{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Blue Emp hasis" -1 262 "Times" 1 12 0 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 271 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 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 }{PSTYLE "Hea ding 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 " Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{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 "Times" 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 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 57 "The shooting method for two point boundary value problems" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Ston e, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24. 3.2007" }}{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 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 271 7 "DEsol.m" }{TEXT -1 32 " is required by this worksheet. " }}{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/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 36 "The two point boundary value problem" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "For a first order differential equation of the form " } {XPPEDIT 18 0 "dy/dx = f(x,y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$%\" xG%\"yG" }{TEXT -1 63 " there is a unique solution curve passing throu gh a given point" }{XPPEDIT 18 0 "``(x[0],y[0]);" "6#-%!G6$&%\"xG6#\" \"!&%\"yG6#F)" }{TEXT -1 17 " with gradient f(" }{XPPEDIT 18 0 "x[0],y [0];" "6$&%\"xG6#\"\"!&%\"yG6#F&" }{TEXT -1 16 ") at this point." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "In this w orksheet we consider the solution of a 2nd order DE of the form" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = F(x, y,dy/dx);" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"-%\"FG6%F+ F'*&%#dyGF(%#dxGF," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 51 "A typical example is a linear differenti al equation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y /(d*x^2)+f(x);" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-% \"fG6#F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+g(x)*y = k(x);" "6#/ ,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"gG6#%\"xGF'%\"yGF'F'-%\"kG6#F." } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 170 "Since the general solut ion of such a differential equation involves two arbitrary constants, \+ we expect to find more than one solution curve passing through a given point." }}{PARA 0 "" 0 "" {TEXT -1 49 "For example, the general solut ion of the equation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\" \"F'" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "is " }{XPPEDIT 18 0 "y(x) = A*exp(x)+B*exp(-x);" "6#/-%\"yG6#%\"xG,&*&%\"AG\"\"\"-%$e xpG6#F'F+F+*&%\"BGF+-F-6#,$F'!\"\"F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Any solution for which the constants " }{TEXT 265 1 "A" }{TEXT -1 5 " and " }{TEXT 264 1 "B" }{TEXT -1 9 " satisfy " } {XPPEDIT 18 0 "A + B = 1" "6#/,&%\"AG\"\"\"%\"BGF&F&" }{TEXT -1 25 " p asses through the point" }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\"\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 60 "For example, the foll owing expressions all define solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "solns := [seq(i/3*exp(x )+(1-i/3)*exp(-x),i=-6..9)]:\nfor i from 1 to 16 do print(solns[i]) en d do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"-%$expG6#%\"xG F&!\"\"*&\"\"$F&-F(6#,$F*F+F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&* &#\"\"&\"\"$\"\"\"-%$expG6#%\"xGF(!\"\"*&#\"\")F'F(-F*6#,$F,F-F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"%\"\"$\"\"\"-%$expG6#%\"xGF(! \"\"*&#\"\"(F'F(-F*6#,$F,F-F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&- %$expG6#%\"xG!\"\"*&\"\"#\"\"\"-F%6#,$F'F(F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"#\"\"$\"\"\"-%$expG6#%\"xGF(!\"\"*&#\"\"&F'F(- F*6#,$F,F-F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"$F&-% $expG6#%\"xGF&!\"\"*&#\"\"%F'F&-F)6#,$F+F,F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*&#\"\"\"\"\"$F&-%$expG6#%\"xGF&F&*&#\"\"#F'F&-F)6#,$F+!\"\"F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"#\"\"$\"\"\"-%$expG6#%\"xGF (F(*&#F(F'F(-F*6#,$F,!\"\"F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$e xpG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"%\"\"$\"\"\"-%$ expG6#%\"xGF(F(*&#F(F'F(-F*6#,$F,!\"\"F(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"&\"\"$\"\"\"-%$expG6#%\"xGF(F(*&#\"\"#F'F(-F*6 #,$F,!\"\"F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"-%$ex pG6#%\"xGF&F&-F(6#,$F*!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&# \"\"(\"\"$\"\"\"-%$expG6#%\"xGF(F(*&#\"\"%F'F(-F*6#,$F,!\"\"F(F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\")\"\"$\"\"\"-%$expG6#%\"xGF(F 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$\"3(ox'3!yjI6&F-7$Fix$\"3f#f$=w@c:aF-7$F^y$\"3<0s:/U'>q&F-7$Fcy$\"3&) )>;)H\">2/'F-7$Fhy$\"3'pSHkuJPN'F-7$F]z$\"3Oj(p**Gr%)p'F-7$Fbz$\"3us9k HHiRqF-7$FgzF+-F\\[l6$F^[l$\"++++v$*Fe^m-%+AXESLABELSG6$%\"xG%%y(x)G-% %VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "However, there is a unique solutio n passing through two given points" }{XPPEDIT 18 0 "``(x[0],y[0]);" "6 #-%!G6$&%\"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``( x[1],y[1]);" "6#-%!G6$&%\"xG6#\"\"\"&%\"yG6#F)" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 59 "For example, the unique solution passing \+ through the points" }{XPPEDIT 18 0 " ``(0,1)" "6#-%!G6$\"\"!\"\"\"" } {TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(1,0)" "6#-%!G6$\"\"\"\"\"!" } {TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y( x) =-0" "6#/-%\"yG6#%\"xG,$\"\"!!\"\"" }{XPPEDIT 18 0 ".1565176430*exp (x)+1.156517643*exp(-x);" "6#,&*&-%&FloatG6$\"+Ik " 0 "" {MPLTEXT 1 0 319 "de := diff(y(x),x$2)=y(x);\nic := y(0)=1,y(1)=0;\ndsolve(\{de,ic \},y(x));\ns := unapply(rhs(%),x):\np1 := plot(s(x),x=-0.2..1.2,color= coral,thickness=1):\np2 := plot([[[0,1],[1,0]]$4],color=[black,green$3 ],\n style=point,symbol=[circle$2,diamond,cross],symbolsize=[12 ,10$3]):\nplots[display]([p1,p2],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#F)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"\"/-F(6#F+F *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(-%$expG6#!\"\" \"\"\",&F*F--F+6#F.F.F--F+F&F.F-*(F0F.,&F*F.F0F-F--F+6#,$F'F-F.F-" }} {PARA 13 "" 1 "" {GLPLOT2D 406 251 251 {PLOTDATA 2 "6)-%'CURVESG6%7S7$ $!35+++++++?!#=$\"3S:p>J!GWG\"!#<7$$!3%omm;%)R[p\"F*$\"3ef&H@?-!Q7F-7$ $!3fLLe>;KH9F*$\"3i[yn\\oa)>\"F-7$$!3$omm\"4'=28\"F*$\"3i7OVZL=b6F-7$$ !3/ommTEO,$)!#>$\"3lt88lEd76F-7$$!3aNL$eMD)4`F@$\"3_G2ML'>*F* 7$$\"3:lmm;lT6$*F@$\"3pNTW?K**=))F*7$$\"39LL$eYp$*>\"F*$\"3%R')>k)yT$ \\)F*7$$\"3^*****\\XI8]\"F*$\"3Q@!edz(>M\")F*7$$\"3Y*****\\KJX!