{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 "Help Bold" -1 39 "Times" 1 12 0 0 0 0 0 1 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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 259 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 259 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 258 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 259 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 259 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {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 "" 259 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 282 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -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 "Text \+ Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 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 "Time s" 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" -1 256 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 72 "Miscellaneous examples involving \+ Runge-Kutta methods for 1st order DE's " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 18.8.2007" }}{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 262 7 "DEsol.m" }{TEXT -1 32 " is required by t his 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 40 "load root-finding procedures i ncluding: " }{TEXT 0 29 "bisect,secant,findmax,findmin" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 7 "roots.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 th at follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/roots.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "load num erical integration procedures including: " }{TEXT 0 5 "NCint" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 6 "intg.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 32 "read \"K:\\\\Maple/procdrs/i ntg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolveRK" }{TEXT -1 16 ": more examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }{TEXT 271 98 " .. a discrete solution obtained by the improved Eu ler method compared with an analytical solution" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 265 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 71 "This \+ question is concerned with the first order differential equation: " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = x^2*y^2-x*y ;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%\"xG\"\"#%\"yGF,F&*&F+F&F-F&F(" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "subject to the initial condition " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 63 "(a) Find an analytical \+ solution for the differential equation. " }}{PARA 0 "" 0 "" {TEXT -1 85 "(b) Find a discrete numerical solution for the differential equati on in the interval " }{XPPEDIT 18 0 "-3 <= x;" "6#1,$\"\"$!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }{TEXT -1 84 " by using the i mproved Euler method with a total of 50 steps, and find the maximum " }{TEXT 262 8 "relative" }{TEXT -1 77 " error of the numerical solution when compared with the analytical solution. " }}{PARA 0 "" 0 "" {TEXT -1 104 "(c) Plot the points of the discrete numerical solution a long with the graph of the analytical solution. " }}{PARA 0 "" 0 "" {TEXT -1 258 "(d) Construct a numerical procedure which interpolates b etween the points of the discrete solution found in (b) and plot a gra ph of the relative error exhibited by this continuous numerical soluti on when compared with the analytical solution on the interval " } {XPPEDIT 18 0 "-3 <= x;" "6#1,$\"\"$!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= \+ 2;" "6#1%!G\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 266 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 12 "(a) Maple's " }{TEXT 0 6 "dsolve" }{TEXT -1 46 " can be used to find the analytical solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "de := diff(y(x),x) =x^2*y(x)^2-x*y(x);\nic := y(0)=1;\ndsolve(\{de,ic\},y(x));\ng := unap ply(rhs(%),x):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-% \"yG6#%\"xGF,,&*&)F,\"\"#\"\"\")F)F0F1F1*&F,F1F)F1!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&\"\"#\"\"\",(*&F*F+F'F+F+**-%$expG6#, $*&F*!\"\"F'F*F+F+%#PiG#F+F*F*F6-%$erfG6#,$*(F*F4F*F6F'F+F+F+F4*&F*F+F /F+F+F4F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "de := diff(y( x),x)=x^2*y(x)^2-x*y(x);\nic := y(0)=1;\npts := desolveRK(\{de,ic\},y( x),x=-3..2,method=impeuler,steps=50):\ncomparewithfcn(pts,g(x),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*&)F ,\"\"#\"\"\")F)F0F1F1*&F,F1F)F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 6 "" 1 "" {TEXT -1 90 " \+ -3 0.005285755191 function val: 0.0050117203022 rel err: 5.4679e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " -2.9 0.00709 6609601 function val: 0.0067656704564 rel err: 4.8914e-02" }} {PARA 6 "" 1 "" {TEXT -1 90 " -2.8 0.009450150746 funct ion val: 0.0090542614 rel err: 4.3724e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " -2.7 0.01248400906 function val: 0.012 014837827 rel err: 3.9049e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " \+ -2.6 0.0163640368 function val: 0.015813174396 rel err: 3.4836e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " -2.5 \+ 0.0212887174 function val: 0.020647962692 rel err: 3.1032e-02 " }}{PARA 6 "" 1 "" {TEXT -1 90 " -2.4 0.02749402964 f unction val: 0.026755779256 rel err: 2.7592e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " -2.3 0.03525870837 function val: 0. 034416484592 rel err: 2.4472e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " \+ -2.2 0.04490973144 function val: 0.043958890418 \+ rel err: 2.1630e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " -2.1 \+ 0.05682766986 function val: 0.055766336492 rel err: 1.9032e- 02" }}{PARA 6 "" 1 "" {TEXT -1 90 " -2 0.07145122873 \+ function val: 0.070281500334 rel err: 1.6643e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " -1.9 0.08927983612 function val: \+ 0.088009275754 rel err: 1.4437e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " -1.8 0.110872458 function val: 0.10951584784 \+ rel err: 1.2387e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " -1.7 0.1368398991 function val: 0.13542112748 rel err: 1.0477 e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " -1.6 0.167826708 function val: 0.16638050666 rel err: 8.6921e-03" }}{PARA 6 " " 1 "" {TEXT -1 90 " -1.5 0.2044775875 function val: \+ 0.2030506107 rel err: 7.0277e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " -1.4 0.2473822912 function val: 0.24603276876 rel err: 5.4851e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " - 1.3 0.2969931417 function val: 0.29578813736 rel err: 4.0 739e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " -1.2 0.3535118 113 function val: 0.3525211694 rel err: 2.8102e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " -1.1 0.4167485507 function va l: 0.41603518826 rel err: 1.7147e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " -1 0.4859690755 function val: 0.4855765 777 rel err: 8.0831e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " \+ -0.9 0.5597614057 function val: 0.5597017472 rel err: \+ 1.0659e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " -0.8 0.6359 724129 function val: 0.63621820916 rel err: 3.8634e-04" }} {PARA 6 "" 1 "" {TEXT -1 90 " -0.7 0.7117710704 funct ion val: 0.71225663836 rel err: 6.8173e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " -0.6 0.7838785698 function val: 0.78 451124268 rel err: 8.0645e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " \+ -0.5 0.848957274 function val: 0.84963573846 rel err: 7.9854e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " -0.4 \+ 0.9040833685 function val: 0.90471623352 rel err: 6.9952e-04 " }}{PARA 6 "" 1 "" {TEXT -1 90 " -0.3 0.9471767168 f unction val: 0.94769485966 rel err: 5.4674e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " -0.2 0.9772625211 function val: 0 .97762274098 rel err: 3.6847e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " \+ -0.1 0.9945 function val: 0.99468191142 \+ rel err: 1.8288e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " 0 \+ 1 function val: 1 rel err: 0.0000e+ 00" }}{PARA 6 "" 1 "" {TEXT -1 90 " 0.1 0.9955 \+ function val: 0.99534326666 rel err: 1.5747e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " 0.2 0.9830991566 function val: \+ 0.98278821616 rel err: 3.1639e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " 0.3 0.9649021251 function val: 0.964446866 \+ rel err: 4.7204e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " 0.4 0.9428657075 function val: 0.94228043978 rel err: 6.2112 e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " 0.5 0.9187016141 function val: 0.91800226284 rel err: 7.6182e-04" }}{PARA 6 " " 1 "" {TEXT -1 90 " 0.6 0.8938501895 function val: \+ 0.8930519125 rel err: 8.9388e-04" }}{PARA 6 "" 1 "" {TEXT -1 90 " 0.7 0.869502419 function val: 0.86861781372 rel err: 1.0184e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " \+ 0.8 0.8466511092 function val: 0.8456890198 rel err: 1.1 376e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " 0.9 0.8261590 924 function val: 0.82512393706 rel err: 1.2545e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " 1 0.8088399829 function va l: 0.80773151822 rel err: 1.3723e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " 1.1 0.795554951 function val: 0.79436851 112 rel err: 1.4936e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " \+ 1.2 0.7873387025 function val: 0.78606628512 rel err: \+ 1.6187e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " 1.3 0.7855 834466 function val: 0.78421689032 rel err: 1.7426e-03" }} {PARA 6 "" 1 "" {TEXT -1 90 " 1.4 0.7923407516 funct ion val: 0.79088067104 rel err: 1.8461e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " 1.5 0.8108724918 function val: 0.80 935408442 rel err: 1.8761e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " \+ 1.6 0.8467647179 function val: 0.84533713992 rel err: 1.6888e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " 1.7 \+ 0.9104457017 function val: 0.90964318436 rel err: 8.8223e-04 " }}{PARA 6 "" 1 "" {TEXT -1 90 " 1.8 1.023721929 f unction val: 1.0255546918 rel err: 1.7871e-03" }}{PARA 6 "" 1 "" {TEXT -1 90 " 1.9 1.240263445 function val: \+ 1.2539118822 rel err: 1.0885e-02" }}{PARA 6 "" 1 "" {TEXT -1 90 " \+ 2 1.730772746 function val: 1.8194492863 \+ rel err: 4.8738e-02" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum relative error: 5.4679e-02 " }}{PARA 6 "" 1 "" {TEXT -1 60 " obtained for the input \+ value: -3" }}}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "plot([pts,g(x)],x=-3..2,col or=[blue,red],style=[point,line],symbol=circle,\n \+ labels=[`x`,`y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7U7$$!\"$\"\"!$\"3;+++\">bdG&!#?7$$ !3!***************G!#<$\"3-+++,'4m4(F-7$$!3#)*************z#F1$\"3a+++ Y2:]%*F-7$$!3;+++++++FF1$\"31+++14S[7!#>7$$!33+++++++EF1$\"3))******zO SO;F>7$$!3++++++++DF1$\"31+++S<()G@F>7$$!3!**************R#F1$\"31+++k HS\\FF>7$$!3#)*************H#F1$\"3E+++P3(e_$F>7$$!3;+++++++AF1$\"3I++ +WJ(4\\%F>7$$!33+++++++@F1$\"3#******f)pw#o&F>7$$!\"#F*$\"3m*****H(G7X 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "de := diff( y(x),x)=x^2*y(x)^2-x*y(x);\nic := y(0)=1;\ngn := desolveRK(\{de,ic\},y (x),x=-3..2,method=impeuler,steps=50,output=procedure);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*&)F,\"\"#\"\" \")F)F0F1F1*&F,F1F)F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(('gn' (x)-g(x))/g(x),x=-3..2,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7eo7$$!\"$\"\"!$\"+&p!)yY&!#67$$!+H U,\"*G!\"*$\"+eOX\\ZF-7$$!+$H'='z#F1$\"+elH9VF-7$$!+gBa*o#F1$\"+qjTrPF -7$$!+B:>#e#F1$\"+Nk\\%H$F-7$$!+04NvCF1$\"+e:'R)GF-7$$!+wfHwBF1$\"+?)) )Rc#F-7$$!+M.ttAF1$\"+E'zZD#F-7$$!+,oln@F1$\"+&GU4'>F-7$$!+ZM#>1#F1$\" +khz)p\"F-7$$!+&pjJ&>F1$\"+5CKc9F-7$$!+[!ot&=F1$\"+S*4uD\"F-7$$!+P[_\\ `M0\"F-7$$!+7)Q7k\"F1$\"++H!zn)!#77$$!+i^)o`\"F1$\"+!oR++(Fjo 7$$!+^?7U9F1$\"+&R6v\\&Fjo7$$!+X&R%H8F1$\"+a0)30%Fjo7$$!+$o#)RB\"F1$\" +a9C6GFjo7$$!+fO%H7\"F1$\"+#yVtq\"Fjo7$$!+3`lC5F1$\"+]0[4\")!#87$$!*4u \"o\"*F1$\"+0xE'3\"Fiq7$$!*F-89)F1$!+i?!)HQFiq7$$!*Ep)pqF1$!+g!o)*z'Fi q7$$!*Hff3'F1$!++LHW!)Fiq7$$!*ZzY-&F1$!+b%o6)zFiq7$$!*+3B#RF1$!+q&))*R mFiq7$$!*S!piHF1$!+S8dj_Fiq7$$!*5xi#>F1$!+F@#zG$Fiq7$$!)ugb&)F1$!+cuiF 6Fiq7$$\")KE>>F1$\"+_8^i6!#:7$$\"*#RU07F1$\"+DA<*e\"Fiq7$$\"*?S2L#F1$ \"+D7>DKFiq7$$\"*$p)=M$F1$\"+KvU%z%Fiq7$$\"**=]@WF1$\"+S/i`jFiq7$$\"*] $z*R&F1$\"+l%ytw(Fiq7$$\"*kC$pkF1$\"+:Ec$=*Fiq7$$\"*3qcZ(F1$\"+\"*Rl[5 Fjo7$$\"*/\"fF&)F1$\"+9qV!=\"Fjo7$$\"*0Ogb*F1$\"+\\%*)*48Fjo7$$\"+nAFj 5F1$\"+!e+&\\9Fjo7$$\"+&)*pp;\"F1$\"+q]D!f\"Fjo7$$\"+ye,t7F1$\"+Av'zt \"Fjo7$$\"+fO=y8F1$\"+!o#[Z\"F1$\"+)\\mC(>Fjo7$$\"+(G!e &e\"F1$\"+g]zk=Fjo7$$\"+OK'zf\"F1$\"+%R(G>Fjo7$$\"+ EM\"3%=F1$!+L5q$[#Fjo7$$\"+op3m=F1$!+j2(>I%Fjo7$$\"+60O\"*=F1$!+:fQv&) Fjo7$$\"+%Q?&=>F1$!+or+)>\"F-7$$\"+c-oX>F1$!+%\\8Bd\"F-7$$\"+#>g#f>F1$ !+vC6l>F-7$$\"+G,%G(>F1$!++8i&e#F-7$$\"+'4I'z>F1$!+#*)>x+$F-7$$\"+k+U' )>F1$!+NCc?NF-7$$\"+K+@$*>F1$!+(=#oPTF-7$$\"\"#F*$!+==\"Q([F--%'COLOUR G6&%$RGBG$F*F*Fhbl$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!F`cl-%%VIEWG6 $;F(F`bl%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 260 4 "Note" }{TEXT -1 80 ": It is clear from this error \+ graph that the maximum relative error occurs when " }{XPPEDIT 18 0 " x =-3" "6#/%\"xG,$\"\"$!\"\"" }{TEXT -1 56 ". This agrees with the resul t already seen in part (b). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "xx := -3.;\nabserr := evalf( abs(gn(xx)-g(xx)));\nrelerr := evalf(abserr/g(xx),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'abserrG$\"**)[.u#!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerr G$\"&!oa!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 2" }{TEXT 272 87 " .. working with a continuous solution to find zer os and turning points of the solution" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 263 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 71 "This question is concerned with the first order differential equat ion: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=x^2- 3*sqrt(x^2+y))" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*$%\"xG\"\"#F&*&\"\"$F&-% %sqrtG6#,&*$F+F,F&%\"yGF&F&F(" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "subject to the initial condition " }{XPPEDIT 18 0 "y(0)=3 " "6#/-%\"yG6#\"\"!\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 161 "(a) Find a continuous numerical solution for the differential eq uation in the from of a Maple procedure which can be evaluated anywher e throughout the interval " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" } {XPPEDIT 18 0 "``<=5" "6#1%!G\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 119 "(b) Plot the graph of this numerical solution along wi th the gradient field associated with the differential equation. " }} {PARA 0 "" 0 "" {TEXT -1 55 "(c) Find the two zeros of the solution in the interval " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=5" "6#1%!G\"\"&" }{TEXT -1 23 " correct to 10 digits. " }} {PARA 0 "" 0 "" {TEXT -1 76 "(d) Find the coordinates of the minimum p oint on the graph of the solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 264 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "de := diff(y(x),x)=x^2-3*sqrt(x^2+y(x));\nic := y(0)=3;\ngn := desolveRK (\{de,ic\},y(x),x=0..5,method=rk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*$)F,\"\"#\"\"\"F1*&\"\"$F1,&F.F1F )F1#F1F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"! \"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 187 "p1 := plot('gn'(x),x=0..5,color=coral,thickness=2) :\np2 := DEtools[DEplot](de,y(x),x=0..5,y=-2..7,\n color=COLOR(R GB,.7,.3,.4),arrows=medium,dirgrid=[25,25]):\nplots[display]([p1,p2]); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6% 7U7$$\"\"!F)$\"\"$F)7$$\"+4x&)*3\"!#5$\"+'=Z/Y#!\"*7$$\"+uq8Q?F/$\"+uG UM?F27$$\"+(RwX5$F/$\"+>a!Hg\"F27$$\"+sZ3yTF/$\"+'*yw<7F27$$\"+]4\\Y_F /$\"+/b?2))F/7$$\"+U-/PiF/$\"+_-)*fgF/7$$\"+fmpisF/$\"+&*))4aNF/7$$\"+ #*>VB$)F/$\"+$3(eq7F/7$$\"+Mbw!Q*F/$!+z'y=_(!#67$$\"+0j$o/\"F2$!+;&*HE EF/7$$\"+_>jU6F2$!+GiFUTF/7$$\"+j^Z]7F2$!+oNXIdF/7$$\"+)=h(e8F2$!+Pe'= A(F/7$$\"+Q[6j9F2$!+6E*Rd)F/7$$\"+\\z(yb\"F2$!+c`!Qt*F/7$$\"+b/cq;F2$! +j7$G5\"F27$$\"+F2$!+>al,9F27$$\"+\"f#=$3#F2$!+)*HR'[\"F27$$\"+t(pe=#F2$!+Km1b:F27$ $\"+uI,$H#F2$!+b^!Hh\"F27$$\"+rSS\"R#F2$!+r1Q_;F27$$\"+`?`(\\#F2$!+]j& *y;F27$$\"++#pxg#F2$!+bDT(o\"F27$$\"+g4t.FF2$!++Cdx;F27$$\"+!Hst!GF2$! +60dZ;F27$$\"+ERW9HF2$!+T8#Rf\"F27$$\"+KE>>IF2$!+)H%f<:F27$$\"+#RU07$F 2$!+Z@#)>9F27$$\"+?S2LKF2$!+Q,&>G\"F27$$\"+$p)=MLF2$!+>7>I6F27$$\"+*=] @W$F2$!+#)3[t$*F/7$$\"+]$z*RNF2$!++i$oL(F/7$$\"+kC$pk$F2$!+`\"R&zZF/7$ $\"+3qcZPF2$!+nD#G/#F/7$$\"+/\"fF&QF2$\"+`zWw6F/7$$\"+0OgbRF2$\"+CK+%p %F/7$$\"+nAFjSF2$\"+\"y7ey)F/7$$\"+&)*pp;%F2$\"+s\\!RJ\"F27$$\"+ye,tUF 2$\"+dnj-=F27$$\"+fO=yVF2$\"+B=^KBF27$$\"+E>#[Z%F2$\"+.+bgGF27$$\"+(G! e&e%F2$\"+u:&f^$F27$$\"+&)Qk%o%F2$\"+Uo9\\TF27$$\"+UjE!z%F2$\"+_Y#[([F 27$$\"+FM\"3%[F2$\"+Be1T_F27$$\"+60O\"*[F2$\"+o/%)>cF27$$\"+c-oX\\F2$ \"+)\\w5/'F27$$\"\"&F)$\"+F27$$\"+.TWA:F2$!+#)f]')>F27$$\"+L%Qg\\\"F2$!+Ezp0>F 2F[]l7'7$$\"+y!zFc\"F2$!+7fr8?F27$$\"+_UbqF27$$\"+&fuus\" F2$!+mE(*Q?F27$$\"+>YmBF2F`^l7'7$$\"+SsLrF27$$ \"+cFmy>F2$!+F0U=?F27$$\"+fq@J>F2$!+YC4d?F27$$\"+sQLO>F2$!+kfzj>F2Fe_l 7'7$$\"+I,:!)>F2$!+5eHu>F27$$\"+Kl^'=#F2$!+!>/d-#F27$$\"+poEQ@F2$!+!R1 51#F27$$\"+ApSX@F2$!+4:9o>F2Fj`l7'7$$\"+o\"**y=#F2$!+*3,O)>F27$$\"+gTV &R#F2$!+6*)R;?F27$$\"+!*QA[BF2$!+ZL*f0#F27$$\"+V\"zFN#F2$!+,Egi>F2F_bl 7'7$$\"+'*=!fR#F2$!+K>!o+#F27$$\"+)4)4/EF2$!+o!)>$*>F27$$\"+6q'*fDF2$! +Pv)H/#F27$$\"+&ex!eDF2$!+X#*H\\>F2Fdcl7'7$$\"+_ir1EF2$!+x*H7/#F27$$\" +3/&*4GF2$!+B+xe>F27$$\"+3`nrFF2$!+73NA?F27$$\"+PDAgFF2$!+Ua*3$>F2Fidl 7'7$$\"+'phD#GF2$!+WnR!3#F27$$\"+I;x5IF2$!+cKg>>F27$$\"+)>*=\")HF2$!+. 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#zw-(F27$$\"+m6E*)RF2$\"+D,)>5(F2Fg`\\n7'7$$\"+B'em5%F2$\"+z$Qn%oF27$$ \"+*pumA%F2$\"+@;E`rF27$$\"+#zw>A%F2$\"+vP*)fqF27$$\"+?TSzTF2$\"+55!R6 (F2F\\b\\n7'7$$\"+g#H/L%F2$\"+\\2`IoF27$$\"+G2d>WF2$\"+^#p%prF27$$\"+I %3QU%F2$\"+$3Ig2(F27$$\"+QOtwVF2$\"+VP9;rF2Fac\\n7'7$$\"+Dy%)[XF2$\"+M K2BoF27$$\"+H)=yh%F2$\"+mn#p<(F27$$\"+J$fui%F2$\"+`nz%3(F27$$\"+2IJyXF 2$\"+0P$e6(F2Ffd\\n7'7$$\"+`@-kZF2$\"+%RB#>oF27$$\"+n6J>[F2$\"+1mx!=(F 27$$\"+1'[C$[F2$\"+6!e+4(F27$$\"+#)GB#y%F2$\"+o!Q\\6(F2F[f\\n7'7$$\"+^ R@x\\F2$\"+14/ " 0 "" {MPLTEXT 1 0 55 "x1 := fsolve('gn'(x),x=1);\nx2 := fsolve('gn'(x),x=3.8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+Xr$Q(*)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+#= " 0 "" {MPLTEXT 1 0 65 "x1 := secant('gn'(x),x=1..1.1);\nx2 := secant('gn'(x) ,x=3.8..3.9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+Xr$Q(*)!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+#= " 0 "" {MPLTEXT 1 0 95 "dgn := unapply(subs(y(x)='gn '(x),rhs(de)),x);\nxmin := fsolve('dgn'(x),x=2..3):\n[xmin,gn(xmin)]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dgnGf*6#%\"xG6\"6$%)operatorG%& arrowGF(,&*$)9$\"\"#\"\"\"F1*&\"\"$F1,&F-F1-%#gnG6#F/F1#F1F0!\"\"F(F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+!\\y!)f#!\"*$!+zS\\(o\"F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Altern atively, the special procedure " }{TEXT 0 7 "findmin" }{TEXT -1 59 " c an be used to find the coordinates of the minimum point. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "findm in('gn'(x),x=2.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+!\\y!)f#! \"*$!+zS\\(o\"F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exa mple 3" }{TEXT 273 88 " .. working with a continuous solution to find \+ zeros and turning points of the solution " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 267 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 71 "This question is concerned wi th the first order differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = 1-x^2-x*y(x)^2;" "6#/*&%#dyG\"\"\"% #dxG!\"\",(F&F&*$%\"xG\"\"#F(*&F+F&*$-%\"yG6#F+F,F&F(" }{TEXT -1 3 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 33 "subject to the initial condition " } {XPPEDIT 18 0 "y(0) = 1/2;" "6#/-%\"yG6#\"\"!*&\"\"\"F)\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 162 "(a) Find a continuous numerical solution for the differential equation in the from of a Map le procedure which can be evaluated anywhere throughout the interval \+ " }{XPPEDIT 18 0 "-2 <= x;" "6#1,$\"\"#!\"\"%\"xG" }{XPPEDIT 18 0 "`` \+ <= 2;" "6#1%!G\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 119 " (b) Plot the graph of this numerical solution along with the gradient \+ field associated with the differential equation. " }}{PARA 0 "" 0 "" {TEXT -1 55 "(c) Find all the zeros of the solution in the interval " }{XPPEDIT 18 0 "-2 <= x;" "6#1,$\"\"#!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }{TEXT -1 23 " correct to 10 digits. " }}{PARA 0 " " 0 "" {TEXT -1 88 "(d) Find the coordinates of any maximum or minimum points on the graph of the solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 268 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "de := diff(y(x),x)=1-x^2-x*y(x)^2; \nic := y(0)=1/2;\ngn := desolveRK(\{de,ic\},y(x),x=-2..2,method=rk78) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,, (\"\"\"F.*$)F,\"\"#F.!\"\"*&F,F.)F)F1F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "p1 := plot('gn'(x),x=-2..2,color=red,thickness= 2):\np2 := DEtools[DEplot](de,y(x),x=-2..2,y=-1..1.5,\n color=bl ue,arrows=medium,dirgrid=[25,25]):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7jn7$$!\"# \"\"!$\"+t=h)z)!#57$$!+$Q6G\">!\"*$\"+/I(yY(F-7$$!+M!\\p$=F1$\"+.H7IiF -7$$!+))Qj^'***F1$!+7/z#y\"F-7$$!++Y0j&*F-$!+[*32v\"F-7$$!*]5*H\"*F1$!+u*GSo \"F-7$$!*I\"3&H)F1$!+N?1l9F-7$$!*Twp`(F1$!+W5vr6F-7$$!*P;bj'F1$!+4Y*e< (FR7$$!*Zh=(eF1$!+d#Hq^#FR7$$!*G\\N)\\F1$\"+%[c\\u$FR7$$!*ZUs>%F1$\"+@ 1`k**FR7$$!*GRXL$F1$\"+eXMUF1$\"+*f^-)\\F-7$$\")f`@') F1$\"+W)y%[eF-7$$\"*nZ)H;F1$\"+VR\\mlF-7$$\"*Ky*eCF1$\"+$yv%ysF-7$$\"* S^bJ$F1$\"+m()G>zF-7$$\"*0TN:%F1$\"+41qH%)F-7$$\"*7RV'\\F1$\"+y'Riz)F- 7$$\"+NcY9aF-$\"+X$\\6%*)F-7$$\"*:#fkeF1$\"+EIfU!*F-7$$\"+S30piF-$\"+G ?:'4*F-7$$\"*`4Nn'F1$\"+Cc/9\"*F-7$$\"+:bN0rF-$\"+^A3%4*F-7$$\"*],s`(F 1$\"+w+XM!*F-7$$\"+X\"=&GzF-$\"+oh*p%*)F-7$$\"*zM)>$)F1$\"+jKtG))F-7$$ \"*qfa<*F1$\"+Q+6o%)F-7$$\"*1O0)**F1$\"+!Hn)4!)F-7$$\"+#G2A3\"F1$\"+*Q H'=uF-7$$\"+$)G[k6F1$\"+5vpQnF-7$$\"+7yh]7F1$\"+(4Aa#fF-7$$\"+()fdL8F1 $\"+xy*e/&F-7$$\"+-FT=9F1$\"+;Y$\\/%F-7$$\"+Epa-:F1$\"+[inRHF-7$$\"+Sv &)z:F1$\"+&RBt!=F-7$$\"+GUYo;F1$\"+%o!)*fLFR7$$\"+o'*33F1$!+M%4BV&F-7$$\"+1j\"[$>F1$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "x1 := fsolve('gn'(x),x=-2..-1);\nx2 := fsolve ('gn'(x),x=-1..0);\nx3 := fsolve('gn'(x),x=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!+^ %#x2G$!+IN\"**\\&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"+ " 0 "" {MPLTEXT 1 0 94 "x1 := secant('gn'( x),x=-1.5..-1);\nx2 := secant('gn'(x),x=-1..0);\nx3 := secant('gn'(x), x=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!+^%#x2G$!+IN\"**\\&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"+ " 0 "" {MPLTEXT 1 0 158 "dgn := unapply(subs(y(x)='gn'(x),rhs(de)),x) ;\nxmin := fsolve('dgn'(x),x=-1):\nminpt:=[xmin,gn(xmin)];\nxmax := fs olve('dgn'(x),x=0.7):\nmaxpt := [xmax,gn(xmax)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dgnGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(\"\"\"F-*$) 9$\"\"#F-!\"\"*&F0F-)-%#gnG6#F0F1F-F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&minptG7$$!+Vm1;5!\"*$!+1cZ&y\"!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&maxptG7$$\"+JX!\\n'!#5$\"+Ne/9\"*F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Alternatively, the special procedures " }{TEXT 0 7 "findmin" }{TEXT -1 5 " and " }{TEXT 0 7 "findmax" }{TEXT -1 14 " can be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "minpt := findmin(' gn'(x),x=-1);\nmaxpt := findmax('gn'(x),x=0.7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&minptG7$$!+Vm1;5!\"*$!+1cZ&y\"!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&maxptG7$$\"+JX!\\n'!#5$\"+Ne/9\"*F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }{TEXT 274 116 " .. a \+ discrete solution obtained by the Runge-Kutta order 4 method compared \+ with a more accurate numerical solution " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 269 8 "Q uestion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 59 "This question is concerned with the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = sin(y^2)/(x+1);" "6#/*&%#dyG\" \"\"%#dxG!\"\"*&-%$sinG6#*$%\"yG\"\"#F&,&%\"xGF&F&F&F(" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 33 "subject to the initial condition " } {XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 66 "(a) Find a discrete numerical solution f or the over the interval " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" } {XPPEDIT 18 0 "``<=2" "6#1%!G\"\"#" }{TEXT -1 11 " using the " }{TEXT 262 29 "classical order 4 Runge-Kutta" }{TEXT -1 23 " method with 20 s teps. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "(b) The analytical solution for the initial value problem given in (a ) given by Maple's " }{TEXT 0 6 "dsolve" }{TEXT -1 101 " procedure can not easily be used to check the accuracy of the discrete solution foun d in (a). Find a " }{TEXT 262 29 "continuous numerical solution" } {TEXT -1 127 " for the initial value problem given in (a) in the form \+ of a procedure which can be evaluated anywhere throughout the interval " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=2" "6# 1%!G\"\"#" }{TEXT -1 69 ", and plot its graph along with the points of the discrete solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 209 "(c) Use this continuous solution to check the acc uracy of the discrete solution found in (a) by tabulating the values g iven by the discrete numerical solution along with those given by the \+ continuous solution." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 28 ": If you use the procedure " }{TEXT 0 9 "desolveRK" }{TEXT -1 42 " with variable step-size (via the option \"" }{TEXT 262 17 "stepsize= variable" }{TEXT -1 56 "\") and an adaptive Runge-Kutta methods via th e options \"" }{TEXT 262 11 "method=rk45" }{TEXT -1 6 "\" or \"" } {TEXT 262 11 "method=rk78" }{TEXT -1 112 "\", for example, the values \+ given by this continuous solution should be (for the most part) correc t to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "(c) Find the maximum relative error in the calculated va lues of the numerical solution found in (a). Where does this maximum r elative error occcur? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "de := diff(y( x),x)=sin(y(x)^2)/(x+1);\nic := y(0)=1;\npts := desolveRK(\{de,ic\},y( x),x=0..2,method=rk4,steps=20):\nfor i to nops(pts) do x[i-1]=pts[i,1] ,` `,y[i-1]=pts[i,2] end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# deG/-%%diffG6$-%\"yG6#%\"xGF,*&-%$sinG6#*$)F)\"\"#\"\"\"F4,&F,F4F4F4! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6#\"\"!$F'F'%%~~~~G/&%\"yGF&$\" \"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6#\"\"\"$\"+++++5!#5% %~~~~G/&%\"yGF&$\"+6N=%3\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&% \"xG6#\"\"#$\"+++++?!#5%%~~~~G/&%\"yGF&$\"+FB3n6!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6#\"\"$$\"+++++I!#5%%~~~~G/&%\"yGF&$\"+^3[Y7! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6#\"\"%$\"+++++S!#5%%~~~ ~G/&%\"yGF&$\"+F*e-K\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6 #\"\"&$\"+++++]!#5%%~~~~G/&%\"yGF&$\"+42!oQ\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6#\"\"'$\"+++++g!#5%%~~~~G/&%\"yGF&$\"+yKCX9!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6#\"\"($\"+++++q!#5%%~~~~G/ &%\"yGF&$\"+%Q%[&\\\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6# \"\")$\"+++++!)!#5%%~~~~G/&%\"yGF&$\"+`]+Q:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6#\"\"*$\"+++++!*!#5%%~~~~G/&%\"yGF&$\"+p\"F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6#\"#;$\"+++++;!\"*%%~~~~G/ &%\"yGF&$\"+so1-$\"+++++>!\"*%%~~~~G/&%\" yGF&$\"+5PAC " 0 "" {MPLTEXT 1 0 299 "de := diff(y(x),x)=sin(y(x) ^2)/(x+1);\nic := y(0)=1;\ngn := desolveRK(\{de,ic\},y(x),x=0..2,metho d=rk78,stepsize=variable);\nplot([pts,'gn'(x)],x=0..2,style=[point,lin e],symbol=circle,color=[blue,red],\n \+ labels=[`x`,`y(x)`]);\ncomparewithfcn(pts,'gn'(x),x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*& -%$sinG6#*$)F)\"\"#\"\"\"F4,&F,F4F4F4!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%777$$\"\"!F)$\"\"\"F) 7$$\"3/+++++++5!#=$\"3)******4^$=%3\"!#<7$$\"35+++++++?F/$\"3#******pK #3n6F27$$\"3))**************HF/$\"31+++^3[Y7F27$$\"3A+++++++SF/$\"3.++ +F*e-K\"F27$$\"3++++++++]F/$\"33+++42!oQ\"F27$$\"3w**************fF/$ \"3'******zFV_W\"F27$$\"3a**************pF/$\"31+++%Q%[&\\\"F27$$\"3U+ ++++++!)F/$\"3,+++`]+Q:F27$$\"3A+++++++!*F/$\"33+++ZI.;F27$$\"33+++++++6F2$\"3#********[xzi\"F27$$\"3%************* *>\"F2$\"3-+++<%*\\[;F27$$\"3/+++++++8F2$\"3)******>g1cm\"F27$$\"3!*** ***********R\"F2$\"3++++P]\"*z;F27$$\"3++++++++:F2$\"35+++*3J>p\"F27$$ \"33+++++++;F2$\"31+++so1-.^<\"F^s7$$\"+'4;[\\#Fgs$\"+Hs%p?\"F^s7$$\"+j'y]!HFgs$ \"+9a:R7F^s7$$\"+'zs$HLFgs$\"+__\\r7F^s7$$\"+8iI_PFgs$\"+%R3EI\"F^s7$$ \"+<_M(=%Fgs$\"+3PJL8F^s7$$\"+4y_qXFgs$\"+&)R=f8F^s7$$\"+]1!>+&Fgs$\"+ +0#pQ\"F^s7$$\"+]Z/NaFgs$\"+6VC89F^s7$$\"+]$fC&eFgs$\"+$pUrV\"F^s7$$\" +'z6:B'Fgs$\"+jcfd9F^s7$$\"+<=C#o'Fgs$\"+&3!R![\"F^s7$$\"+n#pS1(Fgs$\" +-TV)\\\"F^s7$$\"+j`A3vFgs$\"+qV,=:F^s7$$\"+n(y8!zFgs$\"+9[8M:F^s7$$\" +j.tK$)Fgs$\"+:#z0b\"F^s7$$\"+)3zMu)Fgs$\"+Mq4l:F^s7$$\"+#H_?<*Fgs$\"+ ,>8z:F^s7$$\"+!G;cc*Fgs$\"+$z$3\"f\"F^s7$$\"+4#G,***Fgs$\"+.%RIg\"F^s7 $$\"+!o2J/\"F^s$\"+sM]9;F^s7$$\"+%Q#\\\"3\"F^s$\"+P(\\Pi\"F^s7$$\"+;*[ H7\"F^s$\"+5q-L;F^s7$$\"+qvxl6F^s$\"+Nn*=k\"F^s7$$\"+`qn27F^s$\"+In#* \\;F^s7$$\"+cp@[7F^s$\"+R]8d;F^s7$$\"+3'HKH\"F^s$\"+EWak;F^s7$$\"+xanL 8F^s$\"+R;rq;F^s7$$\"+v+'oP\"F^s$\"+,h#on\"F^s7$$\"+S<*fT\"F^s$\"+I,)> o\"F^s7$$\"+&)Hxe9F^s$\"+:yA(o\"F^s7$$\"+.o-*\\\"F^s$\"+*)\\#=p\"F^s7$ $\"+TO5T:F^s$\"+#Q0jp\"F^s7$$\"+U9C#e\"F^s$\"+x&*Q+ezq\"F^s7$$\"+^j?4F^s$\"+F^s$\"+29OF " 0 "" {MPLTEXT 1 0 171 "de := diff(y(x),x)=sin(y(x) ^2)/(x+1);\nic := y(0)=1;\ngn2 := desolveRK(\{de,ic\},y(x),x=0..2,meth od=rk4,steps=20,output=procedure);\nplot(1-'gn2'(x)/'gn'(x),x=0..2,col or=blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6# %\"xGF,*&-%$sinG6#*$)F)\"\"#\"\"\"F4,&F,F4F4F4!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6&-%'CURVESG6#7bv7$$\"\"!F)F(7$$\"+ !#57$$\"+<0dL[F-$\"#HF37$$\"++-r2`F-$\"#[F 37$$\"+!))\\=y&F-$\"#nF37$$\"+l&*)fD'F-$\"$/\"F37$$\"+I*oU?(F-$\"$<#F3 7$$\"+#H[D:)F-$\"$$RF37$$\"+:w)*=#*F-$\"$C(F37$$\"+%pU&G5F3$\"%'3\"F37 $$\"+Em=N6F3FZ7$$\"+e0$=C\"F3$\"%(3\"F37$$\"+'R\"=\\8F3F\\o7$$\"+LA`c9 F3$\"%'4\"F37$$\"+_w?5:F3$\"%46F37$$\"+qI)Qc\"F3$\"%K6F37$$\"+!y?2f\"F 3$\"%Q6F37$$\"+*[evh\"F3$\"%a6F37$$\"+)>'RW;F3$\"%y6F37$$\"+3RBr;F3$\" %-7F37$$\"+W^\"\\)=F3$\"%j:F37$$\"+zjf)4#F3$\"%m>F37$$\"+Qiq'H#F3$\"%c >F37$$\"+'4;[\\#F3$\"%s>F37$$\"+<*)4YDF3$\"%!*>F37$$\"+Q+&F3$\"%njF37$$\"++FZ=_F3$\"%?iF37$$\"+]Z/NaF3$\"%2hF 37$$\"+v:A([&F3$\"%+hF37$$\"++%)RRbF3$\"%3hF37$$\"+D_d\"f&F3$\"%XhF37$ $\"+]?vVcF3$\"%4iF37$$\"++d5[dF3$\"%\\kF37$$\"+]$fC&eF3$\"%WpF37$$\"+g CAZfF3$\"%*o(F37$$\"+vb)>/'F3$\"%@#)F37$$\"+&o[n8'F3$\"%<\")F37$$\"+'z 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$$\"+XA(yx*F3$\"&6+\"F37$$\"+D-+%))*F3$\"&(=5F37$$\"+4#G,***F3$\"&b0\" F37$$\"+]-1@5!\"*$\"&i-\"F37$$\"+!o2J/\"Fb_m$\"%T**F37$$\"+cQq_5Fb_m$ \"%@)*F37$$\"+K+Ii5Fb_m$\"%R(*F37$$\"+?\")4n5Fb_m$\"%-(*F37$$\"+3i*=2 \"Fb_m$\"%*o*F37$$\"+'H%pw5Fb_m$\"%%p*F37$$\"+%Q#\\\"3\"Fb_m$\"%7(*F37 $$\"+>\"Fb_m$\"%$>* F37$$\"+#=-s>\"Fb_m$\"%m#*F37$$\"++4#)*>\"Fb_m$\"%1$*F37$$\"+='RC?\"Fb _m$\"%z#*F37$$\"+O$e]?\"Fb_m$\"%Y#*F37$$\"+`qn27Fb_mFgel7$$\"+/q%zA\"F b_m$\"%Q*)F37$$\"+cp@[7Fb_m$\"%!p)F37$$\"+>,Zf7Fb_m$\"%r&)F37$$\"+#GB2 F\"Fb_m$\"%x%)F37$$\"+k)\\jF\"Fb_m$\"%[%)F37$$\"+Xk(>G\"Fb_m$\"%Q%)F37 $$\"+EIg(G\"Fb_m$\"%L%)F37$$\"+3'HKH\"Fb_m$\"%f%)F37$$\"+xanL8Fb_m$\"% ;\")F37$$\"+wxEb8Fb_m$\"%')yF37$$\"+v+'oP\"Fb_m$\"%BxF37$$\"+M:v\"Q\"F b_m$\"%-xF37$$\"+#*Hk'Q\"Fb_m$\"%$p(F37$$\"+]W`\"R\"Fb_m$\"%'p(F37$$\" +3fU'R\"Fb_m$\"%0xF37$$\"+P;())R\"Fb_m$\"%:xF37$$\"+mtJ,9Fb_m$\"%3xF37 $$\"+&4jPS\"Fb_mFb`l7$$\"+C)3iS\"Fb_m$\"%dwF37$$\"+#G+6T\"Fb_m$\"%2wF3 7$$\"+S<*fT\"Fb_m$\"%dvF37$$\"+iBQP9Fb_m$\"%JtF37$$\"+&)Hxe9Fb_m$\"%Ir F37$$\"+Sk$)o9Fb_m$\"%gqF37$$\"+%*)**)y9Fb_m$\"%'*pF37$$\"+@;$R[\"Fb_m $\"%wpF37$$\"+[L'*)[\"Fb_m$\"%ipF37$$\"+w]*R\\\"Fb_m$\"%fpF37$$\"+.o-* \\\"Fb_m$\"%jpF37$$\"+A_1?:Fb_m$\"%wnF37$$\"+TO5T:Fb_m$\"%\"f'F37$$\"+ U9C#e\"Fb_m$\"%/jF37$$\"+1*3`i\"Fb_m$\"%hgF37$$\"+$*zym;Fb_m$\"%hdF37$ $\"+^j?4Fb_m$\"%_WF3 7$$\"+/Uac>Fb_m$\"%\\UF37$$\"\"#F)$\"%)3%F3-%'COLOURG6&%$RGBGF(F($\"*+ +++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!Fadn-%%VIEWG6$;F(Fbcn%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }{TEXT 276 54 " .. the numerical analogue of an indefinite integral " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 275 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 89 "(a) Construct an initial value problem consisting of a differential e quation of the form " }{XPPEDIT 18 0 "dy/dx = F(x);" "6#/*&%#dyG\"\"\" %#dxG!\"\"-%\"FG6#%\"xG" }{TEXT -1 38 " and an initial condition of th e form " }{XPPEDIT 18 0 "y(-1) = alpha;" "6#/-%\"yG6#,$\"\"\"!\"\"%&al phaG" }{TEXT -1 32 ", whose solution is the function" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = Int(exp(sin(u)),u = -1 .. x );" "6#/-%\"gG6#%\"xG-%$IntG6$-%$expG6#-%$sinG6#%\"uG/F1;,$\"\"\"!\"\" F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 218 "(b) Use a Runge-Kutta method with adaptive step-size to \+ obtain a continuous numerical solution of the initial value problem fo und in (a) in the form of a Maple procedure which can be evaluated any where in 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 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "( c) Plot a graph of " }{XPPEDIT 18 0 "y = g(x);" "6#/%\"yG-%\"gG6#%\"x G" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[-1,3]" "6#7$,$ \"\"\"!\"\"\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "(d) Find the value " }{XPPEDIT 18 0 "x = \+ r" "6#/%\"xG%\"rG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "x \+ = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 3 " "6#/%\"xG\"\"$" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "g(r) = 3; " "6#/-%\"gG6#%\"rG\"\"$" }{TEXT -1 17 ". Your value of " }{TEXT 277 1 "r" }{TEXT -1 32 " should be correct to 10 digits." }}{PARA 0 "" 0 " " {TEXT -1 23 "(e) Find the value of " }{XPPEDIT 18 0 "Int(g(x),x = - 1 .. r);" "6#-%$IntG6$-%\"gG6#%\"xG/F);,$\"\"\"!\"\"%\"rG" }{TEXT -1 43 ", to an accuracy of about 10 digits, where " }{TEXT 278 1 "r" } {TEXT -1 35 " is the number computed in part(d)." }}{PARA 0 "" 0 "" {TEXT 279 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 30 "(a) The differential equation " }{XPPEDIT 18 0 "dy/dx=exp(sin(x))" "6 #/*&%#dyG\"\"\"%#dxG!\"\"-%$expG6#-%$sinG6#%\"xG" }{TEXT -1 28 " with \+ the initial condition " }{XPPEDIT 18 0 "y(-1)=0" "6#/-%\"yG6#,$\"\"\"! \"\"\"\"!" }{TEXT -1 18 " has the solution " }{XPPEDIT 18 0 "g(x) = In t(exp(sin(u)),u = -1 .. x)" "6#/-%\"gG6#%\"xG-%$IntG6$-%$expG6#-%$sinG 6#%\"uG/F1;,$\"\"\"!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "We can check this with Maple's " }{TEXT 0 6 "dsolve" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "u := 'u':\nde := diff(y(x),x)=exp(sin(x));\nic := y( -1)=0;\ndsolve(\{de,ic\},y(x));\ng := unapply(rhs(%),x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,-%$expG6#-%$sin GF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$IntG6$-%$expG6#-%$sin G6#%$_z1G/F1;!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"x G6\"6$%)operatorG%&arrowGF(-%$IntG6$-%$expG6#-%$sinG6#%$_z1G/F5;!\"\"9 $F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 6 "Remark" }{TEXT -1 116 ": We can obtain numerical values for the solution by numerical in tegration and even plot the graph of the solution. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "(b) By constructing a \+ continuous solution for the initial value problem of part (a) in the f orm of a numerical procedure " }{TEXT 262 2 "gn" }{TEXT -1 50 " we obt ain a method of evaluation of the function " }{XPPEDIT 18 0 "g(x)" "6# -%\"gG6#%\"xG" }{TEXT -1 94 " which does not use numerical integration . To achieve the best results we can use the option \"" }{TEXT 262 15 "output=rkinterp" }{TEXT -1 6 "\" for " }{TEXT 0 9 "desolveRK" }{TEXT -1 131 " whereby the resulting procedure uses a fast interpolation met hod. With early versions of Maple the evaluation of such a procedure \+ " }{TEXT 262 2 "gn" }{TEXT -1 35 " was faster than the evaluation of \+ " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 175 " by numerical integration. Maple 8 and later versions have very fast numerical inte gration routines which use hardware floating point arithmetic. These r outines are used when " }{TEXT 262 6 "Digits" }{TEXT -1 112 " has the \+ default value of 10, and in such a situation there is no advantage in \+ using such a numerical procedure " }{TEXT 262 2 "gn" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The fol lowing loop evaluates the function " }{XPPEDIT 18 0 "g(x) = Int(exp(si n(u)),u = -1 .. x)" "6#/-%\"gG6#%\"xG-%$IntG6$-%$expG6#-%$sinG6#%\"uG/ F1;,$\"\"\"!\"\"F'" }{TEXT -1 51 " by numerical integration for 100 ra ndom values of " }{TEXT 280 1 "x" }{TEXT -1 17 " in the interval " } {XPPEDIT 18 0 "[-1,3]" "6#7$,$\"\"\"!\"\"\"\"$" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 80 "In a typical implementation of Maple on a PC the hardware routines are used for " }{TEXT 262 6 "Digits" } {XPPEDIT 18 0 "``<=14" "6#1%!G\"#9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "randomize() :\nDigits := 14:\nst := time():\nfor i to 100 do\n xx := 4*rand()*Fl oat(1,-12)-1;\n evalf(g(xx));\nend do:\ntime()-st;\nDigits := 10:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$c\"!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 262 6 "Digit s" }{TEXT -1 35 " is increased to 15, evaluation of " }{XPPEDIT 18 0 " g(x) = Int(exp(sin(u)),u = -1 .. x)" "6#/-%\"gG6#%\"xG-%$IntG6$-%$expG 6#-%$sinG6#%\"uG/F1;,$\"\"\"!\"\"F'" }{TEXT -1 42 " by numerical integ ration is much slower. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "randomize():\nDigits := 15:\nst := time():\nfor i to 100 do\n xx := 4*rand()*Float(1,-12)-1;\n evalf (g(xx));\nend do:\ntime()-st;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%4^!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Nevertheless we proceed to find a continuous num erical solution for the initial value problem in part (a). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "d e := diff(y(x),x)=exp(sin(x));\nic := y(-1)=0;\ngn := desolveRK(\{de,i c\},y(x),x=-1..3,method=rk67,output=rkinterp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,-%$expG6#-%$sinGF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 51 "A s a check, we may compare values of the procedure " }{TEXT 262 2 "gn" }{TEXT -1 37 " with those obtained by evaluating " }{XPPEDIT 18 0 "I nt(exp(sin(u)),u=-1..x)" "6#-%$IntG6$-%$expG6#-%$sinG6#%\"uG/F,;,$\"\" \"!\"\"%\"xG" }{TEXT -1 14 " numerically. " }}{PARA 0 "" 0 "" {TEXT -1 26 "Here is one such example. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "xx := evalf(2*sqrt(2));\neva lf(evalf[15](g(xx)));\nevalf(gn(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+CrUGG!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z![G\\'! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z![G\\'!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot('gn'(x),x=-1..3);" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"\"\"\"!$F*F *7$$!+LQ6G\"*!#5$\"+2:__Q!#67$$!+U.\\p$)F/$\"+U@&\\P(F27$$!+$))Qj^(F/$ \"+\"*[ob6F/7$$!+$=Kvl'F/$\"+:F\"Qg\"F/7$$!+Us!G!eF/$\"+yf)33#F/7$$!+4 yO5]F/$\"+JbmaDF/7$$!+vE%)*=%F/$\"+([573$F/7$$!+4WDTLF/$\"+u`\"*oOF/7$ $!+vvQ&\\#F/$\"+Gt_.VF/7$$!+n&4`i\"F/$\"+n;*G,&F/7$$!*%QW*e)F/$\"+W*Q, p&F/7$$\"'I,Q!\"*$\"+Z60Fbo7$$\"*cK78*Fbo$\"+H&>l3#Fbo7$$\"*Uc-)**Fbo$\"+'*ohy AFbo7$$\"+f`@'3\"Fbo$\"+@xq([#Fbo7$$\"+nZ)H;\"Fbo$\"+^F*on#Fbo7$$\"+Ky *eC\"Fbo$\"+!>ax)GFbo7$$\"+S^bJ8Fbo$\"+\"R7:6$Fbo7$$\"+0TN:9Fbo$\"+\\: '[L$Fbo7$$\"+7RV'\\\"Fbo$\"+??v`NFbo7$$\"+:#fke\"Fbo$\"+M$y#)z$Fbo7$$ \"+`4Nn;Fbo$\"+h&fx,%Fbo7$$\"+],s`$=Fbo$\"+h 1sdWFbo7$$\"+qfa<>Fbo$\"+%GB)zYFbo7$$\"+1O0)*>Fbo$\"+T?-$)[Fbo7$$\"+#G 2A3#Fbo$\"+@!*G)3&Fbo7$$\"+$)G[k@Fbo$\"+*p#)4G&Fbo7$$\"+7yh]AFbo$\"+.o UtaFbo7$$\"+()fdLBFbo$\"+o)*>\\cFbo7$$\"+-FT=CFbo$\"+iT$)=eFbo7$$\"+Ep a-DFbo$\"+>-vwfFbo7$$\"+Sv&)zDFbo$\"+ " 0 "" {MPLTEXT 1 0 32 "r := secant('gn'(x)=3,x=1..1.3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$\"+T>7*G\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 17 ": The procedure \+ " }{TEXT 0 6 "fsolve" }{TEXT -1 18 " can be used with " }{XPPEDIT 18 0 " g(x)=Int(exp(sin(u)),u = -1 .. x)" "6#/-%\"gG6#%\"xG-%$IntG6$-%$ex pG6#-%$sinG6#%\"uG/F1;,$\"\"\"!\"\"F'" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fsolve('g'(x)=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+T>7*G\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "(e) The problem is to find the double integral " } {XPPEDIT 18 0 "Int(Int(exp(sin(u)),u = -1 .. x),x = -1 .. r)" "6#-%$In tG6$-F$6$-%$expG6#-%$sinG6#%\"uG/F.;,$\"\"\"!\"\"%\"xG/F4;,$F2F3%\"rG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Newton-Cotes integration via the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 36 " along with the numerical procedure " }{TEXT 262 5 "gn(x)" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "Int(exp(sin(u )),u = -1 .. x)" "6#-%$IntG6$-%$expG6#-%$sinG6#%\"uG/F,;,$\"\"\"!\"\"% \"xG" }{TEXT -1 30 " provides one way to do this. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "NCint('gn'(x ),x=-1..r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)31[R#!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Actually, the double integral can be found by using numerical integration alone . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Int(g(x),x=-1..r);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-%$expG6#-%$sinG6#%$_z1G/F.;!\"\"%\"xG/F2 ;F1$\"+T>7*G\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)31[R#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "gn2 := x -> NCint(exp(sin(u)),u=-1..x);\nNCint('gn2'(x),x=-1.. r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gn2Gf*6#%\"xG6\"6$%)operator G%&arrowGF(-%&NCintG6$-%$expG6#-%$sinG6#%\"uG/F5;!\"\"9$F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)31[R#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 6 "Remark" }{TEXT -1 54 ": T he following code constructs a numerical procedure " }{TEXT 262 4 "gn1 5" }{TEXT -1 40 " which can be used to obtain values for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 53 " which are (for the most p art) correct to 15 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "de := diff(y(x),x)=exp(sin(x));\ni c := y(-1)=0;\ngn15 := evalf[15](desolveRK(\{de,ic\},y(x),x=-1..3,meth od=rk67,output=rkinterp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/- %%diffG6$-%\"yG6#%\"xGF,-%$expG6#-%$sinGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The numerical procedure " }{TEXT 39 4 "gn15" }{TEXT -1 58 " can be used to calculate a 15 digit value f or the number " }{TEXT 281 1 "r" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "g(r)=3" "6#/-%\"gG6#%\"rG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "evalf[15]( secant('gn15'(x)=3,x=1..1.3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0 #*)QT>7*G\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "evalf[15](fsolve('g'(x)=3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0#*)QT>7*G\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "With " }{TEXT 262 6 "Digits" }{TEXT -1 35 " set to 15 the numerical procedure " }{TEXT 262 4 "gn15" } {TEXT -1 23 " evaluates faster than " }{XPPEDIT 18 0 "g(x) = Int(exp(s in(u)),u = -1 .. x)" "6#/-%\"gG6#%\"xG-%$IntG6$-%$expG6#-%$sinG6#%\"uG /F1;,$\"\"\"!\"\"F'" }{TEXT -1 26 " by numerical integration." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "randomize():\nDigits := 15: \nst := time():\nfor i to 100 do\n xx := 4*rand()*Float(1,-12)-1;\n \+ evalf(gn15(xx));\nend do:\ntime()-st;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$N#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }