"+s-b&*\\F 87$$\"+%Q#\\\"3\"Fas$\"+(\\&p%)\\F87$$\"+;*[H7\"Fas$\"+g:dm\\F87$$\"+q vxl6Fas$\"+M7vT\\F87$$\"+`qn27Fas$\"+HQG7\\F87$$\"+cp@[7Fas$\"+#zs&z[F 87$$\"+3'HKH\"Fas$\"++#H\"R[F87$$\"+xanL8Fas$\"+I]l*z%F87$$\"+v+'oP\"F as$\"+Z>xaZF87$$\"+S<*fT\"Fas$\"+/y17ZF87$$\"+&)Hxe9Fas$\"+O(oNm%F87$$ \"+.o-*\\\"Fas$\"+Fas$ \"+xds,TF87$$\"+/Uac>Fas$\"+k%RC0%F87$$\"\"#F)$\"+++++SF8-%*THICKNESSG 6#Fdz-%+AXESLABELSG6$%\"xG%%y(x)G-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))e qk\"Fe[lFf[l-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The following segment of code c ompares the value of the numerical solution " }{TEXT 270 5 "gn(x)" } {TEXT -1 43 " with the value of the analytical solution " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 " x = sqrt(3)" "6#/%\"xG-%%sqrtG6#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 165 "The numerical solution is constructed so as to give va lues which are correct to 10 digits, but values which are correct to a couple of extra digits can be obtained. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "g := x -> x/(1+x^2); \nxx := sqrt(3); \n'gn'(xx)=gn(xx);\n'g'(xx)=evalf(g(xx));\n'gn'(xx)=e valf[14](gn(xx));\n'g'(xx)=evalf[14](g(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&*$)9$\"\"# \"\"\"F2F2F2!\"\"F0F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG* $\"\"$#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#gnG6#*$\"\"$ #\"\"\"\"\"#$\"+>q7IV!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#* $\"\"$#\"\"\"\"\"#$\"+?q7IV!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%# gnG6#*$\"\"$#\"\"\"\"\"#$\"/U#*=q7IV!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#*$\"\"$#\"\"\"\"\"#$\"/A#*=q7IV!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "We can construct a soluti on which goes both to the left and right of the initial point" } {XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "de : = diff(y(x),x)=1/(1+x^2)-2*y(x)^2;\nic := y(0)=0;\ngn := desolveTA(\{d e,ic\},x=-4..4);\nplot('gn(x)',x=-4..4,thickness=2,color=brown,labels= [`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-% \"yG6#%\"xGF,,&*&\"\"\"F/,&F/F/*$)F,\"\"#F/F/!\"\"F/*&F3F/)F)F3F/F4" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 13 " " 1 "" {GLPLOT2D 480 269 269 {PLOTDATA 2 "6'-%'CURVESG6#7ao7$$!\"%\"\" !$!+w6%HN#!#57$$!+nFiDQ!\"*$!+N;xYCF-7$$!+o!)*Qn$F1$!+3]:MDF-7$$!+xxE. NF1$!+OpTREF-7$$!+Ok]JLF1$!+KMb`FF-7$$!+[9cgJF1$!+%)[2wGF-7$$!+iN2-IF1 $!+C>M)*HF-7$$!+N&oz$GF1$!+!yoW8$F-7$$!+\")3DoEF1$!+/-?'G$F-7$$!+:v2* \\#F1$!+Ku>\\MF-7$$!+8>1DBF1$!+FfbHOF-7$$!+w))yr@F1$!+Fu/*z$F-7$$!+S(R #**>F1$!+/C\"4+%F-7$$!++@)f#=F1$!+A>&H@%F-7$$!+gi,f;F1$!+G/H@WF-7$$!+ \"G&R2:F1$!+\\^h1YF-7$$!+tK5F8F1$!+D,D1[F-7$$!+$yP2D\"F1$!+\">;u([F-7$ $!+$HsV<\"F1$!+s'*4O\\F-7$$!+u5a&3\"F1$!+F\\?$)\\F-7$$!*a)4n**F1$!+[G( ***\\F-7$$!+S^%Rd*F-$!+qWE&*\\F-7$$!+Nz\")\\F-7$$!+I$Qwy)F -$!+'yJ&e\\F-7$$!*$\\[%R)F1$!+&y$RC\\F-7$$!+N2Vs\"F1$\"+x(>Xn\"F-7$$\"+v7+#\\#F-$\"+ AZHYBF-7$$\"*O&pfKF1$\"+C4gYHF-7$$\"++g#))3%F-$\"+e;:.NF-7$$\"*kcz\"\\ F1$\"+YV9gRF-7$$\"+D(HXx&F-$\"+)370L%F-7$$\"*\"G5JmF1$\"+(HVeg%F-7$$\" +gC4puF-$\"+([:Wz%F-7$$\"*6#32$)F1$\"+z<@:\\F-7$$\"+]6[7()F-$\"+/8)G& \\F-7$$\"+&=!)y6*F-$\"+H_vy\\F-7$$\"+?#zK_*F-$\"+K6/%*\\F-7$$\"*Ey'G** F1$\"+?>()**\\F-7$$\"+GJ*G3\"F1$\"+Xp=%)\\F-7$$\"+J%=H<\"F1$\"+F02P\\F -7$$\"+q,\"QD\"F1$\"+O%pZ([F-7$$\"+3>qM8F1$\"+b!>')z%F-7$$\"+,.W2:F1$ \"+d;c1YF-7$$\"+fp'Rm\"F1$\"+br7:WF-7$$\"+T>4N=F1$\"+;(Q;?%F-7$$\"+8s5 '*>F1$\"+iPn/SF-7$$\"+mXTk@F1$\"+C\\W2QF-7$$\"+od'*GBF1$\"+*on`i$F-7$$ \"+EcB,DF1$\"+J?/ZMF-7$$\"+v>:nEF1$\"+)[?sG$F-7$$\"+0a#o$GF1$\"+*p_a8$ F-7$$\"+`Q40IF1$\"+%eHf*HF-7$$\"+\"3:(fJF1$\"+;]qwGF-7$$\"+e%GpL$F1$\" +&H<)\\FF-7$$\"+:-V&\\$F1$\"+\\1WWEF-7$$\"+ZhUkOF1$\"+M&)zRDF-7$$\"+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }{TEXT 285 46 " .. a differential equation for \+ which Maple's " }{TEXT 0 6 "dsolve" }{TEXT 304 36 " cannot find an ana lytical solution " }}{PARA 0 "" 0 "" {TEXT -1 36 "Consider the differe ntial equation " }{XPPEDIT 18 0 "dy/dx = 1/(1+x^2)-2*y^2+y;" "6#/*&%# dyG\"\"\"%#dxG!\"\",(*&F&F&,&F&F&*$%\"xG\"\"#F&F(F&*&F.F&*$%\"yGF.F&F( F1F&" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "x=0" "6#/%\" xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" } {TEXT -1 28 " with the initial condition " }{XPPEDIT 18 0 "y(0)=0" "6# /-%\"yG6#\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 73 "\nMap le cannot find an analytical solution for this differential equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "de := diff(y(x),x)=1/(1+x^2)-2*y(x)^2+y(x);\nic := y(0)=0;\nds olve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%dif fG6$-%\"yG6#%\"xGF,,(*&\"\"\"F/,&F/F/*$)F,\"\"#F/F/!\"\"F/*&F3F/)F)F3F /F4F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}} {PARA 0 "" 0 "" {TEXT -1 68 "\nMaple can plot a numerical solution alo ng with the direction field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "de := diff(y(x),x)=1/(1+x^2 )-2*y(x)^2+y(x);\nDEtools[DEplot](de,y(x),x=0..2,y=0..1.5,[[0,0]],arro ws=medium,color=blue,linecolor=coral);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,(*&\"\"\"F/,&F/F/*$)F,\"\"#F/F/ !\"\"F/*&F3F/)F)F3F/F4F)F/" }}{PARA 13 "" 1 "" {GLPLOT2D 397 292 292 {PLOTDATA 2 "6'-%'CURVESG6^dl7'7$$!3Q0@%ot%*y:$!#>F(7$$\"3Q0@%ot%*y:$F *F,7$$\"3/b#)\\$y*HRT4Q)>\"F*7$$\"3!*y^.ciayt!#?$\"3I\"y3x Y\"f#Q#F*F+7'7$$\"3KeB*>rghM(F*$!3G'eLH1/`9$F*7$$\"3!4]&p'3Z1P\"!#=$\" 3G'eLH1/`9$F*7$$\"3u3j?,eyP8FB$\"3m!RU`Poq=\"F*7$$\"33.unj))4G6FB$\"3+ ))yRHlizBF*F?7'7$$\"3_T#)[Mzs!y\"FB$!3jryC_%=w5$F*7$$\"3%>B,8Q)zHCFB$ \"3jryC_%=w5$F*7$$\"3='Q'4*ydGR#FB$\"3!fs-fkwM:\"F*7$$\"3u%>8cA$o&=#FB $\"3g#3xOt$[qBF*FU7'7$$\"3+JcgMl+BGFB$!3%['y:j]?XIF*7$$\"3;z&y!RHy#\\$ FB$\"3%['y:j]?XIF*7$$\"3uI\\BH*y#\\MFB$\"3CA1v6sn)4\"F*7$$\"31SxNe_EYK FB$\"3w+u8qz]aBF*Fdo7'7$$\"3OR)HW()o@'QFB$!3/Gr8!4*)*eHF*7$$\"3a2\"\\r v$))eXFB$\"3/Gr8!4*)*eHF*7$$\"3c\\H\"*HNn1XFB$\"3XOzzE!RX-\"F*7$$\"3Hu /d!f2%4VFB$\"3jkl*=Vz3L#F*Fip7'7$$\"3c(Rax`6\"**[FB$!3?=U))f^x]GF*7$$ \"3i'G>$>JXBF F*7$$\"3CK.tU6x'p'FB$\"3w0)>$>JXBFF*7$$\"3*pD*[6IeAmFB$\"3qW'[='pA)H)F 67$$\"3>q7?q%>5W'FB$\"3GBy5bR]eAF*Fcs7'7$$\"3I/oO%)\\>qpFB$!394G6*[i4e #F*7$$\"3y_j*37Zmw(FB$\"394G6*[i4e#F*7$$\"3+NN3eAC!o(FB$\"3]C:B-2*p;(F 67$$\"31\"oU@4y\"3vFB$\"3`bg@Eo/5AF*Fht7'7$$\"3SWX\"zLvg+)FB$!3Gyi'G3( )zU#F*7$$\"3S\\LCD*Hg$))FB$\"3Gyi'G3()zU#F*7$$\"3H9?&))Qsst)FB$\"3C%)= M.Tb&)fF67$$\"33'fFLe1ad)FB$\"3;)pv:E?Z:#F*F]v7'7$$\"3K9\"=Kn^I/*FB$!3 oOhk/\\YpAF*7$$\"3@;X$yu;V!