=F*$\"3 #eR*zw`'4y(F*7$$\"3%)*****\\a@n4#F*$\"3unI=WGLZuF*7$$\"3iKL3d#e?O#F*$ \"3Qj,J'*)y)\\rF*7$$\"3-mmmr#pvn#F*$\"3)=],.R7F!oF*7$$\"3Kmmm'[[[%HF*$ \"3<(zd;1HR^'F*7$$\"3w)**\\PvddD$F*$\"3E!))e&em%Q='F*7$$\"3clmmO^'4`$F *$\"3s7W\"Rklm*eF*7$$\"3I***\\PD6H$QF*$\"3'RBf4\"Hr'e&F*7$$\"3V***\\7O N/7%F*$\"3)Q!QmvMI'H&F*7$$\"3&fmmTgO/U%F*$\"3]![nh%f&z*\\F*7$$\"3[mmT& RJfp%F*$\"3du'\\&z%[zs%F*7$$\"3'GLLeu*3$*\\F*$\"3]*4(*z)fsSWF*7$$\"3uJ L3dPv,`F*$\"3iOGj(RJl9%F*7$$\"31***\\ioY/d&F*$\"3+*3aYWgO*QF*7$$\"3;KL $3TU1'eF*$\"34'=\"3i^qBOF*7$$\"31*******)HWghF*$\"3M^7$3Q\"zF*$\"3S[&>[b0(*y \"F*7$$\"3x*****\\*3T6#)F*$\"3\\P[P/02I:F*7$$\"3_kmT?w=$\\)F*$\"3ELsc0 9.(G\"F*7$$\"3-++v)[Dxy)F*$\"3%RWTC$[2M5F*7$$\"33mmm\"4!pv!*F*$\"3=:Zv ;YKwyF@7$$\"3Y)**\\PMirP*F*$\"3WW=3tdE.`F@7$$\"3OMLL`f^n'*F*$\"3v]V()3 $)oHGF@7$$\"3GKL$eXWW'**F*$\"3rV)oP3#\\DIFP7$$\"3cm;/C9*e-\"F-$!3jl()4 pVR.AF@7$$\"3++++R,&H0\"F-$!3ZQ=dF!Hx]%F@7$$\"3smm\"*zC'R3\"F-$!3mL&3d .\"F-$ !3#)\\XHWX?8 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "Introduction to the shooting method" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 53 ": The example in this section requires the procedure " } {TEXT 0 7 "desolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "Consider the two point boundary value pro blem" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2 )+2*x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(*&F&F(F+F( F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-y = -sinh(2*x),y(0) = 0,y(2) = 1;" "6%/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF),$-%%sinhG6#*&\"\"#F'%\"x GF'F)/-F*6#\"\"!F5/-F*6#F0F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "We try various initial value pr oblems of the form" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2*x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(! \"\"F(*&F&F(F+F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-y = -sinh(2* x),y(0) = 0;" "6$/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF),$-%%sinhG6#*&\"\" #F'%\"xGF'F)/-F*6#\"\"!F5" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y*`'`(0) \+ = c" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&%\"cG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "where we vary the parameter " }{TEXT 266 1 "c" }{TEXT -1 51 ", giving the first derivative of the solution at 0." }} {PARA 0 "" 0 "" {TEXT -1 45 "We have matched the first boundary condit ion " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 21 ", a nd would like the " }{TEXT 267 1 "y" }{TEXT -1 117 " value of our solu tion at the end of the interval from 0 to 2 to coincide with the secon d boundary condition, namely " }{XPPEDIT 18 0 "y(2) = 1" "6#/-%\"yG6# \"\"#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "We can think of each trial as a shot at the tar get consisting of the second boundary condition " }{XPPEDIT 18 0 "y(2) = 1" "6#/-%\"yG6#\"\"#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 10 "First shot" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 346 "de := diff(y(x),x$2)+2*x*diff(y(x),x)-y(x)=-sinh(x); \nic1 := y(0)=0, D(y)(0)=1;\nfn1 := desolveK2(\{de,ic1\},x=0..2,method =rk78);\np1 := plot('f1(x)',x=0..2,color=red,thickness=2):\np2 := plot ([[[2,fn1(2)]]$4],color=[black,green$3],\n style=point,symbol=[ circle$2,diamond,cross],symbolsize=[12,10$3]):\nplots[display]([p1,p2] ,labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%% diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2F-F2-F(6$F*F-F2F2F*!\" \",$-%%sinhGF,F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic1G6$/-%\"yG6# \"\"!F*/--%\"DG6#F(F)\"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 380 238 238 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$\"\"!F)F(7$$\"+M3VfV!#6$\"+o.ncVF-7$$ \"+#H[D:)F-$\"+_N^M\")F-7$$\"+e0$=C\"!#5$\"+L!paB\"F87$$\"+3RBr;F8$\"+ $yrdl\"F87$$\"+zjf)4#F8$\"+H14o?F87$$\"+'4;[\\#F8$\"+n7xVCF87$$\"+j'y] !HF8$\"+-!y[#GF87$$\"+'zs$HLF8$\"+W8N4KF87$$\"+8iI_PF8$\"+RKh\"e$F87$$ \"+<_M(=%F8$\"+G*4>&RF87$$\"+4y_qXF8$\"+ikdmUF87$$\"+]1!>+&F8$\"+?;-2Y F87$$\"+]Z/NaF8$\"+qaGL\\F87$$\"+]$fC&eF8$\"+p0*4qF87$$\"+!G;cc*F8$\"+e_ qYrF87$$\"+4#G,***F8$\"+'*[_wsF87$$\"+!o2J/\"!\"*$\"+&[I?R(F87$$\"+%Q# \\\"3\"Fas$\"+QRowuF87$$\"+;*[H7\"Fas$\"+Mfo^vF87$$\"+qvxl6Fas$\"+oQ\\ 6wF87$$\"+`qn27Fas$\"+.a)Gl(F87$$\"+cp@[7Fas$\"+KE3xwF87$$\"+3'HKH\"Fa s$\"+[&3go(F87$$\"+xanL8Fas$\"+H*=#ywF87$$\"+v+'oP\"Fas$\"+Ecp`wF87$$ \"+S<*fT\"Fas$\"+f>GFas$\"+(fsp/'F87$$ \"+/Uac>Fas$\"+O#zY$eF87$$\"\"#F)$\"+RS`#f&F8-%'COLOURG6&%$RGBG$\"*+++ +\"!\")F(F(-%*THICKNESSG6#Fdz-F$6&7#7$Fcz$\"3-+++RS`#f&!#=-Fhz6&FjzF)F )F)-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&Fc[l-Fhz6&FjzF(F[[ lF(-F[\\l6$F]\\l\"#5F_\\l-F$6&Fc[lFe\\l-F[\\l6$%(DIAMONDGFi\\lF_\\l-F$ 6&Fc[lFe\\l-F[\\l6$%&CROSSGFi\\lF_\\l-%+AXESLABELSG6%%\"xG%%y(x)G-%%FO NTG6#%(DEFAULTG-%%VIEWG6$;F(FczF\\^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "fn1(2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+RS`#f&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 8 "Too low!" }}{PARA 0 "" 0 "" {TEXT -1 37 "Trying again with a larger value for " }{XPPEDIT 18 0 "y*`'`(0)" " 6#*&%\"yG\"\"\"-%\"'G6#\"\"!F%" }{TEXT -1 34 " should make the end poi nt higher." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 379 "de := diff(y(x),x$2)+2*x*diff(y(x),x)-y(x)=-sinh(x );\nic2 := y(0)=0, D(y)(0)=2;\nfn2 := desolveK2(\{de,ic2\},x=0..2,meth od=rk78);\np1 := plot(['fn1(x)','fn2(x)'],x=0..2,color=[red,magenta],t hickness=2):\np2 := plot([[[2,fn1(2)],[2,fn2(2)]]$4],color=[black,gree n$3],\n style=point,symbol=[circle$2,diamond,cross],symbolsize= [12,10$3]):\nplots[display]([p1,p2],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"# \"\"\"*(F1F2F-F2-F(6$F*F-F2F2F*!\"\",$-%%sinhGF,F6" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$ic2G6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"#" }} {PARA 13 "" 1 "" {GLPLOT2D 374 335 335 {PLOTDATA 2 "6*-%'CURVESG6%7S7$ $\"\"!F)F(7$$\"+M3VfV!#6$\"+o.ncVF-7$$\"+#H[D:)F-$\"+_N^M\")F-7$$\"+e0 $=C\"!#5$\"+L!paB\"F87$$\"+3RBr;F8$\"+$yrdl\"F87$$\"+zjf)4#F8$\"+H14o? F87$$\"+'4;[\\#F8$\"+n7xVCF87$$\"+j'y]!HF8$\"+-!y[#GF87$$\"+'zs$HLF8$ \"+W8N4KF87$$\"+8iI_PF8$\"+RKh\"e$F87$$\"+<_M(=%F8$\"+G*4>&RF87$$\"+4y _qXF8$\"+ikdmUF87$$\"+]1!>+&F8$\"+?;-2YF87$$\"+]Z/NaF8$\"+qaGL\\F87$$ \"+]$fC&eF8$\"+p0*4qF87$$\"+!G;cc*F8$\"+e_qYrF87$$\"+4#G,***F8$\"+'*[_ws F87$$\"+!o2J/\"!\"*$\"+&[I?R(F87$$\"+%Q#\\\"3\"Fas$\"+QRowuF87$$\"+;*[ H7\"Fas$\"+Mfo^vF87$$\"+qvxl6Fas$\"+oQ\\6wF87$$\"+`qn27Fas$\"+.a)Gl(F8 7$$\"+cp@[7Fas$\"+KE3xwF87$$\"+3'HKH\"Fas$\"+[&3go(F87$$\"+xanL8Fas$\" +H*=#ywF87$$\"+v+'oP\"Fas$\"+Ecp`wF87$$\"+S<*fT\"Fas$\"+f>GFas$\"+(fsp/'F87$$\"+/Uac>Fas$\"+O#zY$eF87$$\"\"# F)$\"+RS`#f&F8-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#Fdz-F $6%7SF'7$F+$\"+M5s9()F-7$F1$\"+(f/yi\"F87$F6$\"+b+7uCF87$F<$\"+L+G>LF8 7$FA$\"+_4X^TF87$FF$\"+YZ58\\F87$FK$\"+%=T**o&F87$FP$\"+R.))ykF87$FU$ \"+/[()[sF87$FZ$\"+Pl0A!)F87$Fin$\"+)yFfo)F87$F^o$\"+vys7%*F87$Fco$\"+ ?@$>,\"Fas7$Fho$\"+rqrx5Fas7$F]p$\"+_gTN6Fas7$Fbp$\"+)4<9?\"Fas7$Fgp$ \"+Ol0b7Fas7$F\\q$\"+!*yx98Fas7$Faq$\"+h(4_O\"Fas7$Ffq$\"+N7*yT\"Fas7$ F[r$\"+\\(oaY\"Fas7$F`r$\"+E0U7:Fas7$Fer$\"+3o7`:Fas7$Fjr$\"+XqY%f\"Fa s7$F_s$\"+R\"=Yj\"Fas7$Fes$\"+LsGn;Fas7$Fjs$\"+'4B-q\"Fas7$F_t$\"+dbtJ Fas7$Faw$\"+q-w;>Fas7$Ffw$\"+vMcC>Fas7$F[x$\"+ ,mEI>Fas7$F`x$\"+k,IM>Fas7$Fex$\"+u,aO>Fas7$Fjx$\"+f33P>Fas7$F_y$\"+[> %f$>Fas7$Fdy$\"+hSNL>Fas7$Fiy$\"+uB)*G>Fas7$F^z$\"+gUDB>Fas7$Fcz$\"+&) yU:>Fas-Fhz6&FjzF[[lF(F[[lF^[l-F$6&7$7$Fcz$\"3-+++RS`#f&!