{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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3-+++&*4`D9F.7\$\$\"34+++++++OF;\$!3&******RbS4M\"F.7\$\$\"3=+++++++PF;\$!30 +++^;*>?\"F.7\$\$\"3#)*************z\$F;\$!3\$******4]&yB5F.7\$\$\"3\"******* ********QF;\$!3d*****f)*47@)F[p7\$\$\"\"%F(\$!3:+++<7+ygF[p7\$\$\"3k******** *****4%F;\$!3(******>qt3'RF[p7\$\$\"3=+++++++UF;\$!3x******G;Fj>F[p7\$\$\"3# )*************H%F;\$!3\$*******\\GYr;!#?7\$\$\"3O+++++++WF;\$\"3)******4d\"o0DNF[p7\$\$\"3=+++++++ZF;\$\"3*)*****p=D98%F[p7\$\$\"3#)*********** **z%F;\$\"3;+++YqESWF[p7\$\$\"3O+++++++\\F;\$\"36+++t#p2[%F[p7\$\$\"\"&F(\$\" 35+++-x:\"H%F[p-%'COLOURG6&%\$RGBG\$\"#5!\"\"F(F(-%&STYLEG6#%&POINTG-%&T ITLEG6#%@v(t),~50~steps,~step~size~=~0.1G-%+AXESLABELSG6\$%\"tG%\"vG" 2 294 294 294 5 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 6288 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plot([euler_ptsxv],labels = [`x`,` v`],style = POINT,\ntitle = `phase plane, 50 steps, step size = 0.1`); " }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6\$7U7\$\$\"\"\"\"\"!F*7\$F(\$!3 ++++++++]!#=7\$\$\"3c*************\\*F.\$!3A+++++++!*F.7\$\$\"3()********** ***f)F.\$!31++++++&>\"!#<7\$\$\"3Z++++++0uF.\$!3!************fQ\"F97\$\$\"3 \"*************=gF.\$!3)**********\\!z9F97\$\$\"3#)*********\\*RXF.\$!3!** *********=%[\"F97\$\$\"39+++++wbIF.\$!3\$********\\\\VT\"F97\$\$\"36++++0TT; F.\$!3&********fnUG\"F97\$\$\"3F++++!H9d\$!#>\$!3,+++0Y[46F97\$\$!3F+++]qTBvF V\$!32+++!H[W0*F.7\$\$!3.+++Mlyd;F.\$!3T+++zxQnoF.7\$\$!3\")*****>JDXM#F.\$!3 ))*****f&p,lYF.7\$\$!3%******z+F5\"GF.\$!3!)******34vfDF.7\$\$!3?+++*4-q1\$F .\$!3!*******HA(GU'FV7\$\$!30+++@3BJJF.\$\"3\$******>Fr'>5F.7\$\$!31+++%pj#HI F.\$\"3))******GCN\"Q#F.7\$\$!32+++^%G6z#F.\$\"3?+++!z8(>MF.7\$\$!3*******>2 d\"\\CF.\$\"37+++e_LJTF.7\$\$!32+++YN-O?F.\$\"3t*****>uY'HXF.7\$\$!3&******> ()eIe\"F.\$\"3\$******p;HtL;%F.7\$\$!3<+++'et0_#FV\$\"3#)*****R*e*[m\$F .7\$\$\"3.+++3BKW6FV\$\"3=+++&RXz0\$F.7\$\$\"3p*****HqnA?%FV\$\"38+++,-9*Q#F. 7\$\$\"3P+++/zS\"f'FV\$\"3D+++w()>,T9F.7\$\$\"3@+++d^J5SFV\$!3-+++&*4`D9F.7 \$\$\"3\"******>;%y%e#FV\$!3&******RbS4M\"F.7\$\$\"3.+++3O%QC\"FV\$!30+++^;* >?\"F.7\$\$\"33++++d>&=%!#@\$!3\$******4]&yB5F.7\$\$!3>+++SaL>)*!#?\$!3d***** f)*47@)FV7\$\$!3%******HaaI!=FV\$!3:+++<7+ygFV7\$\$!3)******\\ma3T#FV\$!3(** ****>qt3'RFV7\$\$!3*)*****\\.Up!GFV\$!3x******G;Fj>FV7\$\$!3:+++)>pK+\$FV\$!3 \$*******\\GYr;Fhw7\$\$!37+++FQ)*>IFV\$\"3)******4d \$)GFV\$\"3C+++rfK/EFV7\$\$!38+++t%fFi#FV\$\"3!******>\"o0DNFV7\$\$!3;+++#z`- F#FV\$\"3*)*****p=D98%FV7\$\$!3()*****HF6r&=FV\$\"3;+++YqESWFV7\$\$!3.+++oX3 89FV\$\"36+++t#p2[%FV7\$\$!3;+++2k2]'*Fhw\$\"35+++-x:\"H%FV-%'COLOURG6&%\$R GBG\$\"#5!\"\"F*F*-%&STYLEG6#%&POINTG-%&TITLEG6#%Gphase~plane,~50~steps ,~step~size~=~0.1G-%+AXESLABELSG6\$%\"xG%\"vG" 2 312 312 312 5 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 7568 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 221 "There are many examples of differential equations that cannot be solv ed analytically,\nso that approximation techniques are necessary. 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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