{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 "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 270 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 271 "Tim es" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "Grey Emphasis" -1 280 "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 "Heading 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 "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 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" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 46 "A procedure for solving 1st and 2 nd order DE's" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Version: 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 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 280 7 "DE sol.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 simi lar 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 48 "A procedure for solving 1st and 2nd order DE's: " } {TEXT 0 7 "desolve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "de solve: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 261 2 " " }{TEXT -1 66 " desolve( de ),\n desolve( de, y(x) ), \n desolve( \{de,ic\})" }{TEXT 262 2 ", " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 " desolve( \{de,ic\}, y(x))" }{TEXT 269 2 ", " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 " desolve( \{de,ic \}, rng )" }}{PARA 0 "" 0 "" {TEXT -1 34 " desolve( \{de,ic\}, y(x ), rng )" }{TEXT 264 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " de - " }{TEXT -1 98 " a 1st or 2nd order di fferential equation with the derivatives 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 59 " i nitial conditions in the form y(a) = b or D(y)(c)=d. " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+ procedure " }{TEXT 0 7 "desolve" }{TEXT -1 60 " attempts to solve a 1s t or 2nd order differential equation." }}{PARA 0 "" 0 "" {TEXT -1 54 " Both analytic and numerical solutions can be obtained." }}{PARA 0 "" 0 "" {TEXT -1 110 "This procedure provides an interface for the specif ic procedures considered separately in various worksheets. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Options:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "type=anal ytic/numeric/shoot" }}{PARA 0 "" 0 "" {TEXT -1 206 "This gives the cho ice between an analytcal solution and a numerical solution.\nIn additi on, numerical shooting method for solving two point boundary value pr oblems is available with the option \"type=shoot\"." }}{PARA 0 "" 0 " " {TEXT -1 53 "The default option is to give an analytical solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "type=an alytic, method=auto/sepvar/linear/homog/exact/laplace" }}{PARA 0 "" 0 "" {TEXT -1 91 "This option allows a choice of five analytical methods for the solution of 1st order DE's: " }}{PARA 15 "" 0 "" {TEXT 265 6 "sepvar" }{TEXT -1 38 " - separation of variables . . (using " }{TEXT 0 9 "desolveSV" }{TEXT -1 2 ")," }}{PARA 15 "" 0 "" {TEXT 266 6 "linea r" }{TEXT -1 93 " - the standard method of solution of a linear DE usi ng an integrating factor . . \n (using " }{TEXT 0 9 "desolveLN" } {TEXT -1 2 ")," }}{PARA 15 "" 0 "" {TEXT 267 5 "homog" }{TEXT -1 41 " \+ - solution of a homogeneous of the form " }{XPPEDIT 18 0 "dy/dx = f(y/ x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6#*&%\"yGF&%\"xGF(" }{TEXT -1 21 " by the substitution " }{XPPEDIT 18 0 "y = v*x;" "6#/%\"yG*&%\"vG \"\"\"%\"xGF'" }{TEXT -1 16 " . . \n (using " }{TEXT 0 9 "desolveHG " }{TEXT -1 2 ")," }}{PARA 15 "" 0 "" {TEXT 268 5 "exact" }{TEXT -1 29 " - solution of an exact DE: " }{XPPEDIT 18 0 "M(x,y)+N(x,y);" "6# ,&-%\"MG6$%\"xG%\"yG\"\"\"-%\"NG6$F'F(F)" }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "Diff(M(x,y),y) = Diff(N(x,y),x);" "6#/-%%DiffG6$-%\"MG6$%\"xG%\"yGF +-F%6$-%\"NG6$F*F+F*" }{TEXT -1 16 " . .\n (using " }{TEXT 0 9 "des olveEX" }{TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT 273 4 "auto" }{TEXT -1 78 " - the previous 4 methods are attempted in the order in which t hey are listed." }}{PARA 15 "" 0 "" {TEXT 274 7 "laplace" }{TEXT -1 73 " - the Laplace transform method can be attempted for a linear DE o f form " }{XPPEDIT 18 0 "dy/dx+c*y=f(x)" "6#/,&*&%#dyG\"\"\"%#dxG!\"\" F'*&%\"cGF'%\"yGF'F'-%\"fG6#%\"xG" }{TEXT -1 32 ", where c is independ ent of x. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The default method for \"type=analytic\" when the given DE has \+ order 1 is \"method=auto\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Only one type of 2nd order DE can be handled: a linear DE with constant coefficients." }}{PARA 0 "" 0 "" {TEXT -1 162 "The default method is to find the complementary solution from the auxiliary quadratic equation, and to to find the particular solution \+ by variation of parameters." }}{PARA 15 "" 0 "" {TEXT 275 6 "linear" } {TEXT -1 35 " - gives the default method (using " }{TEXT 0 9 "desolveC C" }{TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT 276 7 "laplace" }{TEXT -1 52 " - the Laplace transform method is attempted (using " }{TEXT 0 9 "desolveL2" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 106 "type=numeric, method=euler/impeuler/rk2/ rk3/rk4/rk4g/rk4t/rk5/rk6/rk7/rk8/rk45/rk56/rk78/taylor/finitediff" }} {PARA 15 "" 0 "" {TEXT -1 439 "Runge-Kutta methods with fixed step-siz e are available under the options \"method=euler\", which uses Euler's method, \"method=impeuler\" which uses the improved Euler method, \"m ethod=rk2\" . . \"method=rk8\" which use a Runge-Kutta method of the \+ given order N in rkN.\n\"method=rk4\" uses the classical order 4 Runge -Kutta method while \"method=rk4g\" uses Gill's order 4 Runge-Kutta me thod and \"method=rk4t\" uses the 3/8 order 4 Runge-Kutta method." }} {PARA 15 "" 0 "" {TEXT -1 149 "Runge-Kutta methods with variable step- size are available under the options \"method=rk45\", \"method=rk56\", \"method=rk78\" etc.\nThe procedures used are " }{TEXT 0 9 "desolveRK " }{TEXT -1 24 " for 1st order DE's and " }{TEXT 0 9 "desolveK2" } {TEXT -1 20 " for 2nd order DE's." }}{PARA 15 "" 0 "" {TEXT -1 100 "Ta ylor series methods are available by means of the option \"method=tayl or\".\nThe procedures used are " }{TEXT 0 9 "desolveTA" }{TEXT -1 24 " \022 for 1st order DE's and" }{TEXT 0 9 "desolveT2" }{TEXT -1 43 " for 2nd order DE's.\nThe procedure used is " }{TEXT 0 9 "desolveSH" } {TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 145 "A finite difference me thod for solving two point boundary value problems is available with t he option \"method=finitediff\".\nThe procedure used is " }{TEXT 0 9 " desolveFD" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 270 4 "Note" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 99 "Further details concerning options can be obtained from the inform ation provided with the specific " }{TEXT 0 9 "desolve??" }{TEXT -1 20 " procedure involved." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 16 "How to activate: " }{TEXT -1 134 "\nTo make the procedure active place the cursor anywh ere after the prompt [ > and press [Enter].\nYou can then close up th e subsection." }}{PARA 0 "" 0 "" {TEXT 270 4 "Note" }{TEXT -1 140 ": T his procedure only works in a particular situation if the correspondin g procedure mentioned under the method option above is also loaded." } }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "desolve: implementation " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16544 "desolve := proc()\n local Options,ff,typ,mthd,de,ic0,ic1,init cond,soln,prntflg,\n derivs,degree,j,num,order,rng,hasRange,ee,st artopts,\n xx,yy,x,y,yx,x2,x3,yx2,df,df1,df2,df3,la,a,b,c,pm1,pm2 ,vars,\n nvars,t,opts;\n\n if nargs>0 then \n ff := args[1 ]\n else\n error \"at least one argument must be supplied\"\n \+ end if;\n initcond := false;\n num := nops(ff);\n if type(ff,\{ set(equation),list(equation)\}) then\n if num=3 then\n ff := map(_u -> if has(_u,D@@2) then convert(_u,diff) else _u end if,ff) ;\n de := op(1,ff);\n ic0 := op(2,ff);\n ic1 : = op(3,ff);\n if not has(de,diff) then\n de := op(2 ,ff);\n ic0 := op(1,ff);\n end if;\n if not has(de,diff) then\n de := op(3,ff);\n ic1 := op (2,ff);\n end if;\n initcond := true;\n elif num= 2 then\n ff := map(_u -> if has(_u,D) then convert(_u,diff) el se _u end if,ff);\n de := op(1,ff);\n ic0 := op(2,ff); \n if not has(de,diff) then\n de := op(2,ff);\n \+ ic0 := op(1,ff);\n end if;\n initcond := true; \n end if;\n elif type(ff,equation) then\n if has(de,'D') \+ then de := convert(de,'diff') else de := ff end if;\n else\n er ror \"the 1st argument, %1, is invalid .. it should be an equation or \+ a set (or list) of 2 or 3 equations\",ff;\n end if;\n\n startopts \+ := 2;\n if nargs>1 then\n ee := args[2];\n if type(ee,func tion) 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 := 3;\n else\n error \"the 2nd argument, %1, has incorrect form for the dependent variable\",ee;\n end if;\n end if;\n end if;\n\n # Check out the derivatives in t he DE.\n derivs := indets(de,'specfunc(anything,diff)');\n if deri vs=\{\} then\n error \"the 1st argument, %1, is invalid .. it sho uld be a first or second order differential equation or a set (or list ) containing a differential equation and one or two initial condition \",ff;\n end if;\n nvars := nops(indets(derivs,name));\n if nvar s<>1 then\n if nvars=0 then\n error \"there is a problem \+ with the independent variable occurring in the derivative(s)\";\n \+ else\n error \"there should only be one independent variable \+ in the differential equation\"\n end if;\n end if;\n nvars := nops(indets(derivs,anyfunc(name)));\n if nvars<>1 then\n if nv ars=0 then\n error \"there is a problem with the dependent var iable occurring in the derivative(s)\"\n else\n error \"t here should only be one dependent variable in the differential equatio n\"\n end if;\n end if;\n order := nops(derivs);\n if order <>1 and order<>2 then\n error \"there are too many derivatives in the differential equation .. note that the differential equation shou ld have order 1 or 2\"\n end if; \n\n if order=1 then\n df : = op(1,derivs);\n if type(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 \"the 1st argument %1, in the derivati ve, %2, is invalid .. it should be the 'unknown' dependent variable\", yx,df;\n end if; \n x := op(2,df);\n if not ty pe(x,name) then\n error \"the 2nd argument %1, in the deriv ative, %2, is invalid .. it should be the independent variable\",x,df; \n end if; \n else\n error \"the derivative, %1, \+ does not make sense\",df;\n end if;\n else\n (df2,df1) := \+ selectremove(_U->has([op(_U)],diff),derivs);\n if nops(df2)<>1 or nops(df1)<>1 then \n error \"the derivatives, %1, do not make sense\",derivs;\n end if; \n (df2,df1) := (op(df2),op(df1)) ;\n\n # Get the arguments in the derivatives.\n if type(df1, function) and op(0,df1)=diff and nops(df1)=2 then\n yx := op(1 ,df1);\n if not type(yx,anyfunc(name)) then\n error \"the 1st argument %1, in the derivative, %2, is invalid .. it should be the 'unknown' dependent variable\",yx,df1;\n end if; \n \+ x := op(2,df1);\n if not type(x,name) then\n \+ error \"the 2nd argument %1, in the derivative, %2, is invalid .. it s hould be the independent variable\",x,df1;\n end if; \n e lse\n error \"the derivative, %1, does not make sense\",df1;\n end if;\n\n if type(df2,function) and nops(df2)=2 and op(0, df2)='diff' then\n (df3,x3) := selectremove(has,\{op(df2)\},di ff);\n if nops(df3)<>1 or nops(x3)<>1 then \n error \"the derivative, %1, does not make sense\",df2;\n end if;\n \+ (df3,x3) := (op(df3),op(x3));\n if type(df3,function) \+ and nops(df3)=2 and op(0,df3)='diff' then\n yx2 := op(1,df3 );\n if not type(yx2,anyfunc(name)) then\n er ror \"the 1st argument %1, in the derivative, %2, is invalid .. it sho uld be the 'unknown' dependent variable\",yx2,df3;\n end if ; \n x2 := op(2,df2);\n if not type(x2,name) the n\n error \"the 2nd argument %1, in the derivative, %2, \+ is invalid .. it should be the independent variable\",x2,df3;\n \+ end if; \n if not x2=x3 then\n error \"t he 2nd arguments, %1 and %2 in the derivatives %3 and %4 should be the same\",x2,x3,df2,df3;\n end if;\n else\n \+ error \"the derivative, %1, does not make sense\",df3;\n end if\n else\n error \"the derivative, %1, does not make se nse\",df2;\n end if;\n\n # Arguments in the 2 derivatives mu st be the same.\n if x2<>x or yx2<>yx then\n error \"the \+ differential equation contains inconsistent arguments\"\n end if; \n end if;\n y := op(0,yx);\n vars := indets(de,name);\n if me mber(y,vars) then\n error \"%1 and %2 cannot both appear in the d ifferential equation\",yx,y;\n end if;\n if op(1,yx)<>x then\n \+ error \"the derivatives do not make sense\"\n end if;\n\n if sta rtopts = 3 then \n if x<>xx or y<>yy then\n error \"canno t solve the differential equation for %1\",ee;\n end if;\n end \+ if;\n\n hasRange := false;\n if nargs>startopts and\n (type(a rgs[startopts],range) or\n type(args[startopts],name=ran ge)) then\n rng := args[startopts];\n startopts := start opts + 1;\n hasRange := true;\n end if;\n\n # Get the option s.\n # Set the default values to start with.\n prntflg := false;\n typ := 'analytic';\n mthd := 'auto';\n\n Options:=[];\n if na rgs>=startopts then\n Options:=[args[startopts..nargs]];\n i f not type(Options,list(equation)) then\n error \"each optiona l argument must be an equation\"\n end if;\n if hasoption(Op tions,'info','prntflg') then \n if prntflg<>true then prntflg \+ := false end if;\n end if;\n if hasoption(Options,'type','ty p','Options') then\n if not (typ='analytic' or typ='numeric' o r typ='shoot') then\n error \"\\\"type\\\" must be 'analyti c','numeric' or 'shoot'\"\n end if;\n if typ='numeric' then mthd := 'rk67' end if; # default method\n end if;\n if hasoption(Options,method,'mthd') then\n if not member(mthd,\{ 'auto','sepvar','linear','homog','exact',\n 'bernoulli','inv linear','missingvar','laplace','euler','impeuler',\n 'rk2',' rk2a','rk2b','rk3','rk3a','rk3b','rk3c','rk3d','rk3e','rk3f','rk4',\n \+ 'rk4a','rk4b','rk4c','rk4d','rk4e','rk4f','rk4g','rk4h','rk 4k','rk5',\n 'rk5a','rk5b','rk5c','rk5d','rk5e','rk6','rk6a ','rk6b','rk6c','rk6d',\n 'rk7','rk7a','rk7b','rk7c','rk8', 'rk8a','rk8b','rk8c','rk8d','rk9',\n 'rk9a','rk9b','rk9c',' rk10','rk10a','rk10b','rk45','rk45b','rk56',\n 'rk56b','rk5 6c','rk67','rk67b','rk78','rk78b','rk78c','rk78d','rk89',\n \+ 'rk89b','rk910','taylor','finitediff'\}) then\n error \"\\ \"method\\\" must be 'auto','sepvar','linear','homog','exact','bernoul li','invlinear','missingvar','laplace','euler','impeuler','rk2','rk2a' ,'rk2b','rk3','rk3a','rk3b','rk3c','rk3d','rk3e','rk3f','rk4','rk4a',' rk4b','rk4c','rk4d','rk4e','rk4f','rk4g','rk4h','rk4k','rk5','rk5a','r k5b','rk5c','rk5d','rk5e','rk6','rk6a','rk6b','rk6c','rk6d','rk7','rk7 a','rk7b','rk7c','rk8','rk8a','rk8b','rk8c','rk8d','rk9','rk9a','rk9b' ,'rk9c','rk10','rk10a','rk10b','rk45','rk45b','rk56','rk56b','rk56c',' rk67','rk67b','rk78','rk78b','rk78c','rk78d','rk89','rk89b','rk910','t aylor' or 'finitediff'\"\n end if;\n if member(mthd,\{ 'auto','sepvar','linear','homog','exact',\n 'bernoulli','in vlinear','missingvar','taylor',\n 'finitediff','laplace'\}) then \n Options:=convert(\{op(Options)\}minus\{'method'=mt hd\},list);\n end if;\n end if;\n if hasoption(Optio ns,'alpha','alph') then\n mthd := 'customrk2';\n end if; \n end if;\n\n if typ='numeric 'and mthd<>'finitediff' and not has Range then\n error \"a range for the numerical solution must be s upplied\"\n end if; \n\n if order=1 then #1\n if typ='analyt ic' then #2\n if mthd='auto' then #3\n opts := Options; \n hasoption(opts,'scheme','t','opts');\n soln := trap error(desolveSV(ff,op(opts)));\n if soln=lasterror then #4\n \+ if lasterror<>\"the variables in the ODE cannot be separated \" then #5\n error lasterror\n else\n \+ if prntflg then\n print(``);\n print(`T he DE does not have separable variables`);\n end if;\n \+ opts := Options;\n hasoption(opts,'format','t','o pts');\n hasoption(opts,'scheme','t','opts');\n \+ soln := traperror(desolveLN(ff,op(opts)));\n if soln=las terror then #6 \n if lasterror<>\"the DE is not linear\" then #7\n error lasterror\n else\n \+ if prntflg then\n print(``);\n \+ print(`The DE is not linear`);\n end if;\n \+ opts := Options;\n hasoption(opts,'schem e','t','opts');\n soln := traperror(desolveHG(ff,op(op ts)));\n if soln=lasterror then #8 \n \+ if lasterror<>\"the DE is not homogeneous\" then #9\n \+ error lasterror\n else\n i f prntflg then\n print(``);\n \+ print(`The DE is not homogeneous`);\n end if; \n opts := Options;\n hasoptio n(opts,'format','t','opts');\n soln := traperror(d esolveNL(ff,op(opts)));\n if soln=lasterror then # 10\n if lasterror<>\"the DE is not inverse linea r\" then #11\n error lasterror\n \+ else\n if prntflg then\n \+ print(``);\n print(`The DE is not inverse linear`);\n end if;\n \+ soln := traperror(desolveBN(ff,op(opts)));\n \+ if soln=lasterror then #12\n if \+ lasterror<>\"the DE is not of Bernoulli type\" then #13\n \+ error lasterror\n else\n \+ if prntflg then\n \+ print(``);\n print(`The DE is not of \+ Bernoulli type`);\n end if;\n \+ soln := \n traperror(deso lveEX(ff,op(opts)));\n if soln=lasterror t hen #14\n if lasterror<>\"the DE is not \+ exact\" then #15\n error lasterror\n \+ else\n if prntflg then\n print(``);\n \+ print(`The DE is not exact`);\n \+ end if;\n error \+ \"unable to solve the ODE analytically\"\n \+ end if #15\n end if #14\n \+ end if #13\n end if #12\n \+ end if #11\n end if #10\n \+ end if #9\n end if #8\n end if \+ #7\n end if #6 \n end if #5\n \+ end if; #4\n return soln;\n elif mthd='sepvar' then\n \+ return desolveSV(ff,op(Options));\n elif mthd='linear' the n\n return desolveLN(ff,op(Options));\n elif mthd='homog ' then\n return desolveHG(ff,op(Options));\n elif mthd=' exact' then\n return desolveEX(ff,op(Options));\n elif m thd='bernoulli' then\n return desolveBN(ff,op(Options));\n \+ elif mthd='invlinear' then\n return desolveNL(ff,op(Options ));\n elif mthd='laplace' then\n return desolveLP(ff,op( Options));\n else\n error \"method, %1, is incompatible \+ with type, %2, or order %3 differential equations\",mthd,typ,order;\n \+ end if;\n else # typ='numeric'\n if mthd='taylor' then \n return desolveTA(ff,rng,op(Options));\n elif member(m thd,\{euler,impeuler,customrk2,'rk2','rk2a','rk2b','rk3','rk3a',\n \+ 'rk3b','rk3c','rk3d','rk3e','rk3f','rk4','rk4a','rk4b','rk4c', 'rk4d',\n 'rk4e','rk4f','rk4g','rk4h','rk4k','rk5','rk5a',' rk5b','rk5c','rk5d',\n 'rk5e','rk6','rk6a','rk6b','rk6c','r k6d','rk7','rk7a','rk7b','rk7c',\n 'rk8','rk8a','rk8b','rk8 c','rk8d','rk9','rk9a','rk9b','rk9c','rk10',\n 'rk10a','rk1 0b','rk45','rk45b','rk56','rk56b','rk56c','rk67','rk67b',\n \+ 'rk78','rk78b','rk78c','rk78d','rk89','rk89b','rk910'\}) then\n \+ return desolveRK(ff,rng,op(Options));\n else\n error \+ \"method, %1, is incompatible with type, %2, or order %3 differential \+ equations\",mthd,typ,order;\n end if;\n end if;\n else # o rder=2\n if typ='analytic' then\n if mthd='auto' then\n \+ soln := traperror(desolveCC(ff,op(Options)));\n if \+ soln=lasterror then \n if lasterror<>\"the DE is not line ar\" and\n lasterror<>\"the DE does not have order 2\" \+ and \n [lasterror][1]<>\"the coefficients must be indepe ndent of the variable %1\" then \n error lasterror;\n \+ else\n if prntflg then\n print (``);\n if lasterror=\"the DE is not linear\" then \n \+ print(`The DE is not linear`);\n el if lasterror=\"the DE does not have order 2\" then\n \+ print(`The DE does not have order 2`); \n else\n \+ print(`The DE does not have constant coefficients`);\n end if;\n end if;\n soln \+ := traperror(desolveMV(ff,op(Options)));\n if soln=laste rror then \n if lasterror<>\"the DE does not have a m issing variable\" then \n error lasterror\n \+ else\n if prntflg then\n \+ print(``);\n print(`The DE does not have a missin g variable`);\n end if;\n error \" unable to solve the ODE analytically\"\n end if\n \+ end if\n end if\n end if;\n \+ return soln;\n elif mthd='linear' then\n return des olveCC(ff,op(Options));\n elif mthd='missingvar' then \n \+ return desolveMV(ff,op(Options));\n elif mthd='laplace' \+ then\n return desolveL2(ff,op(Options));\n else\n \+ error \"method, %1, is incompatible with type, %2, or order % 3 differential equations\",mthd,typ,order;\n end if;\n el if typ='numeric' then\n if mthd='taylor' then\n ret urn desolveT2(ff,rng,op(Options));\n elif member(mthd,\{'rk45' ,'rk56','rk67','rk78','rk89'\}) then\n return desolveK2(ff, rng,op(Options));\n elif mthd='finitediff' then\n r eturn desolveFD(ff,op(Options));\n else\n error \"m ethod, %1, is incompatible with type, %2, or order %3 differential equ ations\",mthd,typ,order;\n end if;\n else # typ='shoot'\n if startopts=2 then\n return desolveSH(ff,op(Optio ns))\n else\n return desolveSH(ff,rng,op(Options)) \n end if;\n end if;\n end if;\nend proc: # desolve" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT 0 7 "desolve" }{TEXT -1 18 ": order 1 examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx=-y*sin(x), y(0)=1" "6$/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"yGF&-%$sinG6# %\"xGF&F(/-F+6#\"\"!F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "We find both analytical and numerical solutions, and compare their values when " }{XPPEDIT 18 0 "x = Pi/2;" "6#/%\"xG*&%#PiG\"\"\"\"\"#! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "First we find the analytical solution." }}{PARA 0 "" 0 "" {TEXT -1 81 "The differential equation has separable variables, so we use the default option \"" } {TEXT 280 13 "method=sepvar" }{TEXT -1 37 "\", which could of course, \+ be omitted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 117 "de := diff(y(x),x)=-y(x)*sin(x);\nic := y(0)=1;\nd esolve(\{de,ic\},y(x),method=sepvar,info=true);\nf := unapply(rhs(%),x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF, ,$*&F)\"\"\"-%$sinGF+F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG /-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DThe~DE~has~separable~variables~.~.~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(%\"yG!\"\"F),$-F%6 $-%$sinG6#%\"xGF1F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"yG,&-%$cosG6#%\"xG\"\"\"&%\"CG6# F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG*&&%\"CG6#\"\"#\"\"\"-%$expG6#-%$cosG6#%\"xGF*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~ ~G/&%\"CG6#\"\"#-%$expG6#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,&\"\"\"!\" \"-%$cosGF&F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$% )operatorG%&arrowGF(-%$expG6#,&\"\"\"!\"\"-%$cosG6#9$F0F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Now we co nstruct a numerical solution over the interval " }{XPPEDIT 18 0 "x = 0 " "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2" "6#/%\"xG \"\"#" }{TEXT -1 50 ", using an order 7-8 adaptive Runge-Kutta method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "de := diff(y(x),x)=-y(x)*sin(x);\nic := y(0)=1;\nfn \+ := desolve(\{de,ic\},y(x),x=0..2,type=numeric,method=rk78,info=true); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$ *&F)\"\"\"-%$sinGF+F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]omethod:~Verne r~13~stage,~order~7-8~..~Maple's~dverk78~coefficientsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Ran~embedded~order~7~scheme~provides~error~contro lG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%[o8~extra~stages~are~used~to~co nstruct~the~interpolation~procedureG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&+! [!#A%2abs~err~bound~->~G$\"\"\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6( %%stepG\"\"\"%$~~~G$\"+++++5!#6F%7$F&$\"0Cmm,+&****!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Fincreasing~step-size~by~a~factor~of~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs ~err~estimate~->~G$\"&++'!#@%2abs~err~bound~->~G$\"&&****!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"#%$~~~G$\"+++++]!#6F%7$$\"+++++gF ($\"0yF*z:-#)**!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Fincreasing~ste p-size~by~a~factor~of~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&S1#!#:%2abs ~err~bound~->~G$\"&?)**F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\" \"$%$~~~G$\",++++]#!#6F%7$$\",++++5$F($\"0k(Qox^M&*!#:" }}{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~e stimate~->~G$\"&-1&!#:%2abs~err~bound~->~G$\"&X`*F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6(%%stepG\"\"%%$~~~G$\"0uKc6w*RF!#:F%7$$\"0uKc6w*ReF($ \"0gr6))3FZ)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adj ust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Freducing~step-size~and~repeating~stepG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&tN$!#:%2abs~err~b ound~->~G$\"&FZ)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"&%$~~ ~G$\"0$oOY?\\#G#!#:F%7$$\"0d**>;oC7)F($\"0Y](Q!f)=tF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~e stimate~->~G$\"&'HV!#:%2abs~err~bound~->~G$\"&*=tF&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6(%%stepG\"\"'%$~~~G$\"0(pH*QXiI#!#:F%7$$\"0lH^NrG/\"!# 9$\"0SGT\"\\/)3'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to ~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&~G$\"&!)3'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"(%$~~~G $\"0,8jp%R;A!#:F%7$$\"0&4wC3^k7!#9$\"0f1FR-M(\\F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&lE\"!#:%2abs~err~bound~->~G$\"&M(\\F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\")%$~~~G$\"0'p**>]`RA!#:F%7$$\"0lgnKk%)[\"!#9 $\"0lvLJBT*RF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adj ust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&;&R!#;%2abs~err~bound~- >~G$\"&T*R!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"*%$~~~G$\" 0O'oDKV\"R#!#:F%7$$\"0HH$\\wgF~G$\"&(>J!#;%2abs~err~bound~->~G$\"&p9$!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#5%$~~~G$\"/rq1N#Rs#!#9F%7$$\"0+++++++#F($\"0[ d;xrkU#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*last~stepG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 89 "As far as the eye 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\"+zjf)4#F\\\\l$\"+E4*Hy*F\\\\l7$$\"+'4;[\\#F\\\\l$\"+(z[^p*F\\\\l7$$ \"+j'y]!HF\\\\l$\"+$fU'*e*F\\\\l7$$\"+'zs$HLF\\\\l$\"+N)pcY*F\\\\l7$$ \"+8iI_PF\\\\l$\"+VU)yK*F\\\\l7$$\"+<_M(=%F\\\\l$\"+'Q6B<*F\\\\l7$$\"+ 4y_qXF\\\\l$\"+^5\\C!*F\\\\l7$$\"+]1!>+&F\\\\l$\"+yJ)p%))F\\\\l7$$\"+] Z/NaF\\\\l$\"+[e-e')F\\\\l7$$\"+]$fC&eF\\\\l$\"+\"ztoY)F\\\\l7$$\"+'z6 :B'F\\\\l$\"+,t]'G)F\\\\l7$$\"+<=C#o'F\\\\l$\"+-\\zk!)F\\\\l7$$\"+n#pS 1(F\\\\l$\"+^&z<(yF\\\\l7$$\"+j`A3vF\\\\l$\"+J.TUwF\\\\l7$$\"+n(y8!zF \\\\l$\"+ra*fV(F\\\\l7$$\"+j.tK$)F\\\\l$\"+&4Ap?(F\\\\l7$$\"+)3zMu)F\\ \\l$\"+0p@()pF\\\\l7$$\"+#H_?<*F\\\\l$\"+#H?tv'F\\\\l7$$\"+!G;cc*F\\\\ l$\"+=-UYlF\\\\l7$$\"+4#G,***F\\\\l$\"+97**>jF\\\\l7$$\"+!o2J/\"!\"*$ \"+!3-o3'F\\\\l7$$\"+%Q#\\\"3\"Fccl$\"+$4=h)eF\\\\l7$$\"+;*[H7\"Fccl$ \"+QWRscF\\\\l7$$\"+qvxl6Fccl$\"+WE[baF\\\\l7$$\"+`qn27Fccl$\"+&*>jZ_F \\\\l7$$\"+cp@[7Fccl$\"+3Y1^]F\\\\l7$$\"+3'HKH\"Fccl$\"+-Q[Q[F\\\\l7$$ \"+xanL8Fccl$\"+Vt*Gl%F\\\\l7$$\"+v+'oP\"Fccl$\"+K*42Y%F\\\\l7$$\"+S<* fT\"Fccl$\"+6G3#H%F\\\\l7$$\"+&)Hxe9Fccl$\"+2:\"R6%F\\\\l7$$\"+.o-*\\ \"Fccl$\"+_:G_RF\\\\l7$$\"+TO5T:Fccl$\"+]!\\'*y$F\\\\l7$$\"+U9C#e\"Fcc l$\"+A6$pj$F\\\\l7$$\"+1*3`i\"Fccl$\"+*)er$[$F\\\\l7$$\"+$*zym;Fccl$\" +)osDM$F\\\\l7$$\"+^j?4Fccl$\"+OsTAEF\\\\l7$$\"+/Ua c>Fccl$\"+v>BDDF\\\\l7$Fez$\"+s " 0 "" {MPLTEXT 1 0 33 "xx := Pi/2;\nevalf(f(xx));\nfn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$*&\"\"#!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+7WzyO!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7 WzyO!#5" }}}{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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 272 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = x^3+1, y(1) = 2" "6$/,&*&%#dy G\"\"\"%#dxG!\"\"F'%\"yGF',&*$%\"xG\"\"$F'F'F'/-F*6#F'\"\"#" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "We find both an analytical \+ and a numerical solution, and compare their values when " }{XPPEDIT 18 0 "x = 2*sqrt(2);" "6#/%\"xG*&\"\"#\"\"\"-%%sqrtG6#F&F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "de := x*diff(y(x),x)+y(x)=x^3+1;\nic := y(1)=2;\ndes olve(\{de,ic\},y(x),method=linear,info=true);\nf := unapply(rhs(%),x); \nfn := desolve(\{de,ic\},y(x),x=1..3,type=numeric,method=rk78);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&*&%\"xG\"\"\"-%%diffG6$-%\"yG 6#F(F(F)F)F-F),&*$)F(\"\"$F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#icG/-%\"yG6#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-%\"yG6 #%\"xGF,\"\"\"*&F,!\"\"F)F-F-*&,&*$)F,\"\"$F-F-F-F-F-F,F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integr ating~factor~.~.~~G-%$expG6#-%$IntG6$*&\"\"\"F+%\"xG!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6# %\"xG\"\"\"F(F)-%$IntG6$,&*$)F(\"\"$F)F)F)F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"xG \"\"\"F(F),(*&\"\"%!\"\"F(F,F)F(F)&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~i nitial~condition~.~.~~G/&%\"CG6#\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,( *&\"\"%!\"\"F'\"\"$\"\"\"F-F-*(F,F-F*F+F'F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&#\"\"\"\" \"%F/*$)9$\"\"$F/F/F/F/F/*&#F4F0F/*&F/F/F3!\"\"F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot ([f(x),'fn(x)'],x=1..3,color=[red,green],thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 267 168 168 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$\"\" \"\"\"!$\"\"#F*7$$\"3ALLL3VfV5!#<$\"3zPMam=\"G+#F07$$\"3smm\"H[D:3\"F0 $\"3%eq$[9$H(4?F07$$\"3XLL$e0$=C6F0$\"3\"o\"[/pKLA?F07$$\"3QLL$3RBr;\" 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"" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = sin(y)/(x +1),y(0) = 1;" "6$/*&%#dyG\"\"\"%#dxG!\"\"*&-%$sinG6#%\"yGF&,&%\"xGF&F &F&F(/-F-6#\"\"!F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "We find both analytical and numerical solutions, and compare their value s when " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 38 "First we find the analytical solution." } }{PARA 0 "" 0 "" {TEXT -1 56 "The differential equation again has sepa rable variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 84 "de := diff(y(x),x)=sin(y(x))/(x+1);\nic := y(0 )=1;\nsol := desolve(\{de,ic\},info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&-%$sinG6#F)\"\"\",&F, F1F1F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DThe~DE~has~separable~variables~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(-%$sinG6#%\"yG!\"\"F,-F%6$*&F( F(,&%\"xGF(F(F(F-F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#,&-%$cscG6#%\"yG\"\"\"-%$cotGF*!\" \",&-F%6#,&%\"xGF,F,F,F,&%\"CG6#F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&,&\"\"\"!\"\"-%$cosG6#% \"yGF'F'-%$sinGF+F(F(*&&%\"CG6#\"\"#F',&%\"xGF'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~G/&%\"CG6#\" \"#*&-%$sinG6#\"\"\"F-,&-%$cosGF,F-F-F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/-%\"yG6# %\"xG-%'arctanG6$,$**\"\"#\"\"\",&F)F0F0F0F0-%$sinG6#F0F0,,*(F/F0-%$co sGF4F0F)F0F0*&F7F0)F)F/F0F0*$F:F0!\"\"*&F/F0F)F0F " 0 "" {MPLTEXT 1 0 102 "f := unapply(rhs(%),x):\nplot([f(x),Pi],x=0 ..25,y=0..3.3,\n linestyle=[1,2],color=[red,black]);" }} {PARA 13 "" 1 "" {GLPLOT2D 292 210 210 {PLOTDATA 2 "6&-%'CURVESG6%7Y7$ $\"\"!F)$\"\"\"F)7$$\"3ummTN@Ki8!#=$\"34eN-Z1/66!#<7$$\"3\\LL$3FWYs#F/ 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$FHFe]l7$FRFe]l7$FfnFe]l7$F[oFe]l7$F`oFe]l7$FeoFe]l7$FjoFe]l7$F_pFe]l7 $FdpFe]l7$FipFe]l7$F^qFe]l7$FcqFe]l7$FhqFe]l7$F]rFe]l7$FbrFe]l7$FgrFe] l7$F\\sFe]l7$FasFe]l7$FfsFe]l7$F\\tFe]l7$FatFe]l7$FftFe]l7$F[uFe]l7$F` uFe]l7$FeuFe]l7$FjuFe]l7$F_vFe]l7$FdvFe]l7$FivFe]l7$F^wFe]l7$FcwFe]l7$ FhwFe]l7$F]xFe]l7$FbxFe]l7$FgxFe]l7$F\\yFe]l7$FayFe]l7$FfyFe]l7$F[zFe] l7$F`zFe]l7$FezFe]l7$FjzFe]l7$F_[lFe]l7$Fd[lFe]l7$Fi[lFe]l7$F^\\lFe]l7 $Fc\\lFe]l-Fh\\l6&Fj\\lF)F)F)-F_]l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q\"yF` al-%%VIEWG6$;F(Fc\\l;F($\"#L!\"\"" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Now we construct a numerical so lution over the interval " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 5" "6#/%\"xG\"\"&" }{TEXT -1 50 ", using an order 7-8 adaptive Runge-Kutta method. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "de := dif f(y(x),x)=sin(y(x))/(x+1);\nic := y(0)=1;\nfn := desolve(\{de,ic\},x=0 ..5,type=numeric,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d eG/-%%diffG6$-%\"yG6#%\"xGF,*&-%$sinG6#F)\"\"\",&F,F1F1F1!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The following t able compares some values given by numerical solution \"" }{TEXT 280 2 "fn" }{TEXT -1 48 "\" with values given by the analytical solution \+ \"" }{TEXT 280 1 "f" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "matrix([[`x`,`f(x)`,`fn (x)`,`f(x)-fn(x)`],\nevalf(seq([Pi*i/5,f(Pi*i/5),fn(Pi*i/5),f(Pi*i/5)- fn(Pi*i/5)],\ni=1..7),15)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7*7&%\"xG%%f(x)G%&fn(x)G%+f(x)-fn(x)G7&$\"0ezrI&=$G'!#:$\"0)prZ! GSX\"!#9$\"02;x/GSX\"F2$\"#\"*F27&$\"0#fVhqjc7F2$\"0ktW6y&y " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 4" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -y+sin (x)+cos(x),y(0) = 1;" "6$/*&%#dyG\"\"\"%#dxG!\"\",(%\"yGF(-%$sinG6#%\" xGF&-%$cosG6#F.F&/-F*6#\"\"!F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "We find both analytical and numerical solutions, and comp are their values when " }{XPPEDIT 18 0 "x = sqrt(3);" "6#/%\"xG-%%sqrt G6#\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 38 "First we find the analytical solution." }}{PARA 0 "" 0 "" {TEXT -1 59 "The differential equation is linear, so we use th e option \"" }{TEXT 280 13 "method=linear" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "d e := diff(y(x),x)=-y(x)+sin(x)+cos(x);\nic := y(0)=1;\ndesolve(\{de,ic \},method=linear,info=true);\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,(F)!\"\"-%$sinGF+ \"\"\"-%$cosGF+F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\" \"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-%\"yG6#%\"xGF,\"\"\"F)F -,&-%$sinGF+F--%$cosGF+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-% $IntG6$\"\"\"%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~ ~~=~~~G-%$expG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"xG\"\"\"-%$expGF'F)-%$IntG6$*& ,&-%$sinGF'F)-%$cosGF'F)F)F*F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"xG\"\"\"-%$expGF'F) ,&*&F*F)-%$sinGF'F)F)&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition ~.~.~~G/&%\"CG6#\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&-%$sinGF&\"\"\"-%$expG 6#,$F'!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$% )operatorG%&arrowGF(,&-%$sinG6#9$\"\"\"-%$expG6#,$F0!\"\"F1F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Now we fi nd the numerical solution over the interval from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2" "6#/%\"x G\"\"#" }{TEXT -1 32 " using the Taylor series method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "de := \+ diff(y(x),x)=-y(x)+sin(x)+cos(x);\nic := y(0)=1;\ngn := desolve(\{de,i c\},x=0..2,type=numeric,method=taylor,info=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,(F)!\"\"-%$sinGF+\"\" \"-%$cosGF+F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"! \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"#%%~~~~Gf*6 $%\"xG%\"yG6\"6$%)operatorG%&arrowGF*,&*&$F$\"\"!\"\"\"-%$sinG6#9$F2! \"\"9%F2F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"$%% ~~~~Gf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF*,(*&$\"\"\"\"\"!F1-%$cosG 6#9$F1!\"\"*&$F1F2F19%F1F7-%$sinGF5F1F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"%%%~~~~Gf*6$%\"xG%\"yG6\"6$%)operatorG %&arrowGF*9%F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\" \"&%%~~~~Gf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF*,(*&$\"\"\"\"\"!F19% F1!\"\"-%$sinG6#9$F1-%$cosGF7F1F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"'%%~~~~Gf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF*,& *&$\"\"#\"\"!\"\"\"-%$sinG6#9$F3!\"\"9%F3F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"(%%~~~~Gf*6$%\"xG%\"yG6\"6$%)operatorG %&arrowGF*,(*&$\"\"\"\"\"!F1-%$cosG6#9$F1!\"\"*&$F1F2F19%F1F7-%$sinGF5 F1F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\")%%~~~~Gf *6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF*9%F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"*%%~~~~Gf*6$%\"xG%\"yG6\"6$%)operatorG %&arrowGF*,(*&$\"\"\"\"\"!F19%F1!\"\"-%$sinG6#9$F1-%$cosGF7F1F*F*F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"#5%%~~~~Gf*6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF*,&*&$\"\"#\"\"!\"\"\"-%$sinG6#9$F3!\"\"9%F3 F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&d v#!#J%2abs~err~bound~->~G$\"\"\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 (%%stepG\"\"\"%$~~~G$\"+++++5!#6F%7$F&$\"/M3n'\\++\"!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%Fincreasing~step-size~by~a~factor~of~5G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 &%5abs~err~estimate~->~G$\"&vj#!#C%2abs~err~bound~->~G$\"&++\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"#%$~~~G$\"+++++]!#6F%7$$\"+ ++++gF($\"/Q1S&G<+\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Fincreasin g~step-size~by~a~factor~of~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&uJ#!#<%2a bs~err~bound~->~G$\"&<+\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%step G\"\"$%$~~~G$\",++++]#!#6F%7$$\",++++5$F($\"/xm#f0&Q5!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err ~estimate~->~G$\"&op\"!#:%2abs~err~bound~->~G$\"&&Q5!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6(%%stepG\"\"%%$~~~G$\"/[]xC;GT!#9F%7$$\"/[]xC;GsF ($\"/!y\\p#)o9\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~t o~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&()[\"!#:%2abs~err ~bound~->~G$\"&p9\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"& %$~~~G$\"/Y4!HQKX%!#9F%7$$\"/*fn2S\"o6!#8$\"/n]!4o4B\"F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Freducin g~step-size~and~repeating~stepG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5a bs~err~estimate~->~G$\"&pB%!#:%2abs~err~bound~->~G$\"&5B\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"'%$~~~G$\"/#foF7tO%!#9F%7$$ \"/eW/8([g\"!#8$\"/AH9\\L+7F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@usi ng~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&8/#!#:%2a bs~err~bound~->~G$\"&.?\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%step G\"\"(%$~~~G$\".Ub&pG^R!#8F%7$$\"/++++++?F($\"/g15FjW5F(" }}{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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "As far as the eye can see, the graphs of the analytical and numerical solutions coincide." 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by the two solutions, when " } {XPPEDIT 18 0 "x = sqrt(3)" "6#/%\"xG-%%sqrtG6#\"\"$" }{TEXT -1 29 ", \+ are identical to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "xx := sqrt(3);\nevalf(g(xx));\ngn(x x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG*$\"\"$#\"\"\"\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^y%R;\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^y%R;\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 5" }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx+sin(x)*y = cos(x),y(0) = 1;" "6$/,&*&%#dyG\"\"\"% #dxG!\"\"F'*&-%$sinG6#%\"xGF'%\"yGF'F'-%$cosG6#F./-F/6#\"\"!F'" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "We find both an analyti cal and a numerical solution, and compare their values when " } {XPPEDIT 18 0 "x = 2*sqrt(2);" "6#/%\"xG*&\"\"#\"\"\"-%%sqrtG6#F&F'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "de := diff(y(x),x)+sin(x)*y(x)=cos(x);\nic := y (0)=1;\ndesolve(\{de,ic\},y(x),method=linear,info=true);\nf := unapply (rhs(%),x);\nfn := desolve(\{de,ic\},y(x),x=0..3,type=numeric,method=r k78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#% \"xGF-\"\"\"*&F*F.-%$sinGF,F.F.-%$cosGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%DiffG6$-% \"yG6#%\"xGF,\"\"\"*&F)F--%$sinGF+F-F--%$cosGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~fa ctor~.~.~~G-%$expG6#-%$IntG6$-%$sinG6#%\"xGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~~=~~~G-%$expG6#,$-%$cosG6#%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"xG\"\"\"-%$expG6#,$-%$cosGF'!\"\"F)-%$IntG6$*&F.F)F*F )F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"xG\"\"\"-%$expG6#,$-%$cosGF'!\"\"F),&-%$I ntG6$*&-F/6#%#_uGF)-F+6#,$F6F0F)/F8;%!GF(F)&%\"CG6#F)F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplyin g~the~initial~condition~.~.~~G/&%\"CG6#\"\"\",&-%$expG6#!\"\"F(-%$IntG 6$*&-%$cosG6#%#_vGF(-F+6#,$F2F-F(/F5;%!G\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*& ,&-%$IntG6$*&-%$cosG6#%#_vG\"\"\"-%$expG6#,$F.!\"\"F2/F1;\"\"!F'F2-F46 #F7F2F2-F46#,$-F/F&F7F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(*&,&-%$IntG6$*&-%$cosG6#%#_vG\"\"\"-%$ex pG6#,$F2!\"\"F6/F5;\"\"!9$F6-F86#F;F6F6-F86#,$-F36#F?F;F;F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([f(x),'fn(x)'],x=0..3,color=[red,green],thickness=[1,2]);" }} {PARA 13 "" 1 "" {GLPLOT2D 267 168 168 {PLOTDATA 2 "6&-%'CURVESG6%7S7$ $\"\"!F)$\"+++++5!\"*7$$\"+]i9Rl!#6$\"+Yo6j5F,7$$\"+WA)GA\"!#5$\"+Nw$R 6\"F,7$$\"+Qeui=F6$\"+jT#f;\"F,7$$\"+i3&o]#F6$\"+OD<77F,7$$\"+pX*y9$F6 $\"+2+n^7F,7$$\"+WTAUPF6$\"+Ky:#G\"F,7$$\"+%*zhdVF6$\"+43I28F,7$$\"+%> fS*\\F6$\"+qHJE8F,7$$\"+>$f%GcF6$\"+;/=Q8F,7$$\"+Dy,\"G'F6$\"+T'4JM\"F ,7$$\"+7W7F,7$$\"+!R5'f5F,$\"+glD87F,7$$\"+/QBE6F,$\"+kkHt6F, 7$$\"+:o?&=\"F,$\"+urzM6F,7$$\"+a&4*\\7F,$\"+H&e'*3\"F,7$$\"+j=_68F,$ \"+!4!HW5F,7$$\"+Wy!eP\"F,$\"+TL1\\**F67$$\"+UC%[V\"F,$\"+wKl![*F67$$ 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}{TEXT -1 10 " and when " }{XPPEDIT 18 0 "x = sqrt(2);" "6#/%\"xG-%%sqrtG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "First we find the anal ytical solution." }}{PARA 0 "" 0 "" {TEXT -1 64 "The differential equa tion is homogeneous, so we use the option \"" }{TEXT 280 12 "method=ho mog" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "de := diff(y(x),x)=(y(x)/x)^2+1;\nic := \+ y(1)=2;\ndesolve(\{de,ic\},y(x),method=homog,info=true);\nh := unapply (rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG 6#%\"xGF,,&*&F)\"\"#F,!\"#\"\"\"F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#icG/-%\"yG6#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%%IHomogeneous~DE~for~which~substituti ng~~~G/%\"yG*&%#_vG\"\"\"%\"xGF(%-~~gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(%\"xG!\"\"F)-F%6$*&F(F(,(*$)%#_vG\" \"#F(F(F(F(F1F*F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"xG,&*&#\"\"#\"\"$\"\"\"*&F,#F-F+ -%'arctanG6#,$*(F,!\"\",&*&F+F-%#_vGF-F-F-F5F-F,F/F-F-F-F-&%\"CG6#F-F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#%\"xG,&*&#\"\"#\"\"$\"\"\"*&F,#F-F+-%'arctanG6#,$**F,! \"\",&*&F+F-%\"yGF-F-F'F5F-F'F5F,F/F-F-F-F-&%\"CG6#F-F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~G/&%\"CG6# \"\"\",$**\"\"#F(\"\"*!\"\"%#PiGF(\"\"$#F(F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$ *&#\"\"\"\"\"'F+*(F'F+,&*$\"\"$#F+\"\"#F+*&F0F+-%$tanG6#,$*&#F+\"#=F+* &,&*&\"\"*F+-%#lnGF&F+F+*(F2F+%#PiGF+F0F1F+F+F0F1F+F+F+F+F+F0F1F+F+" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,$*&#\"\"\"\"\"'F/*(9$F/,&*$\"\"$#F/\"\"#F/*&F5F/-%$tanG6#,$*&#F/ \"#=F/*&,&*&\"\"*F/-%#lnG6#F2F/F/*(F7F/%#PiGF/F5F6F/F/F5F6F/F/F/F/F/F5 F6F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Now we find the numerical solution over the interval from " }{TEXT 277 1 "x" }{TEXT -1 10 " = 0.1 to " }{XPPEDIT 18 0 "x = 1.8 " "6#/%\"xG-%&FloatG6$\"#=!\"\"" }{TEXT -1 32 " using the Taylor serie s method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "de := diff(y(x),x)=(y(x)/x)^2+1;\nic := y(1)=2;\nhn \+ := desolve(\{de,ic\},y(x),x=0.1..1.8,type=numeric,method=taylor);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*&F) \"\"#F,!\"#\"\"\"F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG 6#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "As far as the eye can see, the graphs of the analytical a nd numerical solutions coincide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([h(x),'hn(x)'],x=0.1..1 .8,color=[red,green],thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6&-%'CURVESG6%7hn7$$\"3/+++++++5!#=$!3(GHsYTO 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k()Fi^l7$$\"+AGQ&z\"Feel$\"+[]F_$*Fi^l7$$\"+69p(z\"Feel$\"+A\"*H-5!\"( 7$$\"#=Fe_l$\"+Dm^z5F^bm-Fd^l6&Ff^lFj^lFg^lFj^l-F]_l6#\"\"#-%+AXESLABE LSG6$Q\"x6\"Q!F]cm-%%VIEWG6$;Fd_lF`bm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "We compare the value s given by the two solutions, when " }{XPPEDIT 18 0 "x = 1/10;" "6#/% \"xG*&\"\"\"F&\"#5!\"\"" }{TEXT -1 7 " . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "xx := 1/10; \nevalf(evalf(h(xx),15));\nhn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#xxG#\"\"\"\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+:kLJq!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+8kLJq!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 " . . . and the values given by \+ the two solutions, when " }{XPPEDIT 18 0 "x = sqrt(2);" "6#/%\"xG-%%sq rtG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "xx := sqrt(2);\nevalf(evalf(h(xx),1 5));\nhn(xx);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG*$\"\"##\"\" \"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*)pY'4'!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+))pY'4'!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 7" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 34 "We solve the initial value problem" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(x)*sin(x)-x*y^2+[ y*(1-x^2)];" "6#,(*&-%$cosG6#%\"xG\"\"\"-%$sinG6#F(F)F)*&F(F)*$%\"yG\" \"#F)!\"\"7#*&F/F),&F)F)*$F(F0F1F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 " dy/dx = 0,y(0) = 1;" "6$/*&%#dyG\"\"\"%#dxG!\"\"\"\"!/-%\"yG6#F)F&" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 76 "both analytically and nu merically, and plot the graph of both solutions for " }{TEXT 278 1 "x " }{TEXT -1 18 " between -1 and 1." }}{PARA 0 "" 0 "" {TEXT -1 46 "Not e that the differential equation is exact. " }}{PARA 0 "" 0 "" {TEXT -1 69 "We compare the values of the analytical and numerical solutions when " }{XPPEDIT 18 0 "x = exp(1)/3;" "6#/%\"xG*&-%$expG6#\"\"\"F)\" \"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "de := cos(x)*sin(x)-x*y(x)^2+y(x)* (1-x^2)*diff(y(x),x)=0;\nic := y(0)=1;\ndesolve(\{de,ic\},method=exact ,info=true):\nk := unapply(rhs(%),x);\nd := 10^(-8):\nkn := desolve(\{ de,ic\},x=-1+d..1-d,type=numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#deG/,(*&-%$cosG6#%\"xG\"\"\"-%$sinGF*F,F,*&F+F,)-%\"yGF*\"\"#F,!\" \"*(F1F,,&F,F,*$)F+F3F,F4F,-%%diffG6$F1F+F,F,\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Test~for~exact ness~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*&%\"xG\"\" \")%\"yG\"\"#F*!\"\"*&-%$cosG6#F)F*-%$sinGF2F*F*F,,$*(F-F*F)F*F,F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&%\"yG\"\"\",&F)F)*$)%\" xG\"\"#F)!\"\"F)F-,$*(F.F)F-F)F(F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.Exact~DE~.~.~G/,(*&%\"xG\" \"\")%\"yG\"\"#F(!\"\"*&-%$cosG6#F'F(-%$sinGF0F(F(*(F*F(,&F(F(*$)F'F+F (F,F(-%!G6#*&%#dyGF(%#dxGF,F(F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%Let~G/-%\"FG6$%\"xG%\"yG,&- %$IntG6$,&*&F(\"\"\")F)\"\"#F0!\"\"*&-%$cosG6#F(F0-%$sinGF7F0F0F(F0-% \"gG6#F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G,**(\"\"#!\"\"% \"xGF&%\"yGF&F'*&#\"\"\"\"\"%F,-%$cosG6#,$*&F&F,F(F,F,F,F'#F,F-F'-%\"g G6#F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'where~G/-%%DiffG6$-%\"gG6#%\"yGF+,&*&F+\"\"\",&F.F.*$ )%\"xG\"\"#F.!\"\"F.F.-F&6$,(*(F3F4F2F3F+F3F4*&#F.\"\"%F.-%$cosG6#,$*& F3F.F2F.F.F.F4#F.F;F4F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,~~~~~~~ ~~~=G%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%$so~G/-%\"gG6#%\"yG-%$IntG6$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'~~=~~~G,$*&\"\"#!\"\"%\"yGF&\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"FG6$%\"xG %\"yG,(*(\"\"#!\"\"F'F+F(F+F,*&#\"\"\"\"\"%F/-%$cosG6#,$*&F+F/F'F/F/F/ F,*&F+F,F(F+F/%0~~satisfies~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %%DiffG6$-%\"FG6$%\"xG%\"yGF*,&*&F*\"\"\")F+\"\"#F.!\"\"*&-%$cosG6#F*F .-%$sinGF5F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"FG6$% \"xG%\"yGF+*&F+\"\"\",&F-F-*$)F*\"\"#F-!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&#\"\"\"\"\" #F'*&)%\"xGF(F')-%\"yG6#F+F(F'F'!\"\"*&#F'\"\"%F'-%$cosG6#,$*&F(F'F+F' F'F'F0*&#F'F(F'*$F,F'F'F'&%\"CG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~conditi on~.~.~~G/&%\"CG6#\"\"\"#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! 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}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "T he numerical solution is not very accurate when " }{TEXT 279 1 "x" } {TEXT -1 21 " is close to 1 or -1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "xx := 0.99999;\nevalf(k(xx)) ;\nkn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"&*****!\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?X<37!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?W<37!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 8" }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dt+4*y = 6*t^2*exp(-2*t),y(0) = 1;" "6$/,&*&%#dyG\" \"\"%#dtG!\"\"F'*&\"\"%F'%\"yGF'F'*(\"\"'F'*$%\"tG\"\"#F'-%$expG6#,$*& F1F'F0F'F)F'/-F,6#\"\"!F'" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "de := diff(y(t),t)+4*y(t)=6*t^2*exp(-2*t);\nic := y( 0)=1;\ndesolve(\{de,ic\},y(t),method=laplace,info=true);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tGF-\"\"\"*&\"\"%F .F*F.F.,$*(\"\"'F.)F-\"\"#F.-%$expG6#,$*&F5F.F-F.!\"\"F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%0Linear ~DE~.~.~~G\"\"\"-%%DiffG6$-%\"yG6#%\"tGF.F'F'*&\"\"%F'F+F'F',$*(\"\"'F ')F.\"\"#F'-%$expG6#,$*&F5F'F.F'!\"\"F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DTaking~Laplace ~transforms~gives~.~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(%\"sG\" \"\"%\"LGF'7#-%\"yG6#%\"tGF'F'F'!\"\"*(\"\"%F'F(F'F)F'F',$*&\"#7F',&F& F'\"\"#F'!\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%(so~.~.~G\"\"\"%\"LGF&7#-%\"yG6#%\"tGF&,&%\"sG F&\"\"%F&F&,&*&\"#7F&,&F.F&\"\"#F&!\"$F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Then~.~.~G \"\"\"%\"LGF&7#-%\"yG6#%\"tGF&*(,*\"#?F&*$)%\"sG\"\"$F&F&*&\"\"'F&)F2 \"\"#F&F&*&\"#7F&F2F&F&F&,&F2F&F7F&!\"$,&F2F&\"\"%F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**& \"\"'\"\"\",&%\"sGF(\"\"#F(!\"$F(*&\"\"$F(F)!\"#!\"\"*&F(F(*&F+F(,&F*F (\"\"%F(F(F0F0*(F.F(F+F0F)F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,&*&,(*&\"\"$\"\"\") F'\"\"#F-F-*&F,F-F'F-!\"\"#F,F/F-F--%$expG6#,$*&F/F-F'F-F1F-F-*&#F-F/F --F46#,$*&\"\"%F-F'F-F1F-F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "de := diff(y(t),t)+4*y(t)=6* t^2*exp(-2*t);\nic := y(0)=1;\ndesolve(\{de,ic\},y(t),info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tGF-\"\" \"*&\"\"%F.F*F.F.,$*(\"\"'F.)F-\"\"#F.-%$expG6#,$*&F5F.F-F.!\"\"F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%IThe ~DE~does~not~have~separable~variablesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%D iffG6$-%\"yG6#%\"tGF,\"\"\"*&\"\"%F-F)F-F-,$*(\"\"'F-)F,\"\"#F--%$expG 6#,$*&F4F-F,F-!\"\"F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-% $IntG6$\"\"%%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5~~~~~~~~~~~~~~~ ~=~~~G-%$expG6#,$*&\"\"%\"\"\"%\"tGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"tG\"\"\"-%$ expG6#,$*&\"\"%F)F(F)F)F)-%$IntG6$,$*(\"\"'F))F(\"\"#F)-F+6#,$*&F7F)F( F)F)F)F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"tG\"\"\"-%$expG6#,$*&\"\"%F)F(F)F)F),**( \"\"$F))F(\"\"#F)-F+6#,$*&F4F)F(F)F)F)F)*(F2F)F(F)F5F)!\"\"*&#F2F4F)F5 F)F)&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~G/&%\"CG6# \"\"\"#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,&*&#\"\"$\"\"#\"\"\"*&-%$expG6#,$ *&F,F-F'F-!\"\"F-,(*&F,F-)F'F,F-F-*&F,F-F'F-F4F-F-F-F-F-*&#F-F,F--F06# ,$*&\"\"%F-F'F-F4F-F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 0 7 "desolve" }{TEXT -1 18 ": order 2 examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+y = sin(x),y(0) = 1,y*`'`(0) = 2;" "6%/, &*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)F(F)-%$sinG6#F,/-F(6# \"\"!F)/*&F(F)-%\"'G6#F4F)F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "We find both analytical and numerical solutions, and comp are their values when " }{XPPEDIT 18 0 "x = Pi^2/10;" "6#/%\"xG*&%#PiG \"\"#\"#5!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "First w e find the analytical solution." }}{PARA 0 "" 0 "" {TEXT -1 59 "The on ly method available for second order equations using " }{TEXT 0 7 "des olve" }{TEXT -1 36 " is the method for linear equations." }}{PARA 0 " " 0 "" {TEXT -1 35 "Since this is the default option, \"" }{TEXT 280 13 "method=linear" }{TEXT -1 17 "\" can be omitted." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "de := dif f(y(x),x$2)+y(x)=sin(x);\nic := y(0)=1,D(y)(0)=2;\ndesolve(\{de,ic\},x =0..1,method=linear,info=true);\nf := unapply(rhs(%),x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"# \"\"\"F*F2-%$sinGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG 6#\"\"!\"\"\"/--%\"DG6#F(F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8 auxiliary~equation~.~.~G/,&*$)%\"mG\"\"#\"\"\"F*F*F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~ .~.~G^#\"\"\"^#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%\"fGf*6#%\"xG6\"6$%)operat orG%&arrowGF(,(*&#\"\"&\"\"#\"\"\"-%$sinG6#9$F1F1*&#F1F0F1*&-%$cosGF4F 1F5F1F1!\"\"F9F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Now we find a numerical solution, using the Taylor s eries method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "de := diff(y(x),x$2)+y(x)=sin(x);\nic := y(0)=1,D (y)(0)=2;\nfn := desolve(\{de,ic\},x=0..1,type=numeric,method=taylor,i nfo=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6# %\"xG-%\"$G6$F-\"\"#\"\"\"F*F2-%$sinGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"#%%~~~~Gf*6%%\"xG%\"yG%\"uG6\"6$% )operatorG%&arrowGF+,&-%$cosG6#9$\"\"\"*&$F4\"\"!F49&F4!\"\"F+F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"$%%~~~~Gf*6%%\"xG%\"y G%\"uG6\"6$%)operatorG%&arrowGF+,&*&$\"\"#\"\"!\"\"\"-%$sinG6#9$F4!\" \"9%F4F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"%%%~~ ~~Gf*6%%\"xG%\"yG%\"uG6\"6$%)operatorG%&arrowGF+,&*&$\"\"#\"\"!\"\"\"- %$cosG6#9$F4!\"\"9&F4F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+deriv ativeG\"\"&%%~~~~Gf*6%%\"xG%\"yG%\"uG6\"6$%)operatorG%&arrowGF+,&*&$\" \"$\"\"!\"\"\"-%$sinG6#9$F4F4*&$F4F3F49%F4!\"\"F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"'%%~~~~Gf*6%%\"xG%\"yG%\"uG6\"6$% )operatorG%&arrowGF+,&*&$\"\"$\"\"!\"\"\"-%$cosG6#9$F4F4*&$F4F3F49&F4! \"\"F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"(%%~~~~ Gf*6%%\"xG%\"yG%\"uG6\"6$%)operatorG%&arrowGF+,&*&$\"\"%\"\"!\"\"\"-%$ sinG6#9$F4!\"\"9%F4F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivat iveG\"\")%%~~~~Gf*6%%\"xG%\"yG%\"uG6\"6$%)operatorG%&arrowGF+,&*&$\"\" %\"\"!\"\"\"-%$cosG6#9$F4!\"\"9&F4F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"\"*%%~~~~Gf*6%%\"xG%\"yG%\"uG6\"6$%)operatorG%&ar rowGF+,&*&$\"\"&\"\"!\"\"\"-%$sinG6#9$F4F4*&$F4F3F49%F4!\"\"F+F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%+derivativeG\"#5%%~~~~Gf*6%%\"xG%\"yG %\"uG6\"6$%)operatorG%&arrowGF+,&*&$\"\"&\"\"!\"\"\"-%$cosG6#9$F4F4*&$ F4F3F49&F4!\"\"F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~est imate~->~G$\"&dv#!#J%2abs~err~bound~->~G$\"\"\"!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6(%%stepG\"\"\"%$~~~G$\"+++++5!#6F%7$F&$\"/+vL)\\*>5!#8 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Fincreasing~step-size~by~a~factor ~of~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&.h#!#C%2abs~err~bound~->~G$ \"&*>5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"#%$~~~G$\"++++ +]!#6F%7$$\"+++++gF($\"/e$*RX;=6!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Fincreasing~step-size~by~a~factor~of~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima te~->~G$\"&2:#!#<%2abs~err~bound~->~G$\"&#=6!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"$%$~~~G$\",++++]#!#6F%7$$\",++++5$F($\"/@md% otc\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~st ep-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&Y+#!#;%2abs~err~bound~->~G$ \"&uc\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"%%$~~~G$\"/cF oX40U!#9F%7$$\"/cFoX40tF($\"/=UuJ!49#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$ \"&`l'!#<%2abs~err~bound~->~G$\"&49#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"&%$~~~G$\"/WsJa!\\p#!#9F%7$$\"/++++++5!#8$\"/&[\\hGQP #F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*last~stepG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=the~total~nu mber~of~steps~isG\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The graphs of the two solutions coincide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "pl ot([f(x),'fn(x)'],x=0..1,color=[red,green],thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 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++++\"!\")F(F(-%*THICKNESSG6#F+-F$6%7SF'7$$\"+;arz@!#6$\"+M]NV5!\"*7$$ \"+XTFwSFg[l$\"+4No!3\"Fj[l7$$\"+\"z_\"4iFg[l$\"+*4;A7\"Fj[l7$$\"+S&ph N)Fg[l$\"+)*o`j6Fj[l7$$\"+*=)H\\5!#5$\"+.q;/7Fj[l7$$\"+[!3uC\"F]]l$\"+ f!)QT7Fj[l7$$\"+J$RDX\"F]]l$\"+riYz7Fj[l7$$\"+)R'ok;F]]l$\"+([W$=8Fj[l 7$$\"+1J:w=F]]l$\"+uY2aF]]l$\"+)*oY7>Fj[l7$$\"+yXu9cF]]l$\"+F j[l7$$\"+\\y))GeF]]l$\"+<6ln>Fj[l7$$\"+i_QQgF]]l$\"+#e\\T*>Fj[l7$$\"+! y%3TiF]]l$\"+#R#>>?Fj[l7$$\"+O![hY'F]]l$\"+ctIY?Fj[l7$$\"+#Qx$omF]]l$ \"+*oZ+2#Fj[l7$$\"+u.I%)oF]]l$\"+4-u%4#Fj[l7$$\"+(pe*zqF]]l$\"+#yFl6#F j[l7$$\"+C\\'QH(F]]l$\"+8]qR@Fj[l7$$\"+8S8&\\(F]]l$\"+$\\(*3;#Fj[l7$$ \"+0#=bq(F]]l$\"+,!4C=#Fj[l7$$\"+2s?6zF]]l$\"+VY!G?#Fj[l7$$\"+IXaE\")F ]]l$\"+t2[BAFj[l7$$\"+l*RRL)F]]l$\"+&*ztUAFj[l7$$\"+`<.Y&)F]]l$\"+')Gw hAFj[l7$$\"+8tOc()F]]l$\"+j0'*zAFj[l7$$\"+\\Qk\\*)F]]l$\"+XE4'H#Fj[l7$ $\"+p0;r\"*F]]l$\"+C`)QJ#Fj[l7$$\"+lxGp$*F]]l$\"+yu;HBFj[l7$$\"+!oK0e* F]]l$\"+4F![M#Fj[l7$$\"+<5s#y*F]]l$\"+U&H\"fBFj[l7$F*$\"+:'GQP#Fj[l-Fh 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their values when " }{XPPEDIT 18 0 "x = Pi^2/2 ;" "6#/%\"xG*&%#PiG\"\"#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "First we find the analytical solution." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "de := dif f(y(x),x$2)+9*y(x)=3*x;\nic := y(0)=0,D(y)(0)=1;\ndesolve(\{de,ic\},y( x),info=true):\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"*F2F* F2F2,$*&\"\"$F2F-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\" yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8a uxiliary~equation~.~.~G/,&*$)%\"mG\"\"#\"\"\"F*\"\"*F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roo ts~.~.~G^#\"\"$^#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&#\"\"\"\"\"$F/9$F/F/*&# \"\"#\"\"*F/-%$sinG6#,$*&F0F/F1F/F/F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Now we find a numerical s olution over the interval " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 5" "6#/%\"xG\"\"&" }{TEXT -1 39 " by an adaptive 7-8 Runge-Kutta method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "de := diff(y(x),x$2) +9*y(x)=3*x;\nic := y(0)=0,D(y)(0)=1;\ngn := desolve(\{de,ic\},x=0..5, type=numeric,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/, &-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"*F2F*F2F2,$*&\"\"$F 2F-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/-- %\"DG6#F(F)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The graphs of the two solutions coincide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([g (x),'gn(x)'],x=0..5,color=[red,green],thickness=[1,2]);" }}{PARA 13 " " 1 "" {GLPLOT2D 277 180 180 {PLOTDATA 2 "6&-%'CURVESG6%7^o7$$\"\"!F)F 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\"+yIa==Fidl7$Fg_l$\"+w]<6=Fidl-F\\`l6&F^`lF(F_`lF(-Fc`l6#\"\"#-%+AXES LABELSG6$Q\"x6\"Q!Fgem-%%VIEWG6$;F(Fg_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 " " 0 "" {TEXT -1 48 "We compare the values of the two solutions when " }{XPPEDIT 18 0 "x = Pi^2/2" "6#/%\"xG*&%#PiG\"\"#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xx := Pi^2/2;\nevalf(g(xx));\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$*&\"\"#!\"\"%#PiGF'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+8p\\>=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+9p\\>=!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 3" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^ 2)=y,y(0)=2,y(1)=0" "6%/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F '/-F'6#\"\"!F&/-F'6#F(F0" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "We find both analytical and numerical solutions, and compare their values when " }{XPPEDIT 18 0 "x = Pi/4" "6#/%\"xG*&%#PiG\"\"\"\"\"%! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "This is a two poi nt boundary value problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "First we find the analytical solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "d e := diff(y(x),x$2)=y(x);\nic := y(0)=2,y(1)=0;\ndesolve(\{de,ic\},y(x ),info=true);\nh := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"#/-F(6#\"\"\"F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,&*$)%\"mG \"\"#\"\"\"F*F*!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G\"\"\"!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6general ~solution~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&& %\"CG6#\"\"\"F--%$expGF&F-F-*&&F+6#\"\"#F--F/6#,$F'!\"\"F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Afro m~the~initial~conditions~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\" \"#,&&%\"CG6#\"\"\"F)&F'6#F$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\" \"!*&,&*&&%\"CG6#\"\"\"F+-%$expG6#\"\"#F+F+&F)F.F+F+-F-6#!\"\"F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%-so~that~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\",$* &\"\"#F',&-%$expG6#F*F'F'!\"\"F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"CG6#\"\"#,$*(F'\"\"\"-%$expGF&F*,&F+F*F*!\"\"F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"xG,&*(\"\"#\"\"\",&-%$expG6#F*F+F+!\"\"F0-F.F&F+F0**F*F+F-F+F,F0-F.6 #,$F'F0F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)o peratorG%&arrowGF(,&*(\"\"#\"\"\",&-%$expG6#F.F/F/!\"\"F4-F26#9$F/F4** F.F/F1F/F0F4-F26#,$F7F4F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "Now we find a numerical solution over the interval " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 35 ", using the line ar shooting method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "de := diff(y(x),x$2)=y(x);\nic := y(0)=2,y (1)=0;\nhls := desolve(\{de,ic\},y(x),x=0..1,type=shoot,info=1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$ F,\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\" \"#/-F(6#\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>Using~linear~sh ooting~method.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%SShot,~with~initia l~derivative~0,~has~end~value~->~G$\"+q7;'3$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%coHomogeneous~DE,~with~initial~value~0~and~derivative~ 1,~has~end~value~->~G$\"+%>,_<\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%MInitial~derivative~for~required~solution~->~G$!+r02EE!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "We also f ind a numerical solution over the interval " }{XPPEDIT 18 0 "x = 0" "6 #/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\" \"" }{TEXT -1 32 ", by a finite difference method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "de := diff( y(x),x$2)=y(x);\nic := y(0)=2,y(1)=0;\nhfd := desolve(\{de,ic\},y(x),t ype=numeric,method=finitediff);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!\"\"#/-F(6#\"\"\"F*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "We compare the val ues of the three solutions when " }{XPPEDIT 18 0 "x = Pi/4" "6#/%\"xG* &%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "xx := Pi/4;\nevalf(evalf (h(xx),15));\nevalf(evalf[15](hls(xx)));\nevalf(evalf[15](hfd(xx)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$*&\"\"%!\"\"%#PiG\"\"\"F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2$p-o$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2$p-o$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2$ p-o$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We cons ider the initial value problem" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "d^2 *x/(d*t^2)+1/2;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(* &F(F(F&F,F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dx/dt+4*x = F(t),x(0) = \+ 0,x*`'`(0) = 0;" "6%/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"\"%F'%\"xGF'F'-%\" FG6#%\"tG/-F,6#\"\"!F4/*&F,F'-%\"'G6#F4F'F4" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "F(t)" "6#-%\"FG6#%\"tG" }{TEXT -1 45 " is the triangular waveform given as follows." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 " F := t -> 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o$!3ixm;/'e)*R%F-7$$\"3%zmmTvJga)Fdo$!3QzmmTvJgaF-7$$\"3A,]PM&*>^')Fdo $!3@7+vV`*>^'F-7$$\"3]MLe9tOc()Fdo$!3/XL$e9tOc(F-7$$\"3oo;H#e0I&))Fdo$ !3u'o;H#e0I&)F-7$$\"31,++]Qk\\*)Fdo$!3m5+++&Qk\\*F-7$$\"3-oT5SMLx*)Fdo $!31!oT5SMLx*F-7$$\"3%\\L3-.B]+*Fdo$!3c]m\"zpp(\\**F-7$$\"3)=]7.i7F.*F do$!3;\")\\(oztGn*F-7$$\"3#)omT5ASg!*Fdo$!3w6L$e*y(fR*F-7$$\"3q-]i!R\" y:\"*Fdo$!3+t*\\P4'=U))F-7$$\"3![LL3dg6<*Fdo$!3)>lm;H%R)G)F-7$$\"3K,+v oTAq#*Fdo$!3u')**\\7$exH(F-7$$\"3%ymmmw(Gp$*Fdo$!3^@LLLB72jF-7$$\"3/M$ eRA5\\Z*Fdo$!3cfmTgx*3D&F-7$$\"3C++D\"oK0e*Fdo$!3i(***\\(=tY>%F-7$$\"3 m+++]oi\"o*Fdo$!3G$******\\JP=$F-7$$\"35,+v=5s#y*Fdo$!3#*))**\\7)*ys@F -7$$\"3W,]P40O\"*)*Fdo$!3e&)*\\i!\\R'3\"F-7$$\"#5F)F(-%'COLOURG6&%$RGB G$Fb`m!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q!F]am-%%VIEWG6$;F(Fa`m%(DEFAULT G" 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 48 "We fin d a numerical solution over the interval " }{XPPEDIT 18 0 "t = 0" "6# /%\"tG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "t = 10" "6#/%\"tG\"#5 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 147 "de := diff(x(t),t$2)+1/2*diff(x(t),t)+4*x(t)= F(t);\nic := x(0)=0,D(x)(0)=0;\nsn := evalf[16](desolve(\{de,ic\},x(t) ,t=0..10,type=numeric,method=rk78));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&#F2F1F2-F(6$F *F-F2F2*&\"\"%F2F*F2F2*&,&F-F2*&F1F2-%&floorG6#,&*&F1!\"\"F-F2F2F4F2F2 FAF2)FA,&F%#icG6$/-%\"xG 6#\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#snGf*6 #'%-x_rk78interpG%)realconsG6jt%#b1G%#b6G%#b7G%#b8G%#b9G%#bAG%#bBG%#bC G%#bEG%#bFG%#bGG%#bHG%#bIG%#bJG%#bKG%#bLG%#xFG%#xSG%#yFG%#f1G%#f6G%#f7 G%#f8G%#f9G%#fAG%#fBG%#fCG%#fEG%#fFG%#fGG%#fHG%#fIG%#fJG%#fKG%#fLG%$d1 2G%$d62G%$d72G%$d82G%$d92G%$dA2G%$dB2G%$dC2G%$dE2G%$dF2G%$dG2G%$dH2G%$ dI2G%$dJ2G%$dK2G%$dL2G%$d13G%$d63G%$d73G%$d83G%$d93G%$dA3G%$dB3G%$dC3G 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