{VERSION 2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "As an example of Euler's method for seco nd-order equations,\nconsider the following damped, nondriven mass-spr ing model.\n" }{XPPEDIT 18 0 "d^2x/dt^2 + 2*(dx/dt) + 5*x = 0" "/,(*( %\"dG\"\"#%\"xG\"\"\"*$%#dtG\"\"#!\"\"F(*&\"\"#F(*&%#dxGF(F*F,F(F(*&\" \"&F(F'F(F(\"\"!" }{TEXT -1 9 " " }{XPPEDIT 18 0 "x(0) = 1; dx /dt = 0" "C$/-%\"xG6#\"\"!\"\"\"/*&%#dxG\"\"\"%#dtG!\"\"F'" }{MPLTEXT 1 0 1 "\n" }{TEXT -1 193 "\nBy considering velocity to be a second ind ependent variable, we\ncan implement Euler's scheme for systems of sec ond-order equations.\nThe initial-value problem above is equivalent to the system\n" }{XPPEDIT 18 0 "dx/dt = v," "6$/*&%#dxG\"\"\"%#dtG!\"\" %\"vG%(UNKNOWNG" }{TEXT -1 27 " " }{XPPEDIT 18 0 "x(0) = 1" "/-%\"xG6#\"\"!\"\"\"" }{TEXT -1 1 "\n" }{XPPEDIT 18 0 "dv/dt = -2v - 5x," "6$/*&%#dvG\"\"\"%#dtG!\"\",&*&\"\"#F&%\"vGF&F(* &\"\"&F&%\"xGF&F(%(UNKNOWNG" }{TEXT -1 10 " " }{XPPEDIT 18 0 "v(0) = 0" "/-%\"vG6#\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := (x,v,t) -> v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6%% \"xG%\"vG%\"tG6\"6$%)operatorG%&arrowGF*9%F*F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "g := (x,v,t) -> -2*v-5*x;\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"gG:6%%\"xG%\"vG%\"tG6\"6$%)operatorG%&arrowGF*,&9 %!\"#9$!\"&F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Define the ini tial values:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " t[0] := 0; x[0] := 1; v[0] := 0;" }}{PARA 0 "" 0 "" {TEXT -1 42 "and t he number of steps and the step size:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "N := 50; h := 0.1;" }}{PARA 0 "" 0 "" {TEXT -1 195 "As in the predator-prey example, we'll create three sequences \+ of points: one in the t-x plane,\none in the t-v plane and one in the x-v plane. Start by defining the first point in\neach sequence:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "euler_ptsx := [t[0],x[0]];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "euler_ptsv := [t[0],x[0]];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "euler_ptsxv := [x[0],v[0]];" }} {PARA 0 "" 0 "" {TEXT -1 45 "Define the number of steps and the step s ize:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for n fr om 0 to N-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " t[n+1] := t[n ] + h;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " x[n+1] := x[n] + h*f(x [n],v[n],t[n]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " v[n+1] := v[n ] + h*g(x[n],v[n],t[n]);" }}{PARA 0 "" 0 "" {TEXT -1 60 " \+ new value = old value + stepsize * slope" }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " new_ptx := [ t[n+1], x[n+1] ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " new_ptv := [ t[n+1], v[n+1] ]; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " new_ptxv := [ x[n+1], v[n+1] ];" }}{PARA 0 "" 0 "" {TEXT -1 78 " define the new est points in the three sequences of points." }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " euler_ptsx := euler_ptsx, new_pt x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " euler_ptsv := euler_ptsv, \+ new_ptv;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " euler_ptsxv := euler _ptsxv, new_ptxv;" }}{PARA 0 "" 0 "" {TEXT -1 90 " \+ modify each of the three sequences by adding the new point at the end. " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"t G6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\" \"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+euler_ptsxG7$\"\"!\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+euler_ptsvG7$\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%,euler_ptsxvG7$\"\"\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot([euler_ptsx],labels = [ `t`,`x`],style=POINT,\ntitle = `x(t), 10 steps, step size=0.5`);" }} {PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7U7$\"\"!$\"\"\"F(7$$\"31++++ +++5!#=F)7$$\"36+++++++?F.$\"3c*************\\*F.7$$\"3*)************* *HF.$\"3()*************f)F.7$$\"3A+++++++SF.$\"3Z++++++0uF.7$$\"3+++++ +++]F.$\"3\"*************=gF.7$$\"3y**************fF.$\"3#)*********\\ *RXF.7$$\"3c**************pF.$\"39+++++wbIF.7$$\"3W+++++++!)F.$\"36+++ +0TT;F.7$$\"3A+++++++!*F.$\"3F++++!H9d$!#>7$F)$!3F+++]qTBvFW7$$\"34+++ ++++6!#<$!3.+++Mlyd;F.7$$\"3'**************>\"Fhn$!3\")*****>JDXM#F.7$ $\"3/+++++++8Fhn$!3%******z+F5\"GF.7$$\"3\"**************R\"Fhn$!3?+++ *4-q1$F.7$$\"3++++++++:Fhn$!30+++@3BJJF.7$$\"34+++++++;Fhn$!31+++%pj#H IF.7$$\"3'**************p\"Fhn$!32+++^%G6z#F.7$$\"3/+++++++=Fhn$!3**** ***>2d\"\\CF.7$$\"3\"***************=Fhn$!32+++YN-O?F.7$$\"\"#F($!3&** ****>()eIe\"F.7$$\"34+++++++@Fhn$!3$******\\&f))=6F.7$$\"3=+++++++AFhn $!3k******zn%Ro'FW7$$\"3#)*************H#Fhn$!3<+++'et0_#FW7$$\"3\"*** ***********R#Fhn$\"3.+++3BKW6FW7$$\"3++++++++DFhn$\"3p*****HqnA?%FW7$$ \"34+++++++EFhn$\"3P+++/zS\"f'FW7$$\"3=+++++++FFhn$\"3M+++!o1EH)FW7$$ \"3#)*************z#Fhn$\"3a+++1`*RK*FW7$$\"3\"***************GFhn$\"3 X*****H(eZM(*FW7$$\"\"$F($\"3`+++h0m'f*FW7$$\"34+++++++JFhn$\"3p+++=Xo ***)FW7$$\"3=+++++++KFhn$\"30+++b1FU!)FW7$$\"3#)*************H$Fhn$\"3 _******Q`NEoFW7$$\"3\"**************R$Fhn$\"3<+++`&4:X&FW7$$\"3+++++++ +NFhn$\"3@+++d^J5SFW7$$\"34+++++++OFhn$\"3\"******>;%y%e#FW7$$\"3=++++ +++PFhn$\"3.+++3O%QC\"FW7$$\"3#)*************z$Fhn$\"33++++d>&=%!#@7$$ \"3\"***************QFhn$!3>+++SaL>)*!#?7$$\"\"%F($!3%******HaaI!=FW7$ $\"3k*************4%Fhn$!3)******\\ma3T#FW7$$\"3=+++++++UFhn$!3*)***** \\.Up!GFW7$$\"3#)*************H%Fhn$!3:+++)>pK+$FW7$$\"3O+++++++WFhn$! 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Thi s example, however, can be solved\nexactly. You should verify that th e solution is\n" }{XPPEDIT 18 0 "x(t) = exp(-t) (cos(2t)+sin(2t)/2)" " /-%\"xG6#%\"tG--%$expG6#,$F&!\"\"6#,&-%$cosG6#*&\"\"#\"\"\"F&F4F4*&-%$ sinG6#*&\"\"#F4F&F4F4\"\"#F,F4" }{MPLTEXT 1 0 1 "\n" }{XPPEDIT 18 0 "v (t) = -5/2*exp(-t)*(sin(2t)" "/-%\"vG6#%\"tG,$**\"\"&\"\"\"\"\"#!\"\"- %$expG6#,$F&F,F*-%$sinG6#*&\"\"#F*F&F*F*F," }{TEXT -1 68 "\nWe can the refore see how good Euler's scheme is for this step size." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eulerplot := plot([euler_ptsx],styl e=POINT):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "exactplot := p lot(exp(-t)*(cos(2*t)+sin(2*t)/2),t=0..5):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "display(\{eulerplot,exactplot\},labels=[`t`,`x`],\ntitle = `Eu ler method and exact solution`);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CU RVESG6$7^o7$\"\"!$\"\"\"F(7$$\"3hmmTN@Ki8!#>$\"38BX![S-a***!#=7$$\"3@L L$3FWYs#F.$\"3Fe?os*y<)**F17$$\"3#)***\\iSmp3%F.$\"3C\"yGfx%Qf**F17$$ \"3VmmmT&)G\\aF.$\"3$Gkz#fgZG**F17$$\"3k****\\7G$R<)F.$\"38@yKcM9G&4)F17$$\"3qLLL$eI8k$F1$\"3oPI r?sp(\\(F17$$\"32ML$3x%3yTF1$\"3Q]MjAFleoF17$$\"3h+]PfyG7ZF1$\"3*)o$Qu R2V>'F17$$\"3fmm\"z%4\\Y_F1$\"3T3;q4Rb8bF17$$\"31++v$flMLe*)>VB$)F1$\"3#fU*)HV9xv\"F17$$\"3xmmTg()4_))F1$\" 3#47e\\9DR?\"F17$$\"3Y++DJbw!Q*F1$\"37/P&4%*Qv*oF.7$$\"3%ommTIOo/\"!#< $!3.!fk_0gKK#F.7$$\"3YLL3_>jU6Fhr$!3r!)))y!*[Y^))F.7$$\"38++]i^Z]7Fhr$ !34C!4PDC+W\"F17$$\"33++](=h(e8Fhr$!3eRyJh9Y8=F17$$\"3&*****\\7!Q4T\"F hr$!3yhG%*[0BK>F17$$\"3/++]P[6j9Fhr$!3*[PW'>GG9?F17$$\"3%o;HKR'\\5:Fhr $!32p_v0g9f?F17$$\"3VL$e*[z(yb\"Fhr$!3?2\"f'45#z2#F17$$\"34+Dc,#>Uh\"F hr$!3\\x4k`;Gp?F17$$\"3wmm;a/cq;Fhr$!36%)oF]QaI?F17$$\"3\"pm;a)))G=F17$$\"3%ommmJF17$$\"3/+]iSj0x=Fhr $!3ck8nwE6#p\"F17$$\"3gmmm\"pW`(>Fhr$!3h\"GqZIW#f9F17$$\"3L+]i!f#=$3#F hr$!3jH[u*)3#)y6F17$$\"3?+](=xpe=#Fhr$!38np<$yB20*F.7$$\"37nm\"H28IH#F hr$!3@?#yS0I-G'F.7$$\"3vm;zpSS\"R#Fhr$!3B&oYbN1*>RF.7$$\"3HLL3_?`(\\#F hr$!3_E1)3D%)el\"F.7$$\"3fL$e*)>pxg#Fhr$\"3!4@-#QG,aK!#?7$$\"33+]Pf4t. 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