**FB$\"3oOhk/\\YpAF*7$$\"3Mho[>i\\$z*FB$\"3 #\\pG)pw@$z%F67$$\"3w>rU'*FB$\"3'4i))=y$>%4#F*Fbw7'7$$\"3?:Fl\"p] \"35!#<$!3]@5?Wd15@F*7$$\"3_@?CmC6(4\"Fex$\"3]@5?Wd15@F*7$$\"3_qE`H_)[ 3\"Fex$\"3'3v&f$QQOi$F67$$\"3SI8dC\"=32\"Fex$\"3&o14>6(RI?F*Fhx7'7$$\" 3!R3#)eCi@6\"Fex$!37Qy(ygyP&>F*7$$\"3sE@!yAFO?\"Fex$\"37Qy(ygyP&>F*7$$ \"3-footqL!>\"Fex$\"3/Z.\"[[MG]#F67$$\"3ktw'H)=Jx6Fex$\"3IQ=[f=Dl>F*F] z7'7$$\"3*[$yy`?M;7Fex$!3y#z9p%Ro.=F*7$$\"3=\\eoNP(*48Fex$\"3y#z9p%Ro. =F*7$$\"3S7#G<;9dH\"Fex$\"3I0r2*p)o[9F67$$\"3!Q6#3-'*o$G\"Fex$\"3O\"[& \\0HY+>F*Fb[l7'7$$\"3*HNyhV\"o?8Fex$!3]IOP-&4>m\"F*7$$\"3`/[3p1;;9Fex$ \"3]IOP-&4>m\"F*7$$\"3PS$=r]G5S\"Fex$\"3B'e[_*pP;Z!#@7$$\"3Y\\eV5\"\\* *Q\"Fex$\"3fc%[pQ*RP=F*Fg\\l7'7$$\"3Ytwy%Rl^U\"Fex$!3L&*HRn)>(H:F*7$$ \"3^d\\EEI?A:Fex$\"3L&*HRn)>(H:F*7$$\"3u+S**GGH1:Fex$!3)y?fC11'RUFa]l7 $$\"3;9Za'p%4'\\\"Fex$\"3%f[u&H&fqx\"F*F]^l7'7$$\"3FnDfohxH:Fex$!3N%y? hpwwS\"F*7$$\"3;P&\\#o&=\"G;Fex$\"3N%y?hpwwS\"F*7$$\"3he-\")p!>:h\"Fex $!3tFGT->DQ7F67$$\"3j'y-&eX8-;Fex$\"3z/pK.[4?W(>F67$$\"3z/_CQ)y!3y1>Q>\"F*7$$\"3H\"=$)3aS'R=Fex$\"3=^*>y1>Q>\"F*7 $$\"3E[Ei^q*=#=Fex$!3Sz]>\\J[PEF67$$\"3)==PyDQR\"=Fex$\"3#\\20.\")3vh \"F*F\\bl7'7$$\"3Dw^))*\\%>W=Fex$!3vFM*zqL65\"F*7$$\"3_[`K%=z_%>Fex$\" 3vFM*zqL65\"F*7$$\"3AzR&Q:kq#>Fex$!35#>gn!4SLKF67$$\"3OctYiKs>>Fex$\"3 [5sdUx*>d\"F*Facl7'7$$\"3YqY\\Hb9\\>Fex$!3jvl55%*3<5F*7$$\"3wF`]qW&30# Fex$\"3jvl55%*3<5F*7$$\"3#e?io8CA.#Fex$!3M,b/aaWoPF67$$\"3eG[zSNWD?Fex $\"3#yFX&)4)>I:F*Ffdl7'7$$!3r(>P<\"ofFIF*$\"3/UN!fNfem%F*7$$\"3r(>P<\" ofFIF*$\"3F)pIGVhB6\"FB7$$\"3SyrWJ9!Hz#F*$\"3%[4F<'\\zd\"*F*7$$\"3=;.Y 1W;.kF6$\"3IaGhwVJH5FBF[fl7'7$$\"3;rhx#*[zxuF*$\"3%[!eAWE&pn%F*7$$\"3g >rhoO[d8FB$\"38s%)*R5_76\"FB7$$\"30!Qa[JQFL\"FB$\"3!f1CA%>eZ\"*F*7$$\" 3\"G3H$o$>#=6FB$\"3Fj&[ysx!H5FBF`gl7'7$$\"3IL;.(3zUz\"FB$\"35wgD].85ZF *7$$\"3:SyvGsC;CFB$\"3+XaRLV$z5\"FB7$$\"3]sirD2u(Q#FB$\"387qeRJF<\"*F* 7$$\"3?g'fiiLa<#FB$\"32lB`SjMG5FBFehl7'7$$\"3`x,U)3gr$GFB$\"3'HpQ!p?+l ZF*7$$\"3kKSE&QH'yMFB$\"3D$=<:;ZC5\"FB7$$\"3$[&f*4qrSW$FB$\"3OJ\"*3R#z y1*F*7$$\"3b][7;FUNKFB$\"3(f8nqTjq-\"FBFjil7'7$$\"3/>C.\")[2xQFB$\"3Ph oJhfnS[F*7$$\"3'y_Y0vxRa%FB$\"3Km$*Gs(z[4\"FB7$$\"3$>:#>t5R,XFB$\"3Iq1 **G\"H6+*F*7$$\"3?DHZ9qy(H%FB$\"3PiH$4?d^-\"FBF_[m7'7$$\"3BJ[#=8N[\"\\ FB$\"3k!RbO7=b$\\F*7$$\"3'H&)[wl![6cFB$\"3S8b0cbR&3\"FB7$$\"3[*3)*35$H fbFB$\"3r/\"Ht3r%>*)F*7$$\"3eJ9+`=,i`FB$\"3P9\\#[9oD-\"FBFd\\m7'7$$\"3 %eh4`I>8&fFB$\"3MO'4MR4q/&F*7$$\"3_/)e?kf-o'FB$\"3!*)3!3HkCu5FB7$$\"3e j;)znjth'FB$\"3#e>'Go0)f#))F*7$$\"3)[&>S\"=:vU'FB$\"3Y#4p*pVF>5FBFi]m7 '7$$\"3a#)QZ!zut)pFB$\"3EaB[\"3#zr^F*7$$\"3_u#*y9tY\\xFB$\"37G$>]Ts)F*7$$\"3KUB\"zh/P\\(FB$\"3VEtjntI:5 FBF^_m7'7$$\"3B%[$ebPtB!)FB$\"3xcgOI<#fI&F*7$$\"3d4Wd2:P=))FB$\"3%pW%Q &>b$[5FB7$$\"313[iC7iK()FB$\"3Zu$>^B*e<')F*7$$\"3%4')30bL+c)FB$\"34O-, -Nv55FBFc`m7'7$$\"3Sk!fLqe41*FB$\"3uo^dW^DXaF*7$$\"39mNp<(4k))*FB$\"3n NN'R&=UM5FB7$$\"3e3Hh?PM*y*FB$\"37:PhD:q4&)F*7$$\"3MovxS\\/E'*FB$\"3f! 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diff(y(x),x)=1/(1+x^2)-2*y(x)^2 + y(x);\n ic := y(0)=0;\ngn := desolveTA(\{de,ic\},x=0..2);\nplot('gn(x)',x=0..2 ,thickness=2,color=brown,labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,(*&\"\"\"F/,&F/F/*$)F, \"\"#F/F/!\"\"F/*&F3F/)F)F3F/F4F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#icG/-%\"yG6#\"\"!F)" }}{PARA 13 "" 1 "" {GLPLOT2D 451 265 265 {PLOTDATA 2 "6'-%'CURVESG6#7S7$$\"\"!F)F(7$$\"+M3VfV!#6$\"+&[/tW%F-7$$ \"+#H[D:)F-$\"+$GMoV)F-7$$\"+e0$=C\"!#5$\"+Zy],8F87$$\"+3RBr;F8$\"+>IW nJt$o%F87$$\"+]1!>+&F8$\"+o9IS]F8 7$$\"+]Z/NaF8$\"+`MEr`F87$$\"+]$fC&eF8$\"+:A$Qm&F87$$\"+'z6:B'F8$\"+QP %p!fF87$$\"+<=C#o'F8$\"+P9PohF87$$\"+n#pS1(F8$\"+8H!pO'F87$$\"+j`A3vF8 $\"+jfRslF87$$\"+n(y8!zF8$\"+`PaKnF87$$\"+j.tK$)F8$\"+U:9')oF87$$\"+)3 zMu)F8$\"+msJ7qF87$$\"+#H_?<*F8$\"+.PlCrF87$$\"+!G;cc*F8$\"+ae#=@(F87$ $\"+4#G,***F8$\"+f#y,H(F87$$\"+!o2J/\"!\"*$\"+L$QfN(F87$$\"+%Q#\\\"3\" Fas$\"+#fy9S(F87$$\"+;*[H7\"Fas$\"+SFas$\"+&)H(z6(F87 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" }}{PARA 0 "" 0 "" {TEXT -1 92 "This is a linear differential equation which can be solve d by the integrating factor method." }}{PARA 0 "" 0 "" {TEXT -1 42 "Th e integrating factor to be used here is " }{XPPEDIT 18 0 "exp(Int(1/8, x)) = exp(x/8);" "6#/-%$expG6#-%$IntG6$*&\"\"\"F+\"\")!\"\"%\"xG-F%6#* &F.F+F,F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "Multiplying both sides of the differential equation by " }{XPPEDIT 18 0 "exp(x/8) ;" "6#-%$expG6#*&%\"xG\"\"\"\"\")!\"\"" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x/8)" "6#-%$expG6#*&%\" xG\"\"\"\"\")!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+1/8" "6#,&*& %#dyG\"\"\"%#dxG!\"\"F&*&F&F&\"\")F(F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x/8)=cos(x)" "6#/-%$expG6#*&%\"xG\"\"\"\"\")!\"\"-%$cosG6#F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[exp(x/8)*y] = cos(x)" "6#/7#*&-%$expG 6#*&%\"xG\"\"\"\"\")!\"\"F+%\"yGF+-%$cosG6#F*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 12 "which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x/8)*y=sin(x)+c" "6#/*&-%$expG6#*&%\"xG\" \"\"\"\")!\"\"F*%\"yGF*,&-%$sinG6#F)F*%\"cGF*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 48 "Taking the initial condition into account gives " }{XPPEDIT 18 0 "c = 0;" "6#/%\"cG\"\"!" }{TEXT -1 11 ", so th at " }{XPPEDIT 18 0 "y = exp(-x/8)*sin(x);" "6#/%\"yG*&-%$expG6#,$*&% \"xG\"\"\"\"\")!\"\"F.F,-%$sinG6#F+F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "de := di ff(y(x),x)+y(x)/8=exp(-x/8)*cos(x);\nic := y(0)=0;\nsoln := dsolve(\{d e,ic\},y(x));\ng := unapply(rhs(soln),x);\np1 := DEtools[DEplot](de,y( x),x=0..10,y=-1..1,arrows=medium,\n color=blue,dirgrid=[25,25]): \np2 := plot(g(x),x=0..10,y=-1..1,thickness=2,color=coral):\nplots[dis play]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$ -%\"yG6#%\"xGF-\"\"\"*&#F.\"\")F.F*F.F.*&-%$expG6#,$*&F1!\"\"F-F.F8F.- %$cosGF,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-%\"yG6#%\"xG*&-%$expG6#,$* &\"\")!\"\"F)\"\"\"F1F2-%$sinGF(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$expG6#,$*&#\"\"\"\"\")F3 9$F3!\"\"F3-%$sinG6#F5F3F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 505 298 298 {PLOTDATA 2 "6&-%'CURVESG6_bm7'7$$!33pWh1y_YO!#>$!3w7W#yVB5/\"!#<7 $$\"33pWh1y_YOF*$!3Usev@cw*e*!