#=7$Fcz$\"3%* *****\\)yU:>!#<-Fhz6&FjzF)F)F)-%'SYMBOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&PO INTG-F$6&Fhdl-Fhz6&FjzF(F[[lF(-Fdel6$Ffel\"#5Fhel-F$6&FhdlF^fl-Fdel6$% (DIAMONDGFbflFhel-F$6&FhdlF^fl-Fdel6$%&CROSSGFbflFhel-%+AXESLABELSG6%% \"xG%%y(x)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(FczFegl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "This time the end point is " }{TEXT 259 8 "too high" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 251 "We don't have to continue in a tria l and error fashion, because the final value depends in a linear fashi on on the initial value of the derivative, provided that we don't chan ge any other parameter. This is because the differential equation is l inear." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "If the solution satisfying the initial conditions " }{XPPEDIT 18 0 "y (0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`' `(0) = 1" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&F&" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "y = f[1](x)" "6#/%\"yG-&%\"fG6#\"\"\"6#%\"xG" }{TEXT -1 53 ", and the solution satisfying the initial conditions " } {XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y*`'`(0) = 2" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&\"\"#" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "y = f[2](x)" "6#/%\"yG-&%\"fG6#\"\"# 6#%\"xG" }{TEXT -1 29 ", then the linear combination" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y = c*f[1](x) + (1-c)*f[2](x)" " 6#/%\"yG,&*&%\"cG\"\"\"-&%\"fG6#F(6#%\"xGF(F(*&,&F(F(F'!\"\"F(-&F+6#\" \"#6#F.F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "is a solut ion which satisfies " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = c*f[1]*`'`(x)+(1-c)*f[2] *`'`(x);" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&,&*(%\"cGF&&%\"fG6#F&F&-F(6# %\"xGF&F&*(,&F&F&F-!\"\"F&&F/6#\"\"#F&-F(6#F3F&F&" }{XPPEDIT 18 0 " `` =c+2*(1-c)" "6#/%!G,&%\"cG\"\"\"*&\"\"#F',&F'F'F&!\"\"F'F'" }{XPPEDIT 18 0 "``=2-c" "6#/%!G,&\"\"#\"\"\"%\"cG!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 13 "Furthermore, " }{XPPEDIT 18 0 "y(2) = c*f [1](2)+(1-c)*f[2](2)" "6#/-%\"yG6#\"\"#,&*&%\"cG\"\"\"-&%\"fG6#F+6#F'F +F+*&,&F+F+F*!\"\"F+-&F.6#F'6#F'F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 14 "We can choose " }{TEXT 268 1 "c" }{TEXT -1 32 " so that the linear combination " }{XPPEDIT 18 0 "c*f[1](x)+(1-c)*f[2](x)" "6# ,&*&%\"cG\"\"\"-&%\"fG6#F&6#%\"xGF&F&*&,&F&F&F%!\"\"F&-&F)6#\"\"#6#F,F &F&" }{TEXT -1 41 " satisfies the second boundary condition " } {XPPEDIT 18 0 "y(2) = 1" "6#/-%\"yG6#\"\"#\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 8 "Indeed, " }{XPPEDIT 18 0 "c*f[1](2)+(1-c)* f[2](2)=1" "6#/,&*&%\"cG\"\"\"-&%\"fG6#F'6#\"\"#F'F'*&,&F'F'F&!\"\"F'- &F*6#F-6#F-F'F'F'" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "c = (1-f[2](2 ))/(f[1](2)-f[2](2))" "6#/%\"cG*&,&\"\"\"F'-&%\"fG6#\"\"#6#F,!\"\"F',& -&F*6#F'6#F,F'-&F*6#F,6#F,F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "start := eva lf[13](2-(1-fn2(2))/(fn1(2)-fn2(2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&startG$\".4^a#*\\K\"!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 408 "de := diff(y(x),x$2)+2*x*di ff(y(x),x)-y(x)=-sinh(x);\nic3 := y(0)=0, D(y)(0)=start;\nfn3 := desol veK2(\{de,ic3\},x=0..2,method=rk78);\np1 := plot(['fn1(x)','fn2(x)','f n3(x)'],x=0..2,color=[red,magenta,blue],thickness=2):\np2 := plot([[[2 ,fn1(2)],[2,fn2(2)],[2,fn3(2)]]$4],color=[black,green$3],\n sty le=point,symbol=[circle$2,diamond,cross],symbolsize=[12,10$3]):\nplots [display]([p1,p2],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2F-F2- F(6$F*F-F2F2F*!\"\",$-%%sinhGF,F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$ic3G6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)$\".4^a#*\\K\"!#7" }}{PARA 13 " " 1 "" {GLPLOT2D 433 346 346 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)F( 7$$\"+M3VfV!#6$\"+o.ncVF-7$$\"+#H[D:)F-$\"+_N^M\")F-7$$\"+e0$=C\"!#5$ \"+L!paB\"F87$$\"+3RBr;F8$\"+$yrdl\"F87$$\"+zjf)4#F8$\"+H14o?F87$$\"+' 4;[\\#F8$\"+n7xVCF87$$\"+j'y]!HF8$\"+-!y[#GF87$$\"+'zs$HLF8$\"+W8N4KF8 7$$\"+8iI_PF8$\"+RKh\"e$F87$$\"+<_M(=%F8$\"+G*4>&RF87$$\"+4y_qXF8$\"+i kdmUF87$$\"+]1!>+&F8$\"+?;-2YF87$$\"+]Z/NaF8$\"+qaGL\\F87$$\"+]$fC&eF8 $\"+p0*4qF87$$\"+!G;cc*F8$\"+e_qYrF87$$\"+4#G,***F8$\"+'*[_wsF87$$\"+!o2 J/\"!\"*$\"+&[I?R(F87$$\"+%Q#\\\"3\"Fas$\"+QRowuF87$$\"+;*[H7\"Fas$\"+ Mfo^vF87$$\"+qvxl6Fas$\"+oQ\\6wF87$$\"+`qn27Fas$\"+.a)Gl(F87$$\"+cp@[7 Fas$\"+KE3xwF87$$\"+3'HKH\"Fas$\"+[&3go(F87$$\"+xanL8Fas$\"+H*=#ywF87$ $\"+v+'oP\"Fas$\"+Ecp`wF87$$\"+S<*fT\"Fas$\"+f>GFas$\"+(fsp/'F87$$\"+/Uac>Fas$\"+O#zY$eF87$$\"\"#F)$\"+RS`# f&F8-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#Fdz-F$6%7SF'7$F +$\"+M5s9()F-7$F1$\"+(f/yi\"F87$F6$\"+b+7uCF87$F<$\"+L+G>LF87$FA$\"+_4 X^TF87$FF$\"+YZ58\\F87$FK$\"+%=T**o&F87$FP$\"+R.))ykF87$FU$\"+/[()[sF8 7$FZ$\"+Pl0A!)F87$Fin$\"+)yFfo)F87$F^o$\"+vys7%*F87$Fco$\"+?@$>,\"Fas7 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+dEew5Fas7$Fdt$\"+,\"4')3\"Fas7$Fit$\"+A8X)4\"Fas7$F^u$\"+4VN26Fas7$Fc u$\"+GWd86Fas7$Fhu$\"+7qR=6Fas7$F]v$\"+\\\"z67\"Fas7$Fbv$\"+9/`A6Fas7$ Fgv$\"+CGAA6Fas7$F\\w$\"+3$)H?6Fas7$Faw$\"+]L'o6\"Fas7$Ffw$\"+PBl66Fas 7$F[x$\"+v')406Fas7$F`x$\"+8O'o4\"Fas7$Fex$\"+%*p=(3\"Fas7$Fjx$\"+#>&) p2\"Fas7$F_y$\"+#*\\wj5Fas7$Fdy$\"+$yo00\"Fas7$Fiy$\"+]+3N5Fas7$F^z$\" +&\\)))=5Fas7$Fcz$\"+++++5Fas-Fhz6&FjzF(F(F[[lF^[l-F$6&7%7$Fcz$\"3-+++ RS`#f&!#=7$Fcz$\"3%******\\)yU:>!#<7$Fcz$\"\"\"F)-Fhz6&FjzF)F)F)-%'SYM BOLG6$%'CIRCLEG\"#7-%&STYLEG6#%&POINTG-F$6&F]^m-Fhz6&FjzF(F[[lF(-F\\_m 6$F^_m\"#5F`_m-F$6&F]^mFf_m-F\\_m6$%(DIAMONDGFj_mF`_m-F$6&F]^mFf_m-F\\ _m6$%&CROSSGFj_mF`_m-%+AXESLABELSG6%%\"xG%%y(x)G-%%FONTG6#%(DEFAULTG-% %VIEWG6$;F(FczF]am" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "fn3(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++ ++5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Bull's-eye!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 80 "Our t hird numerical solution satisfies both of the required boundary condit ions " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "y(2) = 1" "6#/-%\"yG6#\"\"#\"\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "The linear shooting method" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 139 "The following linear shooting method is similar to \+ the method outlined in the example of the previous section, but is a b it more efficient." }}{PARA 0 "" 0 "" {TEXT -1 65 "We consider a 2nd o rder linear boundary value problem of the form" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+p(x);" "6#,&*(%\"dG\"\"# %\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-%\"pG6#F+F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+q(x)*y = r(x),y(a) = alpha,y(b) = beta;" "6%/,&*& %#dyG\"\"\"%#dxG!\"\"F'*&-%\"qG6#%\"xGF'%\"yGF'F'-%\"rG6#F./-F/6#%\"aG %&alphaG/-F/6#%\"bG%%betaG" }{TEXT -1 14 " ------- (i) " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "We first solve the " }{TEXT 259 32 "associated initial value problem" }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+p(x);" "6#,&*(%\"dG\" \"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-%\"pG6#F+F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+q(x)*y = r(x),y(a) = alpha;" "6$/,&*&%#dyG\"\"\"% #dxG!\"\"F'*&-%\"qG6#%\"xGF'%\"yGF'F'-%\"rG6#F./-F/6#%\"aG%&alphaG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "`y '`(a)=0" "6#/-%$y~'G6#%\"aG\"\"!" } {TEXT -1 14 " ------- (ii)" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "y = u(x)" "6#/%\"yG-%\"uG6#%\"xG" }{TEXT -1 25 " be the solution to (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Now we solve the " }{TEXT 259 33 "homogeneous initial val ue problem" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/ (d*x^2)+p(x);" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-% \"pG6#F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+q(x)*y = 0,y(a) = 0; " "6$/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&-%\"qG6#%\"xGF'%\"yGF'F'\"\"!/-F/6# %\"aGF0" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`y '`(a)=1" "6#/-%$y~'G6#%\" aG\"\"\"" }{TEXT -1 15 " ------- (iii)" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "y = v(x)" "6#/%\"yG-%\"vG6#%\"xG" }{TEXT -1 26 " be the solution to (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Then the " }{TEXT 259 18 "linear combination" } {TEXT -1 1 " " }{XPPEDIT 18 0 "y = u(x) + c*v(x)" "6#/%\"yG,&-%\"uG6#% \"xG\"\"\"*&%\"cGF*-%\"vG6#F)F*F*" }{TEXT -1 29 " is a particular solu tion of " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d *x^2)+p(x);" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-%\"p G6#F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+q(x)*y = r(x);" "6#/,&* &%#dyG\"\"\"%#dxG!\"\"F'*&-%\"qG6#%\"xGF'%\"yGF'F'-%\"rG6#F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "for any choice of the constant " }{TEXT 269 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "Th e boundary values of the solution " }{XPPEDIT 18 0 "y = u(x)+c*v(x)" " 6#/%\"yG,&-%\"uG6#%\"xG\"\"\"*&%\"cGF*-%\"vG6#F)F*F*" }{TEXT -1 5 " ar e " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(a) = alpha" " 6#/-%\"yG6#%\"aG%&alphaG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(b) = u(b)+c*v(b)" "6#/-%\"yG6#%\"bG,&-%\"uG 6#F'\"\"\"*&%\"cGF,-%\"vG6#F'F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Now we choose the constan t " }{TEXT 270 1 "c" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "y(b)=beta " "6#/-%\"yG6#%\"bG%%betaG" }{TEXT -1 13 ", namely set " }{XPPEDIT 18 0 "c=(beta-u(b))/v(b)" "6#/%\"cG*&,&%%betaG\"\"\"-%\"uG6#%\"bG!\"\"F(- %\"vG6#F,F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "In this wa y we obtain the solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y(x) = u(x)+``((beta-u(b))/v(b))*v(x);" "6#/-%\"yG6#%\" xG,&-%\"uG6#F'\"\"\"*&-%!G6#*&,&%%betaGF,-F*6#%\"bG!\"\"F,-%\"vG6#F6F7 F,-F96#F'F,F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 263 17 "________________ " }}{PARA 0 "" 0 "" {TEXT -1 8 "for (i) ." }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 " " 0 "" {TEXT -1 76 "For the method to succeed, we need to be able to f ind both of the solutions " }{XPPEDIT 18 0 "y=u(x)" "6#/%\"yG-%\"uG6#% \"xG" }{TEXT -1 15 " for (ii), and " }{XPPEDIT 18 0 "y=v(x)" "6#/%\"yG -%\"vG6#%\"xG" }{TEXT -1 38 " for (iii). In addition, we must have " } {XPPEDIT 18 0 "v(b)<>0" "6#0-%\"vG6#%\"bG\"\"!" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT -1 40 "The initial conditions for the solution \+ " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 6 " are: " } {XPPEDIT 18 0 "y(a) = alpha" "6#/-%\"yG6#%\"aG%&alphaG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`y '`(a) = (beta-u(b))/v(b);" "6#/-%$y~'G6#%\"aG*&, &%%betaG\"\"\"-%\"uG6#%\"bG!\"\"F+-%\"vG6#F/F0" }{TEXT -1 2 ".\n" } {XPPEDIT 18 0 "y(a) = alpha" "6#/-%\"yG6#%\"aG%&alphaG" }{TEXT -1 29 " follows from the fact that " }{XPPEDIT 18 0 "y(a) = u(a)+``((beta-u( b))/v(b))*v(a);" "6#/-%\"yG6#%\"aG,&-%\"uG6#F'\"\"\"*&-%!G6#*&,&%%beta GF,-F*6#%\"bG!\"\"F,-%\"vG6#F6F7F,-F96#F'F,F," }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u(a) = alpha" "6#/-%\"uG6#%\"aG%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(a) = 0" "6#/-%\"vG6#%\"aG\"\"!" }{TEXT -1 3 " .\n " }{XPPEDIT 18 0 "`y '`(a) = (beta-u(b))/v(b);" "6#/-%$y~'G6#%\"aG *&,&%%betaG\"\"\"-%\"uG6#%\"bG!\"\"F+-%\"vG6#F/F0" }{TEXT -1 30 " fol lows from the fact that " }{XPPEDIT 18 0 "`y '`(a) = `u '`(a)+``((bet a-u(b))/v(b))*`v '`(a);" "6#/-%$y~'G6#%\"aG,&-%$u~'G6#F'\"\"\"*&-%!G6# *&,&%%betaGF,-%\"uG6#%\"bG!\"\"F,-%\"vG6#F7F8F,-%$v~'G6#F'F,F," } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "`u '`(a) = 0;" "6#/-%$u~'G6#%\"a G\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`v '`(a) = 1;" "6#/-%$v~'G 6#%\"aG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 31 "Linear shooting method examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 39 ": The se examples require the procedure " }{TEXT 0 7 "desolve" }{TEXT -1 1 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 1" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$ %\"xGF&F(!\"\"F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 1" "6#/-%\" yG6#\"\"!\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 0" "6#/-%\"y G6#\"\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "First w e find an analytical form for the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "de := diff(y(x),x $2)=y(x);\nic := y(0)=1,y(1)=0;\ndsolve(\{de,ic\},y(x));\ng := unapply (evalf[13](rhs(convert(%,exp))),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"\"/-F(6#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(-%$expG6#\"\"\"\"\"#,&F-!\"\"*$) F*F.F-F-F0-F+6#,$F'F0F-F-*&F/F0-F+F&F-F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&$\".]Fk " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Now we find a numerical solution by the linear shooting method." }}{PARA 0 "" 0 "" {TEXT -1 28 "We start by considering the " }{TEXT 259 32 "associated initial va lue problem" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2) = y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$ %\"xGF&F(!\"\"F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 1" "6#/-%\" yG6#\"\"!\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0" "6#/- %$y~'G6#\"\"!F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 14 "with \+ solution " }{XPPEDIT 18 0 "u(x)" "6#-%\"uG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 35 "We construct a numerical procedure " } {TEXT 271 2 "Un" }{TEXT -1 43 " for this solution defined on the inter val " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x =1" "6#/%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 79 "In this example, the 2nd order differential equation ha ppens to be homogeneous." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "de := diff(y(x),x$2)=y(x);\nic := y (0)=1,D(y)(0)=0;\nUn := desolveK2(\{de,ic\},x=0..1,method=rk78);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$ F,\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\" \"\"/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Next co nsider the " }{TEXT 259 33 "homogeneous initial value problem" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d* x^2) = y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 1" "6#/-%$y~'G6#\"\"!\"\" \"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 14 "with solution " }{XPPEDIT 18 0 "v(x)" "6#-%\"vG6#%\"xG" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 35 "We construct a numerical procedure " }{TEXT 271 2 "Vn" }{TEXT -1 42 " or this solution defined on the interval " } {XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "de := diff(y(x),x$2)= y(x);\nic2 := y(0)=0,D(y)(0)=1;\nVn := desolveK2(\{de,ic2\},x=0..1,met hod=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6 #%\"xG-%\"$G6$F,\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic2G6$/- %\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 54 "The constant needed to form the linear c ombination of " }{XPPEDIT 18 0 "u(x);" "6#-%\"uG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(x);" "6#-%\"vG6#%\"xG" }{TEXT -1 45 " which s olves the boundary value problem is " }{XPPEDIT 18 0 "-u(1)/v(1);" "6 #,$*&-%\"uG6#\"\"\"F(-%\"vG6#F(!\"\"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c := ev alf[13](-Un(1)/Vn(1)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$!.0b GNIJ\"!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The required solution is " }{XPPEDIT 18 0 "g(x) = u(x)+c*v(x); " "6#/-%\"gG6#%\"xG,&-%\"uG6#F'\"\"\"*&%\"cGF,-%\"vG6#F'F,F," }{TEXT -1 62 ". A numerical procedure giving this solution on the interval \+ " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 45 " could be defined in Maple arrow notation by:" }{TEXT 260 25 " gn := x -> Un(x)+c*Vn(x)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "However, since " }{XPPEDIT 18 0 "`g '`(0) = `u '`(0)+c*`v '`(0);" "6#/-%$g~'G6#\"\"!,&-%$u~'G6#F' \"\"\"*&%\"cGF,-%$v~'G6#F'F,F," }{XPPEDIT 18 0 " ``= c" "6#/%!G%\"cG" }{TEXT -1 48 ", it is easy to construct a numerical procedure " } {TEXT 271 2 "gn" }{TEXT -1 35 " giving this solution afresh using " } {TEXT 0 9 "desolveK2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 "W e could also extend the solution to a larger interval, if desired. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "de := diff(y(x),x$2)=y(x);\nic3 := y(0)=1,D(y)(0)=c;\ngn := deso lveK2(\{de,ic3\},x=0..1,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$ic3G6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)$!.0bGNIJ \"!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 271 2 "gn" }{TEXT -1 62 " now provides our sol ution, ( restricted to the interval from " }{XPPEDIT 18 0 "x = 0" "6#/ %\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x =1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 49 "We compare some values of the numerical solution " }{TEXT 271 2 "gn" }{TEXT -1 32 " and the \+ analytical solution g. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "xvals := [seq(i*0.1,i=0..10)]:\nhv als := evalf(evalf[13](map(gn,xvals))):\ngvals := evalf(evalf[13](map( g,xvals))):\nlinalg[matrix]([[x,`gn(x)`,'g(x)'],\n seq([xvals[i], hvals[i],gvals[i]],i=1..11)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7.7%%\"xG%&gn(x)G-%\"gG6#F(7%$\"\"!F/$\"\"\"F/$\"+++++5!\"*7%$ F1!\"\"$\"+3p\"[t)!#5F87%$\"\"#F7$\"++[0dvF:F>7%$\"\"$F7$\"+Pi#\\X'F:F C7%$\"\"%F7$\"+X2S " 0 "" {MPLTEXT 1 0 88 "plot(['Un(x) ','Vn(x)','gn(x)'],x=0..1,y,\n color=[magenta,blue,red],thickn ess=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 374 255 255 {PLOTDATA 2 "6(-%'CU RVESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"+;arz@!#6$\"+nvB+5!\"*7$$\"+XTFwSF/$ \"+;4$3+\"F27$$\"+\"z_\"4iF/$\"+)HG>+\"F27$$\"+S&phN)F/$\"+5L\\.5F27$$ \"+*=)H\\5!#5$\"+'=5b+\"F27$$\"+[!3uC\"FE$\"+G-z25F27$$\"+J$RDX\"FE$\" +9zc55F27$$\"+)R'ok;FE$\"+Jz)Q,\"F27$$\"+1J:w=FE$\"+Q9l<5F27$$\"+3En$4 #FE$\"+/v*>-\"F27$$\"+/RE&G#FE$\"+%*fAE5F27$$\"+D.&4]#FE$\"+6rVJ5F27$$ \"+vB_Y2a FE$\"+\"p+)\\6F27$$\"+yXu9cFE$\"+z:\"=;\"F27$$\"+\\y))GeFE$\"+BVuu6F27 $$\"+i_QQgFE$\"+px\"z=\"F27$$\"+!y%3TiFE$\"+y*f6?\"F27$$\"+O![hY'FE$\" +b;W;7F27$$\"+#Qx$omFE$\"+$G)pI7F27$$\"+u.I%)oFE$\"+PgZY7F27$$\"+(pe*z qFE$\"+JZFh7F27$$\"+C\\'QH(FE$\"+Gj+y7F27$$\"+8S8&\\(FE$\"++LG%H\"F27$ $\"+0#=bq(FE$\"+7!e=J\"F27$$\"+2s?6zFE$\"+D=gH8F27$$\"+IXaE\")FE$\"+G3 y[8F27$$\"+l*RRL)FE$\"+iO%yO\"F27$$\"+`<.Y&)FE$\"+Nn%zQ\"F27$$\"+8tOc( )FE$\"+Z**\\39F27$$\"+\\Qk\\*)FE$\"+!RNzU\"F27$$\"+p0;r\"*FE$\"+&3n3X \"F27$$\"+lxGp$*FE$\"+&H!)>Z\"F27$$\"+!oK0e*FE$\"+u!G^\\\"F27$$\"+<5s# y*FE$\"+L#3z^\"F27$F*$\"+N13V:F2-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F[[l -F$6$7S7$F(F(7$F-$\"+B!)))z@F/7$F4$\"+'4.u2%F/7$F9$\"+6L98iF/7$F>$\"+2 v*eO)F/7$FC$\"+eZA^5FE7$FI$\"+mbk]7FE7$FN$\"+0Dld9FE7$FS$\"+PcQs;FE7$F X$\"+/\"zr)=FE7$Fgn$\"+o?+4@FE7$F\\o$\"+kp?0BFE7$Fao$\"+dL5FDFE7$Ffo$ \"+UR4^FFE7$F[p$\"+M,**FE7$F\\y$\"+RKJ>5F27$Fay$\"+zq>^5F27$Ffy$\"+54:!3\"F27$F[z $\"+s!*[66F27$F`z$\"+u![>9\"F27$F*$\"+%>,_<\"F2-Fhz6&FjzF(F(F[[l-F$6$7 SF'7$F-$\"+%o[hr*FE7$F4$\"+U<$HZ*FE7$F9$\"+V`Z.#*FE7$F>$\"+U7YO*)FE7$F C$\"+jm![n)FE7$FI$\"+?0wN%)FE7$FN$\"+k*H<>)FE7$FS$\"+hz(H%zFE7$FX$\"+H ;e)p(FE7$Fgn$\"+-4y]uFE7$F\\o$\"+f=WNsFE7$Fao$\"+J_>'*pFE7$Ffo$\"+c4Cf nFE7$F[p$\"+)p#*Q`'FE7$F`p$\"+7LsJjFE7$Fep$\"+&oyU4'FE7$Fjp$\"+o%eb*eF E7$F_q$\"+y$)4ncFE7$Fdq$\"+)43sY&FE7$Fiq$\"+HXK]_FE7$F^r$\"+1<2Y]FE7$F cr$\"+%)HAN[FE7$Fhr$\"+VgbVYFE7$F]s$\"+R5$)QWFE7$Fbs$\"+ybHGUFE7$Fgs$ \"+.3qYSFE7$F\\t$\"+CeC_QFE7$Fat$\"+V@4`OFE7$Fft$\"+14))fMFE7$F[u$\"+B aQuKFE7$F`u$\"+*G(**pIFE7$Feu$\"+&*[n()GFE7$Fju$\"+U'3Vp#FE7$F_v$\"+(R x,_#FE7$Fdv$\"+qv!4L#FE7$Fiv$\"+x[z`@FE7$F^w$\"+3Efp>FE7$Fcw$\"+UYM!z \"FE7$Fhw$\"+$)*)\\.;FE7$F]x$\"+b$[UU\"FE7$Fbx$\"+%pr:C\"FE7$Fgx$\"+6& f41\"FE7$F\\y$\"+\"39T&*)F/7$Fay$\"+VG#31(F/7$Ffy$\"+_OSq`F/7$F[z$\"+ \"Gq.d$F/7$F`z$\"+M<,\\=F/7$F*$!$(G!#:-Fhz6&FjzF[[lF(F(-%*THICKNESSG6# \"\"#-%+AXESLABELSG6$Q\"x6\"Q\"yFb^m-%%VIEWG6$;F(F*%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2 " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2; " "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = exp(-x)+sin(2*x);" "6#/,&*&%#dyG\"\" \"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F',&-%$expG6#,$%\"xGF)F'-%$sinG6#*&F+F'F 2F'F'" }{TEXT -1 12 ", y(1) = 1, " }{XPPEDIT 18 0 "y(3) = -1" "6#/-%\" yG6#\"\"$,$\"\"\"!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "First we find an analytical form for the solution." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "de := dif f(y(x),x$2)+2*diff(y(x),x)+2*y(x)=exp(-x)+sin(2*x);\nic := y(1)=1,y(3) =-1;\ndsolve(\{de,ic\},y(x)):\ng := unapply(evalf[13](rhs(%)),x);\n" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$ G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&F1F2F*F2F2,&-%$expG6#,$F-!\"\"F2- %$sinG6#,$*&F1F2F-F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/- %\"yG6#\"\"\"F*/-F(6#\"\"$!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% \"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*($\".py&y+x')!#7\"\"\"-%$exp G6#,$*&$F1\"\"!F19$F1!\"\"F1-%$sinG6#F9F1F:*($\".+nHgKn\"!#6F1F2F1-%$c osGF=F1F1*($\".++++++\"!#8F1,($\"#5F8F:*&-F<6#,$*&$\"\"#F8F1F9F1F1F1-F 3F=F1F1*(FPF1-FCFMF1FRF1F1F1F2F1F:F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Now we find a numerical solution by the linear shooting m ethod." }}{PARA 0 "" 0 "" {TEXT -1 28 "We start by considering the " } {TEXT 259 32 "associated initial value problem" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6 #,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx+2*y = exp(-x)+sin(2*x),y(1) = 1;" "6$/,&*&%#dy G\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F',&-%$expG6#,$%\"xGF)F'-%$sinG6#*& F+F'F2F'F'/-F,6#F'F'" }{TEXT -1 14 ", y '(1) = 0, " }}{PARA 0 "" 0 "" {TEXT -1 14 "with solution " }{XPPEDIT 18 0 "u(x)" "6#-%\"uG6#%\"xG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "We construct a numerica l procedure " }{TEXT 271 2 "Un" }{TEXT -1 43 " for this solution defin ed on the interval " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "de := diff(y(x),x$2)+2*diff(y(x),x)+2*y(x)=exp(-x)+sin(2*x);\nic \+ := y(1)=1,D(y)(1)=0;\nUn := desolveK2(\{de,ic\},x=1..3,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\" $G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&F1F2F*F2F2,&-%$expG6#,$F-!\"\"F2 -%$sinG6#,$*&F1F2F-F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/ -%\"yG6#\"\"\"F*/--%\"DG6#F(F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Next consider the " }{TEXT 259 33 "homogeneous initial va lue problem" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = 0;" " 6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F'\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "y(1) = 0" "6#/-%\"yG6#\"\"\"\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "`y '`(1) = 1" "6#/-%$y~'G6#\"\"\"F'" }{TEXT -1 2 ", \+ " }}{PARA 257 "" 0 "" {TEXT -1 14 "with solution " }{XPPEDIT 18 0 "v(x )" "6#-%\"vG6#%\"xG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 " We construct a numerical procedure " }{TEXT 271 2 "Vn" }{TEXT -1 42 " \+ or this solution defined on the interval " }{XPPEDIT 18 0 "x = 1;" "6# /%\"xG\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\" $" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "de := diff(y(x),x$2)+2*diff(y(x),x)+2*y(x)=0 ;\nic2 := y(1)=0,D(y)(1)=1;\nVn := desolveK2(\{de,ic2\},x=1..3,method= rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#% \"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&F1F2F*F2F2\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic2G6$/-%\"yG6#\"\"\"\"\"!/--%\"DG6 #F(F)F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The constant needed to form the linear combination of " }{XPPEDIT 18 0 "u(x);" "6#-%\"uG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(x );" "6#-%\"vG6#%\"xG" }{TEXT -1 45 " which solves the boundary value p roblem is " }{XPPEDIT 18 0 "(-1-u(3))/v(3);" "6#*&,&\"\"\"!\"\"-%\"uG 6#\"\"$F&F%-%\"vG6#F*F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "c := evalf((-1-Un(3))/V n(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$!+L'*=lu!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We now co nstruct a numerical procedure " }{TEXT 271 2 "gn" }{TEXT -1 41 " for t he solution over the interval from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG \"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 10" "6#/%\"xG\"#5" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "de := diff(y(x),x$2)+2*diff(y(x),x)+2*y(x)=exp( -x)+sin(2*x);\nic3 := y(1)=1,D(y)(1)=c;\ngn := desolveK2(\{de,ic3\},x= 0..10,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%dif fG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&F1F2F*F2F2, &-%$expG6#,$F-!\"\"F2-%$sinG6#,$*&F1F2F-F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic3G6$/-%\"yG6#\"\"\"F*/--%\"DG6#F(F)$!+L'*=lu!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "We com pare some values of the numerical solution " }{TEXT 271 2 "gn" }{TEXT -1 32 " and the analytical solution g. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "xvals := [seq(i,i=0..1 0)]:\nhvals := evalf(evalf[13](map(gn,xvals))):\ngvals := evalf(evalf[ 13](map(g,xvals))):\nlinalg[matrix]([[x,`gn(x)`,'g(x)'],\n seq([x vals[i],hvals[i],gvals[i]],i=1..11)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7%%\"xG%&gn(x)G-%\"gG6#F(7%\"\"!$\"+(HgKv\"!\")F/7 %\"\"\"$F3F.$\"+++++5!\"*7%\"\"#$!+oyTo;F7F:7%\"\"$$!+++++5F7F>7%\"\"% $!+TWm:8!#5FB7%\"\"&$\"+'=\"**pJFDFG7%\"\"'$!+%p.,o'!#6FK7%\"\"($!+A[ \">>\"FDFP7%\"\")$\"+#R6'p@FDFT7%\"\"*$!+FpR;fFMFX7%\"#5$!+lj)Gt\"FDFf nQ(pprint96\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We plot the graphs of the numerical procedures " }{TEXT 271 2 "Un" }{TEXT -1 5 " and " }{TEXT 271 2 "Vn" }{TEXT -1 19 " used t o construct " }{TEXT 271 2 "gn" }{TEXT -1 11 " (shown in " }{TEXT 260 3 "red" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot(['Un(x)','Vn(x)','gn(x)'],x=1. .3,y,\n color=[magenta,blue,red],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 421 270 270 {PLOTDATA 2 "6(-%'CURVESG6$7S7$$\"\"\"\"\"! 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Matthews and Kurtis D. Fink, Prent ice Hall, page 499." }}{PARA 256 "" 0 "" {TEXT -1 1 "(" }{XPPEDIT 18 0 "1+x^2" "6#,&\"\"\"F$*$%\"xG\"\"#F$" }{TEXT -1 3 ") " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-2*x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(! \"\"F(*&F&F(F+F(F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = 1+x^2; " "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F',&F'F'*$%\"xGF+F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 5/4" "6#/-%\"yG6#\"\"!*&\"\"& \"\"\"\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(4) = -19/20" "6# /-%\"yG6#\"\"%,$*&\"#>\"\"\"\"#?!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "de := \+ (1+x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+2*y(x)=1+x^2;\nic := y(0)=5/4, y(4)=-19/20;\ndsolve(\{de,ic\},y(x));\ng := unapply(rhs(%),x):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&,&\"\"\"F)*$)%\"xG\"\"#F)F) F)-%%diffG6$-%\"yG6#F,-%\"$G6$F,F-F)F)*(F-F)F,F)-F/6$F1F,F)!\"\"*&F-F) F1F)F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!#\" \"&\"\"%/-F(6#F-#!#>\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"xG,.*&,(#\"$p\"\"#?\"\"\"*&\"\"#F.-%'arctanG6#\"\"%F.!\"\"*&#\"#:\" \")F.-%#lnG6#\"# " 0 "" {MPLTEXT 1 0 146 "de := (1 +x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+2*y(x)=1+x^2;\nic := y(0)=5/4,D( y)(0)=0;\nUn := desolveK2(\{de,ic\},x=0..4,method=rk78,tolerance=1e-12 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&,&\"\"\"F)*$)%\"xG\" \"#F)F)F)-%%diffG6$-%\"yG6#F,-%\"$G6$F,F-F)F)*(F-F)F,F)-F/6$F1F,F)!\" \"*&F-F)F1F)F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6# \"\"!#\"\"&\"\"%/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "It is easy to check that " }{XPPEDIT 18 0 "v(x) = x;" "6# /-%\"vG6#%\"xGF'" }{TEXT -1 25 " is the solution for the " }{TEXT 259 33 "homogeneous initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 1 "(" }{XPPEDIT 18 0 "1+x^2" "6#,&\"\"\"F$*$%\"xG\"\"#F$" }{TEXT -1 3 ") " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-2*x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*& F%F(*$%\"xGF&F(!\"\"F(*&F&F(F+F(F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx+2*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F'\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0) = 0 " "6#/-%\"yG6#\"\"!F'" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "`y '`(0) = 1" "6#/-%$y~'G6#\"\"!\"\"\" " }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 43 "so we don't need to s et up a procedure for " }{XPPEDIT 18 0 "v(x)" "6#-%\"vG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The constant needed to form the linear combination of " } {XPPEDIT 18 0 "u(x)" "6#-%\"uG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(x)" "6#-%\"vG6#%\"xG" }{TEXT -1 45 " which solves the boundary value problem is " }{XPPEDIT 18 0 "(-19/20-u(4))/4;" "6#*&,&*&\"#>\" \"\"\"#?!\"\"F)-%\"uG6#\"\"%F)F'F-F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "c := evalf [13]((-19/20-Un(4))/4);\ngn := x -> Un(x)+c*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\".#oDl*3'[!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#gnGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%#UnG6#9$\"\"\"*&%\"cGF1 F0F1F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 271 2 "gn" }{TEXT -1 62 " now prov ides our solution, ( restricted to the interval from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 4" "6#/ %\"xG\"\"%" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 49 "We compare some values of the numerical solution " }{TEXT 271 2 "gn" }{TEXT -1 32 " and the analytical solution g. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "xvals := [seq(i*0.2,i=0.. 20)]:\nhvals := evalf(evalf[13](map(gn,xvals))):\ngvals := evalf(evalf [13](map(g,xvals))):\nlinalg[matrix]([[x,`gn(x)`,`g(x)`],\n seq([ xvals[i],hvals[i],gvals[i]],i=1..21)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#787%%\"xG%&gn(x)G%%g(x)G7%$\"\"!F-$\"++++]7!\"*F.7%$ \"\"#!\"\"$\"+7-N<8F0F57%$\"\"%F4$\"+hX]E8F0F:7%$\"\"'F4$\"+)3iT*=\"F0FD7%$\"#5F4$\"+zf)o0\"F0FI7%$\"#7F4$\"+KZ'3#*) !#5FN7%$\"#9F4$\"+n^ZHqFPFT7%$\"#;F4$\"+d5(=(\\FPFY7%$\"#=F4$\"+oR%=#G FPFhn7%$\"#?F4$\"+'\\/J\\'!#6$\"+&\\/J\\'F_o7%$\"#AF4$!+jwwz9FPFeo7%$ \"#CF4$!+:OD.NFPFjo7%$\"#EF4$!+u\"3EO&FPF_p7%$\"#GF4$!+5\\i-qFPFdp7%$ \"#IF4$!+f.;r$)FP$!+g.;r$)FP7%$\"#KF4$!+l^()=%*FPF`q7%$\"#MF4$!+_$**)4 5F0Feq7%$\"#OF4$!+e$3n.\"F0Fjq7%$\"#QF4$!+\"*e3=5F0F_r7%$\"#SF4$!+++++ &*FPFdrQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 47 "We plot the graphs of the numerical procedures " } {TEXT 271 2 "Un" }{TEXT -1 5 " and " }{TEXT 271 2 "Vn" }{TEXT -1 19 " \+ used to construct " }{TEXT 271 2 "gn" }{TEXT -1 11 " (shown in " } {TEXT 260 3 "red" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot(['Un(x)','x','gn(x)'],x =0..4,y,color=[brown,blue,red],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 426 350 350 {PLOTDATA 2 "6(-%'CURVESG6$7S7$$\"\"!F)$\"++++]7 !\"*7$$\"+m;')=()!#6$\"+,B\"F,7$$\"+<6m$ [#F6$\"+s*[S?\"F,7$$\"+(>%F6$\"+OrP?6F,7$ $\"+\">K'*)\\F6$\"+!Q0#o5F,7$$\"+Dt:5eF6$\"+.!Qd+\"F,7$$\"+\"fX(emF6$ \"+s))zD$*F67$$\"+DCh/vF6$\"+4\\Q:&)F67$$\"+L/pu$)F6$\"+1=A.wF67$$\"+; c0T\"*F6$\"+\\q%)QnF67$$\"+I,Q+5F,$\"+kSE.dF67$$\"+]*3q3\"F,$\"+5$RIg% F67$$\"+q=\\q6F,$\"+T,d\"\\$F67$$\"+fBIY7F,$\"+6_iVCF67$$\"+j$[kL\"F,$ \"+#[td:\"F67$$\"+`Q\"GT\"F,$\"+*)HPGM!#77$$\"+s]k,:F,$!+$GX&*H\"F67$$ \"+`dF!e\"F,$!+RD[,DF67$$\"+sgam;F,$!+dfoPQF67$$\"+7&F6 7$$\"+e/TM=F,$!+Os5pkF67$$\"+cK78>F,$!+&fC!3xF67$$\"+Uc-)*>F,$!+5daT!* 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\"+!oD=!pF67$F`q$\"+Jgz**fF67$Feq$\"+zV2!=&F67$Fjq$\"+b?AjUF67$F_r$\"+ O.CyLF67$Fdr$\"+(=sxW#F67$Fir$\"+%fp9f\"F67$F^s$\"*#>]1nF67$Fcs$!)kMvF F,7$Fhs$!*-))=4\"F,7$F]t$!*L[c&>F,7$Fbt$!*Vlj#GF,7$Fgt$!*pNDl$F,7$F\\u $!*,[PU%F,7$Fau$!*,UIC&F,7$Ffu$!*!=/UfF,7$F[v$!**>TXmF,7$F`v$!*2F5C(F, 7$Fev$!*ioI%yF,7$Fjv$!*&)4$f$)F,7$F_w$!*#3OV))F,7$Fdw$!*n]$e#*F,7$Fiw$ !*#HaF'*F,7$F^x$!*enm\"**F,7$Fcx$!+X`:95F,7$Fhx$!+QM1H5F,7$F]y$!+fq*f. \"F,7$Fby$!+xYlN5F,7$Fgy$!+T\"pv-\"F,7$F\\z$!+RUc55F,7$Faz$!*d^\"f)*F, 7$Ffz$!*+++]*F,-F[[l6&F][lFjdlF(F(-%+AXESLABELSG6$Q\"x6\"Q\"yFe^m-%*TH ICKNESSG6#\"\"#-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 4" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)+4;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"xGF&F(!\"\"F(\"\"%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+4*y = si n(2*x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"%F'%\"yGF'F'-%$sinG6#*& \"\"#F'%\"xGF'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 3" "6#/-%\"yG 6#\"\"!\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(2) = 1" "6#/-%\"yG6# \"\"#\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "First we f ind an analytical expression for the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "de := diff( y(x),x$2)+4*diff(y(x),x)+4*y(x)=sin(2*x);\nic := y(0)=3,y(2)=1;\ndsolv e(\{de,ic\},y(x));\nevalf[13](evalf[20](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*& \"\"%F2-F(6$F*F-F2F2*&F4F2F*F2F2-%$sinG6#,$*&F1F2F-F2F2" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"$/-F(6#\"\"#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"#D\"\")\"\"\"-%$e xpG6#,$*&\"\"#F-F'F-!\"\"F-F-*&#F-\"#;F-**F.F-F'F-,**&F+F--%%coshG6#\" \"%F-F-*&F+F--%%sinhGF=F-F4-%$cosGF=F4F,F4F-,&F;F-F@F4F4F-F4*&#F-F,F-- FC6#,$*&F3F-F'F-F-F-F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\" xG,(*&$\".+++++D\"!#8\"\"\"-%$cosG6#,$*&$\"\"#\"\"!F-F'F-F-F-!\"\"*&$ \".++++]7$!#7F--%$expG6#,$*&$F4F5F-F'F-F6F-F-*($\".cO<41N#!#6F-F;F-F'F -F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "L et's try " }{TEXT 0 7 "desolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "de := diff( y(x),x$2)+4*diff(y(x),x)+4*y(x)=sin(2*x);\nic := y(0)=3,y(2)=1;\ndesol ve(\{de,ic\});\ng := unapply(evalf[13](rhs(%)),x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\" \"*&\"\"%F2-F(6$F*F-F2F2*&F4F2F*F2F2-%$sinG6#,$*&F1F2F-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"$/-F(6#\"\"#\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"\"\"\")F+-% $cosG6#,$*&\"\"#F+F'F+F+F+!\"\"*&#\"#DF,F+-%$expG6#,$*&F2F+F'F+F3F+F+* (,(*&#F+F2F+-F86#\"\"%F+F+*&#F+\"#;F+*&-F.FAF+F@F+F+F+#F6FEF3F+F7F+F'F +F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG %&arrowGF(,(*&$\".++++]7$!#7\"\"\"-%$expG6#,$*&$\"\"#\"\"!F19$F1!\"\"F 1F1*($\".cO<41N#!#6F1F2F1F:F1F1*&$\".+++++D\"!#8F1-%$cosG6#,$*&$F8F9F1 F:F1F1F1F;F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Now we fi nd a numerical solution by the linear shooting method." }}{PARA 0 "" 0 "" {TEXT -1 28 "We start by considering the " }{TEXT 259 32 "associa ted initial value problem" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+4;" "6#,&*(%\"dG\"\"#%\"yG\"\" \"*&F%F(*$%\"xGF&F(!\"\"F(\"\"%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx+4*y = sin(2*x),y(0) = 3;" "6$/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"%F'% \"yGF'F'-%$sinG6#*&\"\"#F'%\"xGF'/-F,6#\"\"!\"\"$" }{TEXT -1 15 " , y \+ '(0) = 0, " }}{PARA 0 "" 0 "" {TEXT -1 14 "with solution " }{XPPEDIT 18 0 "u(x)" "6#-%\"uG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "We construct a numerical procedure " }{TEXT 271 2 "Un" } {TEXT -1 43 " for this solution defined on the interval " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2; " "6#/%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "de := diff(y(x),x$2)+4*diff (y(x),x)+4*y(x)=sin(2*x);\nic := y(0)=3,D(y)(0)=0;\nUn := desolveK2(\{ de,ic\},x=0..2,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG /,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2-F(6$F*F-F2F2* &F4F2F*F2F2-%$sinG6#,$*&F1F2F-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#icG6$/-%\"yG6#\"\"!\"\"$/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Next consider the " }{TEXT 259 33 "homogeneous initial value problem" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)+4;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"xGF&F(!\"\"F(\"\"%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+4*y = 0, y(0) = 0;" "6$/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"%F'%\"yGF'F'\"\"!/-F,6 #F-F-" }{TEXT -1 14 ", y '(0) = 1, " }}{PARA 257 "" 0 "" {TEXT -1 14 " with solution " }{XPPEDIT 18 0 "v(x)" "6#-%\"vG6#%\"xG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "We construct a numerical procedure \+ " }{TEXT 271 2 "Vn" }{TEXT -1 25 " defined on the interval " } {XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 19 " for this solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "d e := diff(y(x),x$2)+4*diff(y(x),x)+4*y(x)=0;\nic2 := y(0)=0,D(y)(0)=1; \nVn := desolveK2(\{de,ic2\},x=0..2,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*& \"\"%F2-F(6$F*F-F2F2*&F4F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic2G6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The constant needed to fo rm the linear combination of " }{XPPEDIT 18 0 "u(x)" "6#-%\"uG6#%\"xG " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(x)" "6#-%\"vG6#%\"xG" }{TEXT -1 45 " which solves the boundary value problem is " }{XPPEDIT 18 0 " (1-u(2))/v(2);" "6#*&,&\"\"\"F%-%\"uG6#\"\"#!\"\"F%-%\"vG6#F)F*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "c := evalf[13]((1-Un(2))/Vn(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\".dO<4cs\"!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "de := diff(y(x),x $2)+4*diff(y(x),x)+4*y(x)=sin(2*x);\nic3 := y(0)=3,D(y)(0)=c;\ngn := d esolveK2(\{de,ic3\},x=0..2,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2-F (6$F*F-F2F2*&F4F2F*F2F2-%$sinG6#,$*&F1F2F-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ic3G6$/-%\"yG6#\"\"!\"\"$/--%\"DG6#F(F)$\".dO<4cs\"! #6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Th e procedure " }{TEXT 271 2 "gn" }{TEXT -1 62 " now provides our soluti on, ( restricted to the interval from " }{XPPEDIT 18 0 "x = 0" "6#/%\" xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2" "6#/%\"xG\"\"#" } {TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 49 "We compare some values \+ of the numerical solution " }{TEXT 271 2 "gn" }{TEXT -1 31 " and the a nalytical solution g." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "xvals := [seq(i*0.2,i=0..10)]:\nhvals := evalf(evalf[13](map(gn,xvals))):\ngvals := evalf(evalf[13](map(g,xval s))):\nlinalg[matrix]([[x,`gn(x)`,'g(x)'],\n seq([xvals[i],hvals[ i],gvals[i]],i=1..11)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7.7%%\"xG%&gn(x)G-%\"gG6#F(7%$\"\"!F/$\"\"$F/$\"+++++I!\"*7%$\"\"#! \"\"$\"+>%Q48&F4F97%$\"\"%F8$\"+9=&=a&F4F>7%$\"\"'F8$\"+ek(Q9&F4FC7%$ \"\")F8$\"+=]@JWF4FH7%$\"#5F8$\"+)pWhl$F4FM7%$\"#7F8$\"+dsdMHF4FR7%$\" #9F8$\"+8*y*3BF4FW7%$\"#;F8$\"+/\\A&y\"F4Ffn7%$\"#=F8$\"+-Xd`8F4F[o7%$ \"#?F8$\"+++++5F4F`oQ(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "We plot the graphs of the numerical proce dures " }{TEXT 271 2 "Un" }{TEXT -1 5 " and " }{TEXT 271 2 "Vn" } {TEXT -1 19 " used to construct " }{TEXT 271 2 "gn" }{TEXT -1 11 " (sh own in " }{TEXT 260 3 "red" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot(['Un(x)','Vn( x)','gn(x)'],x=0..2,y,\n color=[magenta,blue,red],thickness=2) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 425 313 313 {PLOTDATA 2 "6(-%'CURVESG6 $7S7$$\"\"!F)$\"\"$F)7$$\"+M3VfV!#6$\"+q_E*)H!\"*7$$\"+#H[D:)F/$\"+S)o V'HF27$$\"+e0$=C\"!#5$\"+>y,AHF27$$\"+3RBr;F;$\"+5DymGF27$$\"+zjf)4#F; 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$\"+921t$)F/7$F9$\"+#*4?(o*F/7$$\"+LA`c9F;$\"+yqW)3\"F;7$F?$\"+v!)R'> \"F;7$$\"+W^\"\\)=F;$\"+Pb\"HH\"F;7$FD$\"+:KEz8F;7$$\"+Qiq'H#F;$\"+4&G 3X\"F;7$FI$\"+0@v9:F;7$FN$\"+)\\(*[i\"F;7$FS$\"+?:r5z\"F;7$Fep$\"+![Tfv\"F;7$Fjp$\"+')*)z> ]&F27$ FS$\"+pphGbF27$$\"+/&R3a$F;$\"+'\\$yVbF27$FX$\"+L<\\[bF27$$\"+:d#)pRF; $\"+B>KVbF27$Fgn$\"+x)f)GbF27$F\\o$\"+BFB$[&F27$Fao$\"+k@R0aF27$Ffo$\" +r&QRI&F27$F[p$\"+(ex#)=&F27$F`p$\"+Zg,r]F27$Fep$\"+@^o>\\F27$Fjp$\"+i Mr$y%F27$F_q$\"+m#G!>YF27$Fdq$\"+]8@pWF27$Fiq$\"+\\!4AI%F27$F^r$\"+[G- UTF27$Fcr$\"+8*R\\(RF27$Fhr$\"+pkZAQF27$F]s$\"+Gk*)fOF27$Fbs$\"+$QKQ\\ $F27$Fgs$\"+^v5_LF27$F\\t$\"+#\\F27$Fcw$\"+ R$3y#=F27$Fhw$\"+8HzD " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T asks" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 " Construct procedures which provide numerical solutions for the followi ng two point boundary value problems." }}{PARA 0 "" 0 "" {TEXT -1 3 "I f " }{TEXT 0 6 "dsolve" }{TEXT -1 85 " can find an analytical solution , compare this solution with your numerical solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " Q1" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2 ) = y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F'" }{TEXT -1 22 ", y(0) = 0, y(1) = 1 " }}{PARA 0 "" 0 "" {TEXT -1 36 "________ ____________________________" }}{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 36 "____________________ ________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\" *&F%F(*$%\"xGF&F(!\"\"F(F&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y \+ = x^2-1;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF',&*$%\"xG\"\"#F'F'F)" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 5" "6#/-%\"yG6#\"\"!\"\"&" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 10" "6#/-%\"yG6#\"\"\"\"#5" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "________________________ ____________" }}{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 36 "____________________________________" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 " 4*x^2" "6#*&\"\"%\"\"\"*$%\"xG\"\"#F%" }{TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+4*x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(! \"\"F(*&\"\"%F(F+F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+(4*x^2-1) *y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&,&*&\"\"%F'*$%\"xG\"\"#F'F'F 'F)F'%\"yGF'F'\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(1) = 1" "6#/- %\"yG6#\"\"\"F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(6) = 0" "6#/-%\"y G6#\"\"'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "________ ____________________________" }}{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 36 "____________________ ________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }