{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 268 "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 1 0 1 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 "Map le 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 "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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 64 "Checking the accuracy of a numeri cal solution by \"back-tracking\"" }}{PARA 0 "" 0 "" {TEXT -1 37 "by P eter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Vers ion: 13.8.2007" }}{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 "de solve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 268 7 "DEsol.m" }{TEXT -1 32 " is required by this worksheet . " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple sessio n by a command similar to the one that follows, where the file path gi ves its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \" K:\\\\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "Back-tracking to check the accuracy of a \+ numerical solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "First we construct a numerical solution for the different ial equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx =1-x^2-y^2" "6#/*&%#dyG\"\"\"%#dxG!\"\",(F&F&*$%\"xG\"\"#F(*$%\"yGF,F( " }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 27 "with the initial co ndition " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "de := diff(y(x),x)=1-x^2-y(x)^2;\nic := y(0)=0;\nfn \+ := desolveRK(\{de,ic\},x=0..2,method=rk78):\np1 := DEtools[DEplot](de, y(x),x=0..2,y=-1..1,arrows=medium,color=blue):\np2 := plot('fn(x)',x=0 ..2,y=-1..1,thickness=2,color=red):\nplots[display]([p1,p2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,(\"\" \"F.*$)F,\"\"#F.!\"\"*$)F)F1F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #icG/-%\"yG6#\"\"!F)" }}{PARA 13 "" 1 "" {GLPLOT2D 416 327 327 {PLOTDATA 2 "6&-%'CURVESG6^dl7'7$$!3#=%ot%*y:j_!#>$!\"\"\"\"!7$$\"3#=% ot%*y:j_F*F+7$$\"35x9DuO9%)HF*$!3/Uot%*y:85!#<7$F2$!3ay:j_5Uo)*!#=F.7' 7$$\"33$[[Q&4[j_F*$!3a\\-1/'oT***F:7$$\"3c)))4D1:*y:F:$!30vRfRJe+5F67$ $\"3s%HY.yp&\\8F:$!3eq=$)>x[85F67$$\"3%*phJya[_8F:$!3+Yu`catr)*F:FA7'7 $$\"3E`jB]NYz:F:$!3g[WEiepw**F:7$$\"3\"*>JblF1JEF:$!39bNx8/L-5F67$$\"3 c1KNA)evR#F:$!3#3S+_ImW,\"F67$$\"3`#)4A\"*3@4CF:$!3M;[<[Kw\"))*F:FV7'7 $$\"3n$eH%3m$owKFXF:$!3PgBv4N`A**F :Fjp7'7$$\"3g%Rt\")oUfv%F:$!3!f5(psr\\f)*F:7$$\"3-*G+85t.x&F:$!3I*GIFG ]S,\"F67$$\"3>2(Ruw;c^&F:$!3J$)p+7nk?5F67$$\"3o`64\"=oee&F:$!3a4\")y;& fG&**F:F_r7'7$$\"3\\i<&4(4$p#eF:$!3Q#>y.F)*\\!)*F:7$$\"3'yl;k(zk/oF:$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "yend := evalf[15](fn(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%yendG$!09HG3YW+\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Now we construct the backward solution with " } {XPPEDIT 18 0 "y(2)=-1.00444608282914" "6#/-%\"yG6#\"\"#,$-%&FloatG6$ \"09HG3YW+\"!#9!\"\"" }{TEXT -1 26 " as the initial condition." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "de := diff(y(x),x)=1-x^2-y(x)^2;\nic := y(2)=yend;\nfb := desolveR K(\{de,ic\},x=0..2,method=rk78):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #deG/-%%diffG6$-%\"yG6#%\"xGF,,(\"\"\"F.*$)F,\"\"#F.!\"\"*$)F)F1F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"#$!09HG3YW+\"!#9 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 "It we back-track over the interval from \+ " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2" "6#/%\"xG\"\"#" }{TEXT -1 85 ", we return approximately t o the initial condition of the original forward solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "fb(0) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!'U9z!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The difference from the o riginal initial " }{TEXT 263 1 "y" }{TEXT -1 134 " value is only of th e order the machine precision, when working with 10 digits, which is a bout as accurate as we could ever hope for. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "Sensititivity problems in connection with \"back-tracking\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "First we construct a numerical solution for the different ial equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = exp(2*y-y^2)+sin(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\",&-%$expG6#,&*&\" \"#F&%\"yGF&F&*$F0F/F(F&-%$sinG6#%\"xGF&" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 27 "with the initial condition " }{XPPEDIT 18 0 "y(0) = -2" "6#/-%\"yG6#\"\"!,$\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 319 "de := dif f(y(x),x)=exp(2*y(x)-y(x)^2)+sin(x);\nic := y(0)=-2;\ngn := desolveRK( \{de,ic\},x=0..8,method=rk78):\np1 := DEtools[DEplot](de,y(x),x=0..8,y =-2.5..3.8,arrows=medium,\n dirgrid=[ 25,25],color=blue):\np2 := plot('gn(x)',x=0..8,y=-2.5..3.8,thickness=2 ,color=red):\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$expG6#,&*&\"\"#\"\"\"F)F4F4 *$)F)F3F4!\"\"F4-%$sinGF+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/ -%\"yG6#\"\"!!\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 528 371 371 {PLOTDATA 2 "6&-%'CURVESG6_bm7'7$$!3;=$Rkmmmm\"!#=$!30UH)y;-+]#!#<7$$\"3;=$Rkmmm m\"F*$!3%z0<@$y***\\#F-7$$\"3M2!)=Ar&)\\%*!#>$!3FKM$3F6G`#F-7$$\"3-2m# )y%>(\\%*F6$!3'RQU3F'=nCF-F.7'7$$\"3K8cHmgB%z\"F*$!3CYCJ[/O]DF-7$$\"3q _5P+1Vs[F*$!3w`vo^&R'\\CF-7$$\"3g]KQtj&eO%F*$!3dMY#HCZ<]#F-7$$\"3g()*3 _\\2h/%F*$!3RC].)GX6W#F-FD7'7$$\"3!H209BReN&F*$!3?uBL:$f5e#F-7$$\"3qf# G>5%\\xzF*$!3zDwm%oS*=CF-7$$\"3o/lDK.AnwF*$!3?V\\%HJZ)zCF-7$$\"3Wteo7! 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The difference is a bit la rger than is desirable, but this may not mean that the forward solutio n is inaccurate. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "gn(0)-gb(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"'!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The difference is a bit larger than is desirable, but th is may not mean that the forward solution is inaccurate." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 199 "If we can constru ct a more accurate forward solution using 15 digits of precision, wher e 20 digits are used as a working precision, the end point should cert ainly be accurate to more than 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "de := diff(y(x),x )=exp(2*y(x)-y(x)^2)+sin(x);\nic := y(0)=-2;\ngn2 := evalf[15](desolve RK(\{de,ic\},x=0..8,method=rk78)):\nevalf[15](gn2(8));\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$expG6#,&*& \"\"#\"\"\"F)F4F4*$)F)F3F4!\"\"F4-%$sinGF+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0(3rX1@,P!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Comparing the " }{TEXT 265 1 "y" }{TEXT -1 177 " coo rdinate of the new end point 3.70121064571087 with the value 3.701210 64581390 found previously, we see that the previous end point is prob ably accurate to about 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf[15]((gn(8)-gn2(8))/gn2 (8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0$)o*oPo$y#!#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 375 "Looking at the direction field, notice that, at one stage along the solution curve, \+ the arrows appear to converge towards the curve when going from left t o right, but going from right to left we would have divergence away fr om the solution curve. This means that the backward solution is more l ikely to diverge from the true solution curve than the original forwar d solution." }}{PARA 0 "" 0 "" {TEXT -1 127 "If we back-track from a p oint preceding the region where this backward divergence occurs, we re turn very close to the original " }{TEXT 266 1 "y" }{TEXT -1 7 " value ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "de := diff(y(x),x)=exp(2*y(x)-y(x)^2)+sin(x);\nic := y(4)=evalf[15](gn(4));\ngb2 := desolveRK(\{de,ic\},x=0..4,method=rk78 ):\nevalf[15](gn(0)-gb2(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG /-%%diffG6$-%\"yG6#%\"xGF,,&-%$expG6#,&*&\"\"#\"\"\"F)F4F4*$)F)F3F4!\" \"F4-%$sinGF+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\" %$\"0L%p&eVG=#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%&)R!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "An interesting 1st orde r linear DE" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 119 "The differential equation in this example comes \+ from \"Mathematica in Action\" by Stan Wagon, Springer-Verlag, page 29 7. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We solve the initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx+x*cos(Pi*x)^2*y = x*cos(x),y(0) = 0;" "6$/,&*&%# dyG\"\"\"%#dxG!\"\"F'*(%\"xGF'*$-%$cosG6#*&%#PiGF'F+F'\"\"#F'%\"yGF'F' *&F+F'-F.6#F+F'/-F36#\"\"!F:" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 76 "both analytically and numerically, and plot the graph of both s olutions for " }{TEXT 267 1 "x" }{TEXT -1 18 " between 0 and 10." }} {PARA 0 "" 0 "" {TEXT -1 69 "We compare the values of the analytical a nd numerical solutions when " }{XPPEDIT 18 0 "x = 10;" "6#/%\"xG\"#5" }{TEXT -1 25 ", and solve the equation " }{XPPEDIT 18 0 "y(x)=-3" "6#/ -%\"yG6#%\"xG,$\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "Note that the differential equation is linear." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "de := diff( y(x),x)+x*cos(Pi*x)^2*y(x)=x*cos(x);\nic := y(0)=0;\ndesolve(\{de,ic\} ,method=linear,info=true):\nk := unapply(rhs(%),x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(F-F.)-%$cos G6#*&%#PiGF.F-F.\"\"#F.F*F.F.*&F-F.-F2F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Linear~DE~.~.~~G/,&-%%Diff G6$-%\"yG6#%\"xGF,\"\"\"*(F,F-)-%$cosG6#*&%#PiGF-F,F-\"\"#F-F)F-F-*&F, F--F1F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9Integrating~factor~.~.~~G-%$expG6#-%$IntG6$*&%\"xG\" \"\")-%$cosG6#*&%#PiGF,F+F,\"\"#F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%5~~~~~~~~~~~~~~~~=~~~G-%$expG6#,$*&#\"\"\"\"\"%F**&,**,\"\"#F*%#PiG F*%\"xGF*-%$cosG6#*&F0F*F1F*F*-%$sinGF4F*F**&)F0F/F*)F1F/F*F**$)F2F/F* F*F*!\"\"F*F0!\"#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"yG6#%\"xG\"\"\"-%$expG6#,$*&#F)\"\"%F )*&,**,\"\"#F)%#PiGF)F(F)-%$cosG6#*&F5F)F(F)F)-%$sinGF8F)F)*&)F5F4F))F (F4F)F)*$)F6F4F)F)F)!\"\"F)F5!\"#F)F)F)-%$IntG6$*(F(F)-F7F'F)F*F)F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#/*&-%\"yG6#%\"xG\"\"\"-%$expG6#,$*&#F)\"\"%F)*&,**,\"\"#F)%#PiGF)F(F )-%$cosG6#*&F5F)F(F)F)-%$sinGF8F)F)*&)F5F4F))F(F4F)F)*$)F6F4F)F)F)!\" \"F)F5!\"#F)F)F),&-%$IntG6$*(%#_uGF)-F76#FHF)-F+6#,$*&F/F)*&,**,F4F)F5 F)FHF)-F76#*&F5F)FHF)F)-F;FSF)F)*&F=F))FHF4F)F)*$)FRF4F)F)F)FAF)F5FBF) F)F)/FH;%!GF(F)&%\"CG6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%EApplying~the~initial~condition~.~.~~ G/&%\"CG6#\"\"\",$-%$IntG6$*(%#_vGF(-%$cosG6#F.F(-%$expG6#,$*&#F(\"\"% F(*&,**,\"\"#F(%#PiGF(F.F(-F06#*&F=F(F.F(F(-%$sinGF?F(F(*&)F=FF%\"kGf*6#%\"x G6\"6$%)operatorG%&arrowGF(,$-%$IntG6$*(%#_vG\"\"\"-%$cosG6#F1F2-%$exp G6#,$*&#F2\"\"%F2*&,**,\"\"#F2%#PiGF2F1F2-F46#*&FAF2F1F2F2-%$sinGFCF2F 2*&)FAF@F2)F1F@F2F2*$)FBF@F2F2F2!\"\"F2FA!\"#F2F2F2/F1;\"\"!9$*$-F76#, $*&F;F2*&,**,F@F2FAF2FQF2-F46#*&FAF2FQF2F2-FFFenF2F2*&FHF2)FQF@F2F2*$) FZF@F2F2F2FLF2FAFMF2F2FLF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 65 "Because of the unevaluated integral the c omputation of values of " }{XPPEDIT 18 0 "k(x)" "6#-%\"kG6#%\"xG" } {TEXT -1 100 " was slow with Maple 7. 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" 0 "" {MPLTEXT 1 0 214 "xx := 10 ;\nk:=x->Int(_u*cos(_u)*exp((2*Pi*_u*cos(Pi*_u)*sin(Pi*_u)+\nPi^2*_u^2 -1+cos(Pi*_u)^2)/(4*Pi^2)),_u=0..x)/\n exp((2*Pi*x*cos(Pi*x)*sin(Pi*x )+Pi^2*x^2-1+cos(Pi*x)^2)/(4*Pi^2));\nevalf(evalf[14](k(xx)));\nkn(xx) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%$IntG6$*(% #_uG\"\"\"-%$cosG6#F1F2-%$expG6#,$*&#F2\"\"%F2*&,**,\"\"#F2%#PiGF2F1F2 -F46#*&FAF2F1F2F2-%$sinGFCF2F2*&)FAF@F2)F1F@F2F2F2!\"\"*$)FBF@F2F2F2FA !\"#F2F2F2/F1;\"\"!9$*$-F76#,$*&F;F2*&,**,F@F2FAF2FQF2-F46#*&FAF2FQF2F 2-FFFenF2F2*&FHF2)FQF@F2F2*$)FZF@F2F2F2FJF2FAFMF2F2FJF(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+D<&\\K\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+D<&\\K\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "The next computation is slow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fsol ve(k(x)=-3,x=9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+HZ*f]*!\"*" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Using th e numerical solution produces the same result with little delay." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "fsolve('kn(x)'=-3,x=9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+HZ* f]*!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "It we " }{TEXT 261 10 "back-track" }{TEXT -1 24 " over the interval from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x = 5" "6#/%\"xG\"\"&" }{TEXT -1 84 ", we return approx imately to the initial condition of the original forward solution." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "de := diff(y(x),x)+x*cos(Pi*x)^2*y(x)=x*cos(x);\nic := y(5)=evalf [15](kn(5));\nrn := desolveRK(\{de,ic\},x=0..5):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\"\"*(F-F.)-%$cosG6 #*&%#PiGF.F-F.\"\"#F.F*F.F.*&F-F.-F2F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"&$!04#pC\")RpC!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "rn(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"* " 0 "" {MPLTEXT 1 0 112 "de := diff(y(x),x)+x*cos(Pi*x)^2*y(x)=x*cos(x);\nic \+ := y(10)=evalf[15](kn(10));\nsn := desolveRK(\{de,ic\},x=0..10):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xGF-\"\" \"*(F-F.)-%$cosG6#*&%#PiGF.F-F.\"\"#F.F*F.F.*&F-F.-F2F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"#5$!0Q+Xs^\\K\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "sn( 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4N#*eo!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 180 "The problem here is n ot so much the accuracy of the original forward solution, but that the backward solution is sensitive to its initial condition, and is conse quently inaccurate." }}{PARA 0 "" 0 "" {TEXT -1 78 "The reason for thi s is apparent when we draw a portion of the direction field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "d e := diff(y(x),x)+x*cos(Pi*x)^2*y(x)=x*cos(x);\np1 := DEtools[DEplot]( de,y(x),x=0..4,y=-2.4..0.6,arrows=medium,\n dirgri d=[40,30],color=blue):\np2 := plot(kn,0..4,color=red,thickness=2):\npl ots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%% diffG6$-%\"yG6#%\"xGF-\"\"\"*(F-F.)-%$cosG6#*&%#PiGF.F-F.\"\"#F.F*F.F. *&F-F.-F2F,F." }}{PARA 13 "" 1 "" {GLPLOT2D 486 342 342 {PLOTDATA 2 "6 &-%'CURVESG6]fo7'7$$!3.G^?G^?G^!#>$!3!**************R#!#<7$$\"3.G^?G^? 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" }}{PARA 0 "" 0 "" {TEXT -1 44 "Compare the values of the two solutio ns for " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "(d) Does any improvement either the forward or backward solution result from u sing the error condition " }{TEXT 268 27 "\"errorcontrol=accumulative \"" }{TEXT -1 20 " with the procedure " }{TEXT 0 9 "desolveRK" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 42 "______________________________ ____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 42 "_______________________________________ ___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Use the p rocedure " }{TEXT 0 9 "desolveRK" }{TEXT -1 71 " to find a continuous \+ numerical solution for the differential equation " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = sin(x*y);" "6#/*&%#dyG\"\"\"% #dxG!\"\"-%$sinG6#*&%\"xGF&%\"yGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "with the initial condition " }{XPPEDIT 18 0 "y(0) = 1;" " 6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 5 " "6#/%\"xG\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 "(b) Plot the graph of the solution along \+ with the associated gradient field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 128 "(c) Check the solution by back-tracking , that is, by constructing a second continuous numerical solution over the interval from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 5" "6#/%\"xG\"\"&" }{TEXT -1 69 ", by usi ng the end point of the first solution as the starting point." }} {PARA 0 "" 0 "" {TEXT -1 44 "Compare the values of the two solutions f or " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "(d) Does any imp rovement in either the forward or backward solution result from using \+ the error condition " }{TEXT 268 27 "\"errorcontrol=accumulative\"" } {TEXT -1 20 " with the procedure " }{TEXT 0 9 "desolveRK" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 42 "___________________________________ _______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 42 "__________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 }