#=7$$\"3iD))fwYY&>(F*$!3c4:Kd\\j&y*F37$$ !3CmV]zZRgIF*$!3/l`Dz'p\"\\(*F3F.7'7$$\"3w\"RJ#zpLbPF3$!3zim*=ZY3/\"F- 7$$\"3%>%>5aj*zd%F3$!3#4PL5GN:f*F37$$\"3m(R&[tiY5\\F3$!3)3_#*pN&>>*=S 5F-7$$\"3/cBmko>$)))F3$!3A-[Y!33\")f*F37$$\"3[w&32(QYZ\"*F3$!3[Ry(HcD' *z*F37$$\"3&4eq=2MF9)F3$!3ZP\\%)4#RYu*F3FY7'7$$\"3ATsxza$e:\"F-$!3M,yc Jw;P5F-7$$\"3yeFA?X;W8F-$!3r()>K%oB$G'*F37$$\"3%*o%eji\\)\\8F-$!3s[W65 lMO)*F37$$\"3#e'*QuaIpD\"F-$!3[*o\"*)*)>=U(*F3Fho7'7$$\"3%[\"H!))>@RY \"F-$!3f.vB;be45F-7$$\"3m=/`M@Tp=F-$!36k\\iP[9/**F37$$\"3%G@\"*pm-Oz\" F-$!3:<2W2Cq/5F-7$$\"3w`uRw)Q'pMa1'yU%)*F3F]q7'7$$\"37C]A#)H7a >F-$!3O:m%zj^Jn*F37$$\"32V;W%oVD@#F-$!3YQ`?O[oK5F-7$$\"3`=\"Rf\"zt:@F- $!3uvX_/C*\\-\"F-7$$\"3okCX1+X(>#F-$!3lWPTp8275F-Fbr7'7$$\"3U\"z$eLK5< CF-$!3X-?N7)Qxh*F37$$\"3e3iTmn*Ge#F-$!3k*zk(=hAQ5F-7$$\"3M)>c^w=#*\\#F -$!3K6k6i&=e-\"F-7$$\"3W(>o?1%y%f#F-$!3P!zua))Gv,\"F-Fgs7'7$$\"3A]P)[H (pXGF-$!3fk%)eTXD3'*F37$$\"3#Ge\\%Qgj()HF-$!3V`6%eau\"R5F-7$$\"31F3s;s $z!HF-$!3u?-\\Q*fd-\"F-7$$\"3u5PK\"ete+$F-$!36H>J,Im=5F-F\\u7'7$$\"339 B?nc\")eKF-$!3\">'yW4!**3h*F37$$\"3*>Nk%**4&yS$F-$!3\"Q@b!*45*Q5F-7$$ \"3#)R_pzi%pK$F-$!3xs*Q_T(yD5F-7$$\"3cuKLF:ACMF-$!3yqei[cL=5F-Fav7'7$$ \"3OIxG)QUYl$F-$!3o[HGDCaH'*F37$$\"3woAr6wNXQF-$!380kB5F-7$$\"3OfFK1l$*)G%F-$!3B!)G/#3%G35F-F[y7'7$$\" 3w]t-3'4=Q%F-$!3M;zj=fc55F-7$$\"3K:$R'eq&[y%F-$!3iO3i83M%*)*F37$$\"31# eYT7-3r%F-$!3CgZgWa3/5F-7$$\"35\"z^vK(Q%o%F-$!3I?;u?2LR)*F3F`z7'7$$\"3 w]^GXo*z([F-$!3R\"ow8buP.\"F-7$$\"3C\\[raJ+A^F-$!31'=Li[aAm*F37$$\"3C \\KFGCR6^F-$!3=a(e5;/&p)*F37$$\"3?Y:$)\\g&p-&F-$!3#\\!RM15]Z(*F3Fe[l7' 7$$\"395)eh#fnK`F-$!3;4Tsx/8Q5F-7$$\"3yBX<2ul+bF-$!3U3*eFA&p='*F37$$\" 3vC%[oW^>^&F-$!3G8/&[S+e#)*F37$$\"3(>:QDDDmT&F-$!3dbDMk'4=u*F3Fj\\l7'7 $$\"3IU-:A%)=jdF-$!34\")p?)*QBR5F-7$$\"3bDk^W#yM!fF-$!3,)=Iz,hwg*F37$$ \"3nr(f(zp9AfF-$!3[Da?O7i7)*F37$$\"3R>BCMA1CeF-$!3^MB-DjZU(*F3F_^l7'7$ $\"3K!eR3SaB='F-$!3'fDB-14%R5F-7$$\"3Y@/;*fXwJ'F-$!3NSuw(R4fg*F37$$\"3 s>fx?bhPjF-$!3i.2\"QVy.\")*F37$$\"3#*zx@qG4RiF-$!3$GG]Y$GtU(*F3Fd_l7'7 $$\"3'H))o\\JtGf'F-$!3ic&[jFl*Q5F-7$$\"3v_WO=+YSnF-$!3zLW^OsM5'*F37$$ \"3#y!)*>bJ@dnF-$!3\">)*z#3&of\")*F37$$\"3#eTGV'**zfmF-$!3z'>#ec^u*F3F^bl7'7$$\"3O6KS)40KP(F-$!38#z'f,\\zEwF-$!3oGed p%*Qp'*F37$$\"3zGK(\\T>\"=$)ph95F-7$$\"33N'=! 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$$\"3;++DJ'\\;h\"F-$\"3S`*o9Vs&o\")F37$$\"3%om;zR'ok;F-$\"3c%G/wX.c3)F 37$$\"3OLL3_(>/x\"F-$\"390!\\:/!ebyF37$$\"33++D1J:w=F-$\"3'yp#fN]hVvF3 7$$\"3oLLL3En$4#F-$\"3Idh#3!*y)omF37$$\"3#pmmT!RE&G#F-$\"3U5\"yG?'HxcF 37$$\"3eLLe9r5$R#F-$\"3sWD=1\\$e/&F37$$\"3D+++D.&4]#F-$\"3[_nv)>CCP%F3 7$$\"3?+++]jB4EF-$\"3_i=:XT1jOF37$$\"3;+++vB_+]7.%Q%GKF-$!3iNdUG$)\\$z&F*7$$\"3_LLL347TLF-$!3r4n'>^*Q08F37$ $\"3#QLL3xxlV$F-$!3I?h%=s0?*=F37$$\"3nLLLLY.KNF-$!3wwqAq(*\\ZCF37$$\"3 3++D\"o7Tv$F-$!3#\\:wCCyff$F37$$\"3?LLL$Q*o]RF-$!3.7?*pe#Q;WF37$$\"3m+ +D\"=lj;%F-$!3Ir+>f%3n2&F37$$\"33++]iB0pUF-$!3Y_)>@TWxH&F37$$\"3S++vV& R6`U%F-$!3O&z,!zU!f^&F37$$\"3Kn;zWG)) yWF-$!3a5Bi.o%yb&F37$$\"3y+DJ&\\aC`%F-$!3As/AMM9$e&F37$$\"3CML$e9Ege%F -$!3?!R'=(RU>f&F37$$\"3hL3FW;ANYF-$!3w)p\\W\"ol&e&F37$$\"3'QL3F9Y2aF-$!3*>)Ggg7r1 RF37$$\"3Znm;zXu9cF-$!3C:sYq])>2$F37$$\"34+++]y))GeF-$!3\"*4Os*>1x6#F3 7$$\"3H++]i_QQgF-$!3&\\//>le$R6F37$$\"3b++D\"y%3TiF-$!3kk@Yc<3H>F*7$$ \"3+++]P![hY'F-$\"3$Rg')>gOz5)F*7$$\"3iKLL$Qx$omF-$\"3[W6a_xfK;F37$$\" 3Y+++v.I%)oF-$\"3cfVN3'f>R#F37$$\"3?mm\"zpe*zqF-$\"3#[j/dG_8&HF37$$\"3 ;,++D\\'QH(F-$\"3C)ePwT_US$F37$$\"3gm;zp%*\\%R(F-$\"3c:,Y\"G&[cNF37$$ \"3%HL$e9S8&\\(F-$\"3a/^i9G%)oOF37$$\"3%om;/6E.g(F-$\"3HcerLl]VPF37$$ \"3s++D1#=bq(F-$\"3%)*4iv-\\Zx$F37$$\"3#om\"H2FO3yF-$\"3)=wV-fiSw$F37$ $\"3\"HLL$3s?6zF-$\"371f%z)Rw8PF37$$\"3yl;zpe()=!)F-$\"3b&*Rc(*>M?OF37 $$\"3a***\\7`Wl7)F-$\"3cwsS&>'Q([$F37$$\"3enmmm*RRL)F-$\"3rUB:&yM(HJF3 7$$\"3%zmmTvJga)F-$\"3O-!4@+tbk#F37$$\"3]MLe9tOc()F-$\"3ebG(>_:U2#F37$ $\"31,++]Qk\\*)F-$\"3!f)p!e]DX\\\"F37$$\"3![LL3dg6<*F-$\"3ss0'3@QL(zF* 7$$\"3%ymmmw(Gp$*F-$\"32IM9.7N> " 0 "" {MPLTEXT 1 0 293 "de : = diff(y(x),x)+y(x)/8=exp(-x/8)*cos(x);\nic := y(0)=0;\nsoln := desolv eTA(\{de,ic\},x=0..10,stepsize=variable,output=points,order=9):\ncompa rewithfcn(soln,g(x),x);\nplot([g(x),soln],x=0..10,style=[line,point],c olor=[red,blue],\n symbol=circle,label s=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6 $-%\"yG6#%\"xGF-\"\"\"*&#F.\"\")F.F*F.F.*&-%$expG6#,$*&F1!\"\"F-F.F8F. -%$cosGF,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F) " }}{PARA 6 "" 1 "" {TEXT -1 85 " 0 0 fun ction val: 0 rel err: infinity" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.01 0.009987341352 function val: 0.009987 341351 rel err: 1.0013e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 0.06 0.05951595871 function val: 0.05951595871 rel err: \+ 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.2858989767 0.272119 3616 function val: 0.2721193615 rel err: 3.6749e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.5561438437 0.4924621038 function val: 0.4924621038 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.8605114479 0.6808566724 function val: 0.6808566723 \+ rel err: 1.4687e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.168472904 \+ 0.795110927 function val: 0.795110927 rel err: 0.0000e+0 0" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.476551896 0.8277727795 fu nction val: 0.8277727794 rel err: 1.2081e-10" }}{PARA 6 "" 1 " " {TEXT -1 87 " 1.774245151 0.7845690265 function val: 0.78 45690266 rel err: 1.2746e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2 .065906205 0.6796573343 function val: 0.6796573343 rel err : 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.352992053 0.5286 094531 function val: 0.5286094533 rel err: 3.7835e-10" }} {PARA 6 "" 1 "" {TEXT -1 87 " 2.635732977 0.3485472752 functio n val: 0.348547275 rel err: 5.7381e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.913469338 0.1571206846 function val: 0.1571 206849 rel err: 1.9094e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 3.1 84029954 -0.02849486206 function val: -0.02849486187 rel err: \+ 6.6679e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 3.44521765 -0.194363 2803 function val: -0.1943632803 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 3.718687453 -0.3427614282 function val: -0.3427614284 rel err: 5.8350e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 4.028880078 -0.4685859036 function val: -0.4685859036 \+ rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 4.36660194 \+ -0.5450703597 function val: -0.5450703597 rel err: 0.0000e+0 0" }}{PARA 6 "" 1 "" {TEXT -1 87 " 4.67341023 -0.5571414383 fu nction val: -0.5571414382 rel err: 1.7949e-10" }}{PARA 6 "" 1 " " {TEXT -1 87 " 4.969812376 -0.5195810374 function val: -0.51 95810373 rel err: 1.9246e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 5 .260572783 -0.4421908731 function val: -0.4421908731 rel err : 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 5.546868487 -0.3357 107457 function val: -0.3357107457 rel err: 0.0000e+00" }} {PARA 6 "" 1 "" {TEXT -1 87 " 5.82876906 -0.2118252748 functio n val: -0.2118252748 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.105418975 -0.0824358329 function val: -0.08243 583266 rel err: 2.9114e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.3 74162652 0.04095444389 function val: 0.04095444395 rel err: \+ 1.4650e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.637181741 0.151209 3955 function val: 0.1512093953 rel err: 1.3227e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.914247923 0.2486003813 function val: 0.2486003812 rel err: 4.0225e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 7.244427645 0.3315017616 function val: 0.3315017616 \+ rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 7.571924372 \+ 0.3727648091 function val: 0.3727648092 rel err: 2.6827e-1 0" }}{PARA 6 "" 1 "" {TEXT -1 87 " 7.875710344 0.3735513203 fu nction val: 0.3735513204 rel err: 2.6770e-10" }}{PARA 6 "" 1 " " {TEXT -1 87 " 8.170778811 0.3421896718 function val: 0.3 42189672 rel err: 5.8447e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 8 .460585081 0.2853359361 function val: 0.285335936 rel err : 3.5046e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 8.746010648 0.2104 029574 function val: 0.2104029575 rel err: 4.7528e-10" }} {PARA 6 "" 1 "" {TEXT -1 87 " 9.026951796 0.1253522762 functio n val: 0.1253522762 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 9.302307616 0.03819021691 function val: 0.03819 021698 rel err: 1.8329e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 9.5 68736961 -0.04337919706 function val: -0.04337919711 rel err: \+ 1.1526e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 9.83390897 -0.116366 1207 function val: -0.1163661208 rel err: 8.5936e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 10 -0.1558646579 function val: -0.1558646579 rel err: 0.0000e+00" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum relati ve error: 6.6679e-09" }}{PARA 6 "" 1 "" {TEXT -1 59 " obt ained for the input value: 3.184029954" }}{PARA 6 "" 1 "" {TEXT -1 67 " excluding any cases where the function value is z ero." }}{PARA 13 "" 1 "" {GLPLOT2D 498 305 305 {PLOTDATA 2 "6'-%'CURVE SG6%7[p7$$\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=$\"3Gic!G$R)H2\"F-7$$\"3emmm;a rz@F-$\"3kdn*)*>rV5#F-7$$\"3v***\\7y%*z7$F-$\"3I(4+N&pBfHF-7$$\"3[LL$e 9ui2%F-$\"3o$**\\3n'QnPF-7$$\"3z***\\(oMrU^F-$\"3%HqY_L$Q&F-7$$\"3Unmmm6m#G(F-$\"3%)*H_e)*Hm2'F-7$$\"3 9ommT&phN)F-$\"3kuNLn2Y\"o'F-7$$\"3KLLe*=)H\\5!#<$\"3!)*=?sT\"*[g(F-7$ $\"3-++v=JN[6FV$\"3D)>d*yjF,zF-7$$\"3smm\"z/3uC\"FV$\"3\\w8?Tcq7\")F-7 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" }}{PARA 0 "" 0 "" {TEXT -1 33 "(You could just omit the optio n \"" }{TEXT 270 13 "output=points" }{TEXT -1 4 "\". )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "de := \+ diff(y(x),x)+y(x)/8=exp(-x/8)*cos(x);\nic := y(0)=0;\ngn := desolveTA( \{de,ic\},x=0..10,order=9,output=localtaylor,info=2);\nplot('gn(x)',x= 0..10,color=brown,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*&#F.\"\")F.F*F.F.*&-%$expG6#,$*& F1!\"\"F-F.F8F.-%$cosGF,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~- >~G$\"&&*G\"!#G%2abs~err~bound~->~G$\"&++\"!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"\"%$~~~G$\"+++++5!#6F%7$F&$\"/Xh^8M()**!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Fincreasing~step-size~by~a~factor~o f~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&NY#!#A%2abs~err~bound~->~G$ \"&t)\\!#;" }}{PARA 11 "" 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"dy/dx+``(2/x)*y^2 = sin(x)/x;" "6#/ ,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%!G6#*&\"\"#F'%\"xGF)F'*$%\"yGF/F'F'*&-% $sinG6#F0F'F0F)" }{TEXT -1 34 " subject to the initial condition " } {XPPEDIT 18 0 "y(1) = 0;" "6#/-%\"yG6#\"\"\"\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 47 "This differential equation is not linear \+ - the " }{TEXT 291 1 "y" }{TEXT -1 47 " in the previous example has be en replaced by " }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple' s " }{TEXT 0 6 "dsolve" }{TEXT -1 68 " cannot find an analytical solut ion for this differential equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "de := diff(y(x),x)+2/x*y(x )^2=sin(x)/x;\nic := y(1)=0;\ndsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(\"\"#F.F -!\"\"F*F0F.*&-%$sinGF,F.F-F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ic G/-%\"yG6#\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "desolveTA" }{TEXT -1 74 " can be used to obtain a (discrete) solution by the Taylor series met hod. " }}{PARA 0 "" 0 "" {TEXT -1 52 "We construct a numerical solutio n over the interval " }{XPPEDIT 18 0 "1 <= x;" "6#1\"\"\"%\"xG" } {XPPEDIT 18 0 "`` <= 16;" "6#1%!G\"#;" }{TEXT -1 63 " using the order \+ 8 Taylor series method with 75 steps of width " }{XPPEDIT 18 0 "h = 1/ 5;" "6#/%\"hG*&\"\"\"F&\"\"&!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "de := dif f(y(x),x)+2/x*y(x)^2=sin(x)/x;\nic := y(1)=0;\nsoln := desolveTA(\{de, ic\},y(x),x=1..16,steps=75,order=8,output=points):" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(\"\"#F.F-! \"\"F*F0F.*&-%$sinGF,F.F-F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/ -%\"yG6#\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 119 "The following picture shows the points of the discrete solution along with a \"curve\" obtained by joining these points. 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"6#%Freducing~step-size~and~repeating~stepG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&6o'!#;%2abs~err~b ound~->~G$\"&6'>!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#O%$~~ ~G$\"/G+*e+*fO!#9F%7$$\"/v\"3ApaN*!#8$\"/iLs]`G?F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~esti mate~->~G$\"&!Hj!#;%2abs~err~bound~->~G$\"&&G?!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#P%$~~~G$\"/#*>MMToO!#9F%7$$\"/uBk0JA(*!#8$\"/ Yp!HnX&>F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~ step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&Dy$!#;%2abs~err~bound~->~G$ \"&Y&>!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#Q%$~~~G$\"/)y!G QT4P!#9F%7$$\"/Xq%>D$45!#7$\"/S9fy'fv\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&-L\"!#;%2abs~err~bound~->~G$\"&gv\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#R%$~~~G$\"/3mB3PMR!#9F%7$$\"/6%H!*o'[5!#7$\"/ Dp&R,sW\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust ~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&s(**!#;%2abs~err~bound~->~ G$\"&sW\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#S%$~~~G$\"/$ *3*G\"G$e%!#9F%7$$\"/?$er,X4\"!#7$\"/*)>c(Ql-\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&$p\"*!#;%2abs~err~bound~->~G$\"&l-\"!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6(%%stepG\"#T%$~~~G$\"/YB]dD\"G%!#9F%7$$\"/VLtUJP6!#7$ \"/M5C!*H`k!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adj ust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&:d$!#;%2abs~err~bound~- >~G$\"&LX'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#U%$~~~G$\"/m #f*H)o*Q!#9F%7$$\"/OH.JGw6!#7$\"/!o)Q24?O!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&XE\"!#;%2abs~err~bound~->~G$\"&,i$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#V%$~~~G$\"/R9a=%4s$!#9F%7$$\"/]$=_#\\87!#7$\" /A.k*e.#=!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjus t~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&&*p&!#<%2abs~err~bound~->~ G$\"&/#=!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#W%$~~~G$\"/># e7l-s$!#9F%7$$\"/K4t^p]7!#7$\"/\"y$yz)p3\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Freducing~step- size~and~repeating~stepG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~ estimate~->~G$\"&YR$!#<%2abs~err~bound~->~G$\"&q3\"!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#X%$~~~G$\"/:=0(z7.$!#9F%7$$\"/]9qz+\"G\" !#7$\"/Ud0/W/8!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~ adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&qE&!#<%2abs~err~bound ~->~G$\"&WI\"!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#Y%$~~~G$ \"/&Q=Et[1$!#9F%7$$\"/Mw-nl68!#7$\"/B6/(RA@#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&XK%!#<%2abs~err~bound~->~G$\"&A@#!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#Z%$~~~G$\"/e#3\"pB?I!#9F%7$$\"/<(=2f=M\"!#7$ \"/l*zVI/n$!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adj ust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&F^%!#<%2abs~err~bound~- >~G$\"&/n$!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#[%$~~~G$\"/ 2o@C<+K!#9F%7$$\"/&)3'zgQP\"!#7$\"/*4bnb&\\c!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&yk\"!#<%2abs~err~bound~->~G$\"&'\\c!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#\\%$~~~G$\"/hK'Qk~G$\"&Ud#!#;%2abs~err~b ound~->~G$\"&@5)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#]%$~~~ G$\"/=QAN,LP!#9F%7$$\"/c~G$\"&Tv$!#;%2abs~err~bound~->~G$\"&H1\"!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6(%%stepG\"#^%$~~~G$\"/-82[)yw$!#9F%7$$\"/pCBuQ%[\"!#7$ \"/lX`#pyF\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adj ust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&uu#!#;%2abs~err~bound~- >~G$\"&zF\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#_%$~~~G$\" 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function val: \+ 0 abs err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.2 0.1589261239 function val: 0.1589260343 ab s err: 8.9600e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.4 0 .2909407625 function val: 0.290940696 abs err: 6.6500e-08" } }{PARA 6 "" 1 "" {TEXT -1 87 " 1.6 0.3922492574 functi on val: 0.3922492187 abs err: 3.8700e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.8 0.4651844486 function val: 0.4651 844214 abs err: 2.7200e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 2 0.5139441211 function val: 0.5139440989 abs err: \+ 2.2200e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.2 0.542679 3434 function val: 0.5426793246 abs err: 1.8800e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.4 0.5549051841 function val: 0.5549051681 abs err: 1.6000e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.6 0.553462916 function val: 0.5534629024 \+ abs err: 1.3600e-08" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2.8 \+ 0.5406531784 function val: 0.5406531668 abs err: 1.1600e-0 8" }}{PARA 6 "" 1 "" {TEXT -1 87 " 3 0.5183897655 fu nction val: 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0.08956728024 abs err: 5.1100e-0 9" }}{PARA 6 "" 1 "" {TEXT -1 87 " 5.2 0.05290327054 fu nction val: 0.05290326549 abs err: 5.0500e-09" }}{PARA 6 "" 1 " " {TEXT -1 87 " 5.4 0.02142393969 function val: 0.021 42393466 abs err: 5.0300e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 5.6 -0.004216609147 function val: -0.004216614165 abs err : 5.0180e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 5.8 -0.02354 248301 function val: -0.02354248804 abs err: 5.0300e-09" }} {PARA 6 "" 1 "" {TEXT -1 87 " 6 -0.03627781732 functio n val: -0.03627782237 abs err: 5.0500e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.2 -0.0423629166 function val: -0.04236 292168 abs err: 5.0800e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 6.4 -0.04196249872 function val: -0.04196250382 abs err: \+ 5.1000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.6 -0.0354641 9262 function val: -0.03546419775 abs err: 5.1300e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 6.8 -0.02346610525 function val: -0.02346611039 abs err: 5.1400e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 7 -0.006753199021 function val: -0.006753204179 \+ abs err: 5.1580e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 7.2 \+ 0.01373670078 function val: 0.01373669563 abs err: 5.1500e-0 9" }}{PARA 6 "" 1 "" {TEXT -1 87 " 7.4 0.03695537863 fu nction val: 0.03695537349 abs err: 5.1400e-09" }}{PARA 6 "" 1 " " {TEXT -1 87 " 7.6 0.06179240093 function val: 0.061 79239582 abs err: 5.1100e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 7.8 0.087125649 function val: 0.08712564393 abs err : 5.0700e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 8 0.1118 707341 function val: 0.1118707291 abs err: 5.0000e-09" }} {PARA 6 "" 1 "" {TEXT -1 87 " 8.2 0.1350260987 functio n val: 0.1350260937 abs err: 5.0000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 8.4 0.1557112373 function val: 0.1557 112324 abs err: 4.9000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 8.6 0.1731963153 function val: 0.1731963105 abs err: \+ 4.8000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 8.8 0.186922 3666 function val: 0.1869223618 abs err: 4.8000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 9 0.1965120856 function val: 0.196512081 abs err: 4.6000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 9.2 0.2017718852 function val: 0.2017718806 \+ abs err: 4.6000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 9.4 \+ 0.2026863344 function val: 0.2026863299 abs err: 4.5000e-0 9" }}{PARA 6 "" 1 "" {TEXT -1 87 " 9.6 0.1994063166 fu nction val: 0.1994063122 abs err: 4.4000e-09" }}{PARA 6 "" 1 " " {TEXT -1 87 " 9.8 0.1922322898 function val: 0.19 22322854 abs err: 4.4000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 10 0.1815939417 function val: 0.1815939374 abs err : 4.3000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 10.2 0.1680 273716 function val: 0.1680273674 abs err: 4.2000e-09" }} {PARA 6 "" 1 "" {TEXT -1 87 " 10.4 0.1521507355 functio n val: 0.1521507313 abs err: 4.2000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 10.6 0.1346391149 function val: 0.1346 391108 abs err: 4.1000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 10.8 0.1161992193 function val: 0.1161992152 abs err: \+ 4.1000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 11 0.0975444 3265 function val: 0.09754442859 abs err: 4.0600e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 11.2 0.07937065812 function val: 0.07937065409 abs err: 4.0300e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 11.4 0.06233340347 function val: 0.06233339946 \+ abs err: 4.0100e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 11.6 \+ 0.04702656493 function val: 0.04702656093 abs err: 4.0000e-0 9" }}{PARA 6 "" 1 "" {TEXT -1 87 " 11.8 0.03396340143 fu nction val: 0.03396339745 abs err: 3.9800e-09" }}{PARA 6 "" 1 " " {TEXT -1 87 " 12 0.02356022344 function val: 0.023 56021946 abs err: 3.9800e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 12.2 0.01612333624 function val: 0.01612333227 abs err : 3.9700e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 12.4 0.0118 397615 function val: 0.01183975753 abs err: 3.9700e-09" }} {PARA 6 "" 1 "" {TEXT -1 87 " 12.6 0.01077219922 functio n val: 0.01077219526 abs err: 3.9600e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 12.8 0.01285857943 function val: 0.01285 857547 abs err: 3.9600e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 13 0.01791638667 function val: 0.01791638271 abs err: \+ 3.9600e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 13.2 0.025651 7286 function val: 0.02565172465 abs err: 3.9500e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 13.4 0.03567287706 function val: 0.03567287311 abs err: 3.9500e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 13.6 0.04750775675 function val: 0.04750775281 \+ abs err: 3.9400e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 13.8 \+ 0.06062461913 function val: 0.0606246152 abs err: 3.9300e-0 9" }}{PARA 6 "" 1 "" {TEXT -1 87 " 14 0.07445494149 fu nction val: 0.07445493758 abs err: 3.9100e-09" }}{PARA 6 "" 1 " " {TEXT -1 87 " 14.2 0.08841745615 function val: 0.088 41745226 abs err: 3.8900e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 14.4 0.1019421567 function val: 0.1019421528 abs err : 3.9000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 14.6 0.1144 931523 function val: 0.1144931485 abs err: 3.8000e-09" }} {PARA 6 "" 1 "" {TEXT -1 87 " 14.8 0.1255893438 functio n val: 0.12558934 abs err: 3.8000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 15 0.1348220596 function val: 0.1348 220558 abs err: 3.8000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+ 15.2 0.1418690032 function val: 0.1418689995 abs err: \+ 3.7000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 15.4 0.146504 0933 function val: 0.1465040895 abs err: 3.8000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 15.6 0.1486030121 function val: 0.1486030084 abs err: 3.7000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 15.8 0.1481444916 function val: 0.1481444879 \+ abs err: 3.7000e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 16 \+ 0.1452075489 function val: 0.1452075453 abs err: 3.6000e-0 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum absolute error: 8.9600e-08" }}{PARA 6 "" 1 " " {TEXT -1 59 " obtained for the input value: \+ 1.2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 45 " Mean absolute error: 8.1371e-09" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }{TEXT 290 65 " .. a ro ot-finding problem associated with a continuous solution " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "C onsider again the differential equation " }{XPPEDIT 18 0 "dy/dx+``(2/ x)*y^2 = sin(x)/x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%!G6#*&\"\"#F'% \"xGF)F'*$%\"yGF/F'F'*&-%$sinG6#F0F'F0F)" }{TEXT -1 34 " subject to th e initial condition " }{XPPEDIT 18 0 "y(1) = 0;" "6#/-%\"yG6#\"\"\"\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 14 "The procedure " }{TEXT 0 9 "desolveTA" }{TEXT -1 73 " c an be used to obtain a continuous numerical solution over the interval " }{XPPEDIT 18 0 "1 <= x;" "6#1\"\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 100 ;" "6#1%!G\"$+\"" }{TEXT -1 30 " by the Taylor series method. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "de := diff(y(x),x)+2/x*y(x)^2=sin(x)/x;\nic := y(1)=0;\ngn := des olveTA(\{de,ic\},y(x),x=1..100);\nplot('gn(x)',x=1..100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(\"\"# F.F-!\"\"F*F0F.*&-%$sinGF,F.F-F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #icG/-%\"yG6#\"\"\"\"\"!" }}{PARA 13 "" 1 "" {GLPLOT2D 712 402 402 {PLOTDATA 2 "6%-%'CURVESG6$7]\\l7$$\"\"\"\"\"!$F*F*7$$\"+Y\\Vn5!\"*$\" +d%z7f&!#67$$\"+#*)p[8\"F/$\"+h*>l4\"!#57$$\"+Q[I-7F/$\"+lJ<1;F87$$\"+ $yR(p7F/$\"+J;0%3#F87$$\"+GZOHF 87$$\"+?Y/s9F/$\"+VLE4LF87$$\"+m&z%R:F/$\"+[\"pvk$F87$$\"+7X\"pg\"F/$ \"+96A_RF87$$\"+d%\\Vn\"F/$\"+*yUYA%F87$$\"+-WyT#4=F/$\"+F.IzYF87$$\"+S#*3W>F/$\"+b/)\\-&F87$$\"+J\"f*y?F/$\"+Kopu_F 87$$\"+A!HQ@#F/$\"+a*)>SaF87$$\"+9*)p[BF/$\"+]IvJbF87$$\"+gQ8;CF/$\"+$ z=Eb&F87$$\"+0)oN[#F/$\"+*R@#ebF87$$\"+]P+^DF/$\"+h?[\\bF87$$\"+'pQ%=E F/$\"+D4EFbF87$$\"+z%y\"))GF/$\"+@))**=`F87$$\"+i#=z:$F/$\"+.qC_\\F87$ $\"+0uh#R$F/$\"+O1\"R`%F87$$\"+[lJFOF/$\"+8d+ZSF87$$\"+>hmWPF/$\"+wH`% y$F87$$\"+!p:?'QF/$\"+/J%G^$F87$$\"+i_OzRF/$\"+zF[MKF87$$\"+L[r'4%F/$ \"+T<'>&HF87$$\"+/W19UF/$\"+#G^xm#F87$$\"+wRTJVF/$\"+e&zUQ#F87$$\"+ZNw [WF/$\"+tS#R5#F87$$\"+=J6mXF/$\"+'o,!H=F87$$\"+!piMo%F/$\"+okvh:F87$$ \"+hA\"3![F/$\"+$pXVI\"F87$$\"+K=;=\\F/$\"+>U#)e5F87$$\"+/9^N]F/$\"+mh Fr#)F27$$\"+N^X*H&F/$\"+E'fcl$F27$$\"+q))RjbF/$\"+O,&[^\"!#97$$\"++EMF eF/$!+sJhnDF27$$\"+NjG\"4'F/$!+?^b()RF27$$\"+qAFdhF/$!+z1nhTF27$$\"++# eKA'F/$!+bvTkUF27$$\"+NTC*G'F/$!+*3anH%F27$$\"+q+BbjF/$!+fO!*fUF27$$\" +N>?(['F/$!+Zh$[)RF27$$\"++Q<>mF/$!+C(>TX$F27$$\"+Iv6$)oF/$!+.(GUq\"F2 7$$\"+k71ZrF/$\"+4K3.!)!#77$$\"+&eaFT(F/$\"+I;H]QF27$$\"+5zWywF/$\"+F87$$\"+'*H .L5!\")$\"+tG-z:F87$$\"+-\">f3\"Fa]l$\"+>wo16F87$$\"+3_!)Q6Fa]l$\"+;Q \"3L'F27$$\"+)RPy=\"Fa]l$\"++i-bHF27$$\"+)epoB\"Fa]l$\"+Oc!)H7F27$$\"+ Ow7\\7Fa]l$\"+qjW&4\"F27$$\"+$o&Qh7Fa]l$\"+3/l\"3\"F27$$\"+IPkt7Fa]l$ \"+*eNi=\"F27$$\"+y`S\"F27$$\"+syT58Fa]l$\"+$F27$$\"+c+!e'=Fa]l$\"+kKe=IF27$$\"+fX))y=Fa]l$\"+83XHHF27$$\"+i!p>*= Fa]l$\"+k.[JHF27$$\"+kN00>Fa]l$\"+spGBIF27$$\"+qDAJ>Fa]l$\"+>@EjMF27$$ \"+v:Rd>Fa]l$\"+xYi4UF27$$\"+GP1l?Fa]l$\"+M,t9!*F27$$\"+#)ets@Fa]l$\"+ B#Fa]l$\"+t8*3D\"F87$$\"+GZ:?AFa]l$\"+CJvP7F87$$\"+^T 'QC#Fa]l$\"+26P+7F87$$\"+uNdnAFa]l$\"++8ET6F87$$\"+?C*\\J#Fa]l$\"+#zaN t*F27$$\"+l7TiBFa]l$\"+GdmCxF27$$\"+aOz:CFa]l$\"+N$>Xh&F27$$\"+Wg%F27$$\"+T;_#[#Fa]l$\"+hk]**RF27$$\"+Qs'e\\#Fa]l$\"+bu0qQF 27$$\"+OG@4DFa]l$\"+zmO6QF27$$\"+L%eD_#Fa]l$\"+7^jBQF27$$\"+G'\\#\\DFa ]l$\"+zdibSF27$$\"+A3%fd#Fa]l$\"+hXFVXF27$$\"+')R9$o#Fa]l$\"+T\\^r!)F2 7$$\"+^rM!z#Fa]l$\"+jc\\%4\"F87$$\"+!ouh\"GFa]l$\"+A)Q*)F27$$\"+yt'p*HFa]l$\"+\"4!>7sF27$$\"+=^(Q /$Fa]l$\"+uh7edF27$$\"+eGy!4$Fa]l$\"+@,YCZF27$$\"+$z4D5$Fa]l$\"+3YPeXF 27$$\"+GnB9JFa]l$\"+hX(QV%F27$$\"+jO'f7$Fa]l$\"+4$*H_VF27$$\"+)f!pPJFa ]l$\"+QVR9VF27$$\"+LvT\\JFa]l$\"+WZI?VF27$$\"+oW9hJFa]l$\"+modpVF27$$ \"+/9(G<$Fa]l$\"+y!o6Y%F27$$\"+R$)f%=$Fa]l$\"+FXY$f%F27$$\"+>S:'H$Fa]l $\"+%f&z\"H(F27$$\"+*p4xS$Fa]l$\"+i)*3m**F27$$\"+'[A&>MFa]l$\"++\"e+, \"F87$$\"+s_LJMFa]l$\"+*)*R'>5F87$$\"+f![JW$Fa]l$\"+;OCD5F87$$\"+Y3'\\ X$Fa]l$\"+yR\"o-\"F87$$\"+LOxmMFa]l$\"+=CNC5F87$$\"+?keyMFa]l$\"+fh\"z ,\"F87$$\"+1#*R!\\$Fa]l$\"+lqh25F87$$\"+$*>@-NFa]l$\"+\"p'>O**F27$$\"+ SJY\\NFa]l$\"+-*Gh0*F27$$\"+(G9nf$Fa]l$\"+1rOHyF27$$\"+/'y;l$Fa]l$\"+d **3EjF27$$\"+?Hk1PFa]l$\"+ycH`^F27$$\"+**RQ?PFa]l$\"+#e]P&\\F27$$\"+y] 7MPFa]l$\"+07d*z%F27$$\"+eh'yu$Fa]l$\"+6.D$p%F27$$\"+PsghPFa]l$\"+N!oj j%F27$$\"+;$[`x$Fa]l$\"+MzcHYF27$$\"+'R*3*y$Fa]l$\"+=2bsYF27$$\"+v/$G! 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It would be rather difficult either computationally or analytically to determine whether there are any more zeros. In any case, the two z eros of the solution corresponding to the " }{TEXT 293 1 "x" }{TEXT -1 54 " intercepts of the solution (greater than the initial " }{TEXT 294 1 "x" }{TEXT -1 7 " value " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 56 ") visible in this picture can be calculated numerically. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot('gn(x)',x=1..8);" }}{PARA 13 "" 1 "" {GLPLOT2D 345 318 318 {PLOTDATA 2 "6%-%'CURVESG6$7in7$$\"\"\"\"\"!$F*F*7$$\"+?]9Q5!\"*$ \"+fP9&=$!#67$$\"+S+Hw5F/$\"+y!\\3J'F27$$\"+g]V96F/$\"+)o)oj$*F27$$\"+ z+e_6F/$\"+,4DL7!#57$$\"+M'f*=7F/$\"+!4qss\"FB7$$\"+!>R`G\"F/$\"+w))y* =#FB7$$\"+U**)*f8F/$\"+&Q:*pEFB7$$\"+&pSYV\"F/$\"+_?o1JFB7$$\"+#G'y4:F /$\"+Zg#G]$FB7$$\"+o=$\\e\"F/$\"+$[Cl&QFB7$$\"++.sf;F/$\"+=F?oTFB7$$\" +K(3Xt\"F/$\"+fZxTWFB7$$\"+#=ZQ!=F/$\"+dOOjYFB7$$\"+Lc=t=F/$\"+R^,c[FB 7$$\"+Kvx;?F/$\"+y^qq^FB7$$\"+y/Gl@F/$\"+bnk*Q&FB7$$\"+EQHRAF/$\"+a5'G Y&FB7$$\"+urI8BF/$\"+LXO9bFB7$$\"++!R%*Q#F/$\"+QuBYbFB7$$\"+D3dlCF/$\" +*3m\"ebFB7$$\"+zxiKDF/$\"+))\\B`bFB7$$\"+LZo*f#F/$\"+tmuMbFB7$$\"+!) \\\"*> FB7$$\"+yo(3l%F/$\"+h'p]j\"FB7$$\"+ULA&y%F/$\"+Z$**yL\"FB7$$\"+07KI\\F /$\"+gE9M5FB7$$\"+%\\@-3&F/$\"+AzoGuF27$$\"+$opoA&F/$\"+3:lN[F27$$\"+Y $f(o`F/$\"+sq4(f#F27$$\"+DOIEbF/$\"+([sy^%!#77$$\"+nT'ym&F/$!+U^-]6F27 $$\"+h-,>eF/$!+WQ!R]#F27$$\"+(3rf&fF/$!+0Tk/MF27$$\"+Yaq0hF/$!+0atJSF2 7$$\"+4QfYiF/$!+GH%QG%F27$$\"+VF'QR'F/$!+S4!o?%F27$$\"+X]%y`'F/$!+cdY5 QF27$$\"+r6e)o'F/$!+#[3'yIF27$$\"+vzvLoF/$!+Kb>&4#F27$$\"+FAA#)pF/$!+O TV1%)F`y7$$\"+>rXHrF/$\"+w?MUhF`y7$$\"+%p]ZE(F/$\"+=ez+@F27$$\"+)R7)>u F/$\"+1!)>ORF27$$\"+N9]evF/$\"+v-7ccF27$$\"+vGP1xF/$\"+%e%[FvF27$$\"+7 Z!z%yF/$\"+]8c9$*F27$$\"\")F*$\"+\"H2(=6FB-%'COLOURG6&%$RGBG$\"#5!\"\" F+F+-%+AXESLABELSG6$Q\"x6\"Q!Fa_l-%%VIEWG6$;F(Fb^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The two zeros of \+ the solution under consideration can be calculated numerically using \+ " }{TEXT 0 6 "fsolve" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fsolve('gn(x)',x=5..6); \nfsolve('gn(x)',x=7..7.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Cn_ jb!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Tkfpq!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Alternatively, the special procedures " }{TEXT 0 6 "bisect" }{TEXT -1 4 " or " }{TEXT 0 6 "secant" }{TEXT -1 14 " may be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "bisect('gn(x)',x=5..6); \nsecant('gn(x)',x=7..7.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Cn_ jb!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Tkfpq!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 120 "Use the Taylor series method with a fixed step-size to find a discrete numerical solution for the \+ differential equation " }{XPPEDIT 18 0 "dy/dx = y*cos(x);" "6#/*&%#dyG \"\"\"%#dxG!\"\"*&%\"yGF&-%$cosG6#%\"xGF&" }{TEXT -1 19 " over the int erval " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x=10" "6#/%\"xG\"#5" }{TEXT -1 28 " with the initial co ndition " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 104 "Choose the number of steps and t he order of the method so as to give a maximum relative error less tha n " }{XPPEDIT 18 0 "10^(-8);" "6#)\"#5,$\"\")!\"\"" }{TEXT -1 7 "at th e " }{TEXT 278 1 "x" }{TEXT -1 34 " values of the discrete solution. \+ " }}{PARA 0 "" 0 "" {TEXT -1 60 "Compare the numerical solution with t he analytical solution " }{XPPEDIT 18 0 "y = exp(sin(x));" "6#/%\"yG-% $expG6#-%$sinG6#%\"xG" }{TEXT -1 33 " both in a table and graphically. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Use t he procedures " }{TEXT 0 9 "desolveTA" }{TEXT -1 4 " . ." }{HYPERLNK 17 "desolveTA" 1 "" "desolveTA" }{TEXT -1 6 " and " }{TEXT 0 14 "comp arewithfcn" }{TEXT -1 5 " . . " }{HYPERLNK 17 "comparewithfcn" 1 "" "c omparewithfcn" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "________ __________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 63 "(a) Find an analytic al solution for the differential equation " }{XPPEDIT 18 0 "dy/dx = y* (1-x^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"yGF&,&F&F&*$%\"xG\"\"#F(F&" }{TEXT -1 29 " with the initial condition " }{XPPEDIT 18 0 "y(0)=1" " 6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "(b) Find a discrete numerical solu tion to the initial value problem given in (a) over the interval from \+ " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 54 " using an order 8 Taylor seri es method with 30 steps. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 115 "(c) Plot the points of the discrete solution and \+ the graph of the analytical solution together in the same picture." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "(d) What is the maximum relative error in the calculated values of the numeric al solution found in (b), and where does this maximum relative error o ccur?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 " (e) What is the absolute error in the numerical solution when " } {XPPEDIT 18 0 "x = 3" "6#/%\"xG\"\"$" }{TEXT -1 1 "?" }}{PARA 0 "" 0 " " {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 63 "(a) Find an analytical solution for the differential \+ equation " }{XPPEDIT 18 0 "dy/dx = y/(x^2+1);" "6#/*&%#dyG\"\"\"%#dxG! \"\"*&%\"yGF&,&*$%\"xG\"\"#F&F&F&F(" }{TEXT -1 29 " with the initial \+ condition " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "(b) Find a discrete numerical solution to the initial value prob lem given in (a) over the interval from " }{XPPEDIT 18 0 "x=0" "6#/%\" xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" } {TEXT -1 189 " using a Taylor series method with fixed step-size. Choo se the step-size and order of the method so that the maximum relative \+ error in the calculated values has a magnitude no greater than " } {XPPEDIT 18 0 "10^(-8);" "6#)\"#5,$\"\")!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 115 "(a) Use the Taylor ser ies method to obtain an order 10 continuous numerical solution for the differential equation " }{XPPEDIT 18 0 "dy/dx = y*cos(x);" "6#/*&%#dy G\"\"\"%#dxG!\"\"*&%\"yGF&-%$cosG6#%\"xGF&" }{TEXT -1 19 " over the in terval " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x=10" "6#/%\"xG\"#5" }{TEXT -1 28 " with the initial co ndition " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 122 ". The solution should be in the form of a procedure which can be \+ evaluated at any point of the interval. Use the procedure" }{TEXT 0 10 " desolveTA" }{TEXT -1 5 " . ." }{HYPERLNK 17 "desolveTA" 1 "" "de solveTA" }{TEXT -1 54 " with adaptive step-size control by using the o ption \"" }{TEXT 270 13 "adaptive=true" }{TEXT -1 8 "\" (or \"" } {TEXT 270 17 "stepsize=variable" }{TEXT -1 3 "\")." }}{PARA 0 "" 0 "" {TEXT -1 105 "Plot the graph of the numerical solution and compare the numerical solution with the analytical solution " }{XPPEDIT 18 0 "y = exp(sin(x));" "6#/%\"yG-%$expG6#-%$sinG6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "x = 3*exp(1);" "6#/%\"xG*&\"\"$\"\"\"-%$expG6#F'F'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "(b) By using the procedure " }{TEXT 0 9 "desolveTA" } {TEXT -1 18 " with the option \"" }{TEXT 270 14 "stepsize=fixed" } {TEXT -1 7 "\" (or \"" }{TEXT 270 14 "adaptive=false" }{TEXT -1 193 " \"), construct an order 10 continuous solution based on equally spaced local Taylor series which has similar error characteristics to those \+ of the variable step-size continuous solution in (a). " }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }} {PARA 0 "" 0 "" {TEXT -1 59 "This question is concerned with the initi al value problem: " }{XPPEDIT 18 0 "dy/dx = x/(1+sin(x+y)^2);" "6#/*&% #dyG\"\"\"%#dxG!\"\"*&%\"xGF&,&F&F&*$-%$sinG6#,&F*F&%\"yGF&\"\"#F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = -1" "6#/-%\"yG6#\"\"!,$\"\"\" !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 "(a) Use a Taylor series method with adaptive step-s ize control to obtain a continuous numerical solution of this equation in the form of a Maple procedure which can be evaluated anywhere in t he interval from " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " \+ to " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "(b) Plot a graph \+ of the solution along with the gradient field associated with the diff erential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "(c) Find the value " }{XPPEDIT 18 0 "x = r" "6#/%\"xG%\"r G" }{TEXT -1 22 " in the interval from " }{XPPEDIT 18 0 "x=0" "6#/%\"x G\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" } {TEXT -1 37 " such that the solution is zero when " }{XPPEDIT 18 0 "x \+ = r" "6#/%\"xG%\"rG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "Yo ur value of " }{TEXT 283 1 "r" }{TEXT -1 39 " should be accurate to a bout 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "(d) Find the value of the derivative of the solution at t he point given by " }{XPPEDIT 18 0 "x = r" "6#/%\"xG%\"rG" }{TEXT -1 8 ", where " }{TEXT 284 1 "r" }{TEXT -1 39 " is zero of the solution f ound in (c). " }}{PARA 0 "" 0 "" {TEXT -1 34 "________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 72 "(a) Find a continuo us numerical solution for the differential equation " }{XPPEDIT 18 0 " dy/dx = (x-y^2)/(x+y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&%\"xGF&*$%\"yG \"\"#F(F&,&F+F&F-F&F(" }{TEXT -1 29 " with the initial condition " } {XPPEDIT 18 0 "y(0)=2" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 96 ", in the \+ from of a Maple procedure which can be evaluated anywhere throughout t he interval from " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " \+ to " }{XPPEDIT 18 0 "x=8" "6#/%\"xG\"\")" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "(b) Plot the graph of the solution found in (a)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "(c) Find the value " }{XPPEDIT 18 0 "x=al pha" "6#/%\"xG%&alphaG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0,8]" "6#7$\"\"!\"\")" }{TEXT -1 12 ", such that " }{XPPEDIT 18 0 "alpha <> 0;" "6#0%&alphaG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y (alpha)=2" "6#/-%\"yG6#%&alphaG\"\"#" }{TEXT -1 71 ", that is, such th at the solution has the same value which it has when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "_ _________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK " 4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }