{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 "" -1 256 "" 0 1 0 0 0 0 0 2 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 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 } 1 1 0 0 0 0 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 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 "Title " -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 } 3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 47 "Euler's Method for \nNumerical I ntegrating ODE's" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 14 "Initialization" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:\nwith(plots):\nwith(DEtools):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Derive Euler's Method:" }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 "First order differential equation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "diff(y(x),x) = f(x,y(x)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF*-%\"fG6 $F*F'" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "Finite difference appr oximation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "finite_diff_eq \+ := (y[i+1]-y[i])/h = f(x[i],y[i]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%/finite_diff_eqG/*&,&&%\"yG6#,&%\"iG\"\"\"F-F-F-&F)6#F,!\"\"F-%\"hGF 0-%\"fG6$&%\"xGF/F." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Solve for " }{TEXT 256 1 " " }{XPPEDIT 18 0 "y[i+1];" "6#&%\"yG6#,&%\"iG\"\"\"F( F(" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "y[i+1] := solve(finite _diff_eq,y[i+1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#,&%\"iG \"\"\"F)F),&&F%6#F(F)*&-%\"fG6$&%\"xGF,F+F)%\"hGF)F)" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 "Set up initial conditions:" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 58 "Step size, number of intervals, initial c onditions, f(x,y)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "h := 0. 1;\nN := 20;\nx0 := 0.0;\ny0 := 1.0;\nf := (x,y) -> cos(x+y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"NG\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$ \"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$\"#5!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF )-%$cosG6#,&9$\"\"\"9%F2F)F)F)" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Initialize arrays" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x[0 ] := x0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!$F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "y[0] := y0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"!$\"#5!\"\"" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Apply Euler's method" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "for i from 0 to N do\n x[i+1] := x[i] + h:\n y[i+1] := y[i] + f(x[i],y[i])*h:\nod:" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Plot the data points" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "Creat e array of data points" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "l \+ := [[x[n],y[n]] $n=0..N]:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "No w, plot it!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot(l,style= point);\nplot1 := 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