{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 } {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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 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 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 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 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 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 " This worksheet models the mo tion of a pendulum whose rod is a spring." }}{PARA 0 "" 0 "" {TEXT -1 71 " At the end of the worksheet is an animation of the springy-pend ulum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl ots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords h as been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot setup(gdi);" }}{PARA 7 "" 1 "" {TEXT -1 29 "Warning, unknown plot devi ce\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " The spring pen dulum system is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "diff(theta(t),`$`(t,2)) = -(2*diff(r (t),t)*diff(theta(t),t)+g*sin(theta))/r;" "6#/-%%diffG6$-%&thetaG6#%\" tG-%\"$G6$F*\"\"#,$*&,&*(F.\"\"\"-F%6$-%\"rG6#F*F*F3-F%6$-F(6#F*F*F3F3 *&%\"gGF3-%$sinG6#F(F3F3F3F7!\"\"FB" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "diff (r(t),`$`(t,2)) = r*diff(theta(t),t)^2+g*cos(theta)-K*r+K*r[eq];" "6#/ -%%diffG6$-%\"rG6#%\"tG-%\"$G6$F*\"\"#,**&F(\"\"\"*$-F%6$-%&thetaG6#F* F*F.F1F1*&%\"gGF1-%$cosG6#F6F1F1*&%\"KGF1F(F1!\"\"*&F>F1&F(6#%#eqGF1F1 " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 53 " In the spirit of Park er the system is projected as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "y[1] = theta;" "6#/& %\"yG6#\"\"\"%&thetaG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y[2] = diff(t heta(t),t);" "6#/&%\"yG6#\"\"#-%%diffG6$-%&thetaG6#%\"tGF." }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y[3] = r;" "6#/&%\"yG6#\"\"$%\"rG" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y[4] = diff(r(t),t);" "6#/&%\"yG6#\"\"%-%%dif fG6$-%\"rG6#%\"tGF." }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y[5] = sin(theta );" "6#/&%\"yG6#\"\"&-%$sinG6#%&thetaG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "y[6] = cos(theta);" "6#/&%\"yG6#\"\"'-%$cosG6#%&thetaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "y[7] = 1/r;" "6#/&%\"yG6#\"\"(*&\"\"\"F)%\"rG!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y[8] = y[4]*y[7];" "6#/&%\"yG6#\"\") *&&F%6#\"\"%\"\"\"&F%6#\"\"(F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y[9] = y[2]*y[7];" "6#/&%\"yG6#\"\"**&&F%6#\"\"#\"\"\"&F%6#\"\"(F," } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y[10] = y[5]*y[7];" "6#/&%\"yG6#\"#5* &&F%6#\"\"&\"\"\"&F%6#\"\"(F," }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y[ 11] = y[2]*y[3];" "6#/&%\"yG6#\"#6*&&F%6#\"\"#\"\"\"&F%6#\"\"$F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 7 " ' = " }{XPPEDIT 18 0 "y[2];" "6#&%\"yG6#\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y[2]" "6#&%\"yG6#\"\"#" }{TEXT -1 5 "' = " }{XPPEDIT 18 0 "-2*y[2]*y[8]-g*y [10];" "6#,&*(\"\"#\"\"\"&%\"yG6#F%F&&F(6#\"\")F&!\"\"*&%\"gGF&&F(6#\" #5F&F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y[3]" "6#&%\"yG6#\"\"$" }{TEXT -1 6 "' = " } {XPPEDIT 18 0 "y[4];" "6#&%\"yG6#\"\"%" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y[4];" "6#&%\"yG6# \"\"%" }{TEXT -1 6 "' = " }{XPPEDIT 18 0 "y[2]*y[11]+g*y[6]-K*y[3]+K *r[eq];" "6#,**&&%\"yG6#\"\"#\"\"\"&F&6#\"#6F)F)*&%\"gGF)&F&6#\"\"'F)F )*&%\"KGF)&F&6#\"\"$F)!\"\"*&F3F)&%\"rG6#%#eqGF)F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y[5]; " "6#&%\"yG6#\"\"&" }{TEXT -1 6 "' = " }{XPPEDIT 18 0 "y[2]*y[6];" " 6#*&&%\"yG6#\"\"#\"\"\"&F%6#\"\"'F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y[6];" "6#&%\"yG6#\" \"'" }{TEXT -1 6 "' = " }{XPPEDIT 18 0 "-y[2]*y[5];" "6#,$*&&%\"yG6# \"\"#\"\"\"&F&6#\"\"&F)!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[7];" "6#&% \"yG6#\"\"(" }{TEXT -1 6 "' = " }{XPPEDIT 18 0 "-y[7]*y[8];" "6#,$*& &%\"yG6#\"\"(\"\"\"&F&6#\"\")F)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[8]; " "6#&%\"yG6#\"\")" }{TEXT -1 6 "' = " }{XPPEDIT 18 0 "K*r[eq]*y[7]- K*y[3]*y[7]+g*y[6]*y[7]+y[9]*y[11]-y[8]^2;" "6#,,*(%\"KG\"\"\"&%\"rG6# %#eqGF&&%\"yG6#\"\"(F&F&*(F%F&&F,6#\"\"$F&&F,6#F.F&!\"\"*(%\"gGF&&F,6# \"\"'F&&F,6#F.F&F&*&&F,6#\"\"*F&&F,6#\"#6F&F&*$&F,6#\"\")\"\"#F5" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y[9];" "6#&%\"yG6#\"\"*" }{TEXT -1 6 "' = " } {XPPEDIT 18 0 "-3*y[8]*y[9]-g*y[7]*y[10];" "6#,&*(\"\"$\"\"\"&%\"yG6# \"\")F&&F(6#\"\"*F&!\"\"*(%\"gGF&&F(6#\"\"(F&&F(6#\"#5F&F." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[10];" "6#&%\"yG6#\"#5" }{TEXT -1 6 "' = " }{XPPEDIT 18 0 "y[6]*y[ 9]-y[8]*y[10];" "6#,&*&&%\"yG6#\"\"'\"\"\"&F&6#\"\"*F)F)*&&F&6#\"\")F) &F&6#\"#5F)!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[11];" "6#&%\"yG6#\"#6 " }{TEXT -1 6 "' = " }{XPPEDIT 18 0 "-2*y[8]*y[11]-g*y[3]*y[10]+y[2] *y[4];" "6#,(*(\"\"#\"\"\"&%\"yG6#\"\")F&&F(6#\"#6F&!\"\"*(%\"gGF&&F(6 #\"\"$F&&F(6#\"#5F&F.*&&F(6#F%F&&F(6#\"\"%F&F&" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " Do this projection in Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "y[1] := theta(t); y[2] := diff(y[1],t); y[3] := r(t); y[4] := diff(y[3],t); y[5] := sin(y[1]); y[6] := cos (y[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"\"-%&thetaG6#% \"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"#-%%diffG6$-%&the taG6#%\"tGF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"$-%\"rG6# %\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"%-%%diffG6$-%\"r G6#%\"tGF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"&-%$sinG6#- %&thetaG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"'-%$co sG6#-%&thetaG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "y[7 ] := 1/y[3]; y[8] := y[4]*y[7]; y[9] := y[2]*y[7]; y[10] := y[5]*y[ 7]; y[11] := y[2]*y[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\" \"(*&\"\"\"F)-%\"rG6#%\"tG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"yG6#\"\")*&-%%diffG6$-%\"rG6#%\"tGF/\"\"\"F,!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"yG6#\"\"**&-%%diffG6$-%&thetaG6#%\"tGF/\"\"\"-% \"rGF.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"#5*&-%$sinG6 #-%&thetaG6#%\"tG\"\"\"-%\"rGF.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%\"yG6#\"#6*&-%%diffG6$-%&thetaG6#%\"tGF/\"\"\"-%\"rGF.F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 " Check to see the proje ction matches the original system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "-2*y[2]*y[8]-g*y[10];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&*&-%%diffG6$-%&thetaG6#%\"tGF,\"\"\"-F'6$-%\"rGF+F,F-F-F0!\" \"!\"#*&*&%\"gGF--%$sinG6#F)F-F-F0F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "y[2]*y[11]+g*y[6]-K*y[3]+K*r[eq];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,**&)-%%diffG6$-%&thetaG6#%\"tGF,\"\"#\"\"\"-%\"rGF+F .F.*&%\"gGF.-%$cosG6#F)F.F.*&%\"KGF.F/F.!\"\"*&F7F.&F06#%#eqGF.F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " Look at the projected \+ system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "for k f rom 1 by 1 to 11 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " print(diff( y[k],t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%%diffG6$-%&thetaG6#%\"tGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-%&thetaG6#%\"tG-%\"$G6$F)\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%%diffG6$-%\"rG6#%\"tGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-%\"rG6#%\"tG-%\"$G6$F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$cosG6#-%&thetaG6#%\"tG\"\"\"-%%diffG6$F'F*F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#-%&thetaG6#%\"tG\"\" \"-%%diffG6$F(F+F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%%diff G6$-%\"rG6#%\"tGF+\"\"\"*$)F(\"\"#F,!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%diffG6$-%\"rG6#%\"tG-%\"$G6$F+\"\"#\"\"\"F(!\"\" F0*&*$)-F&6$F(F+F/F0F0*$)F(F/F0F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&*&-%%diffG6$-%&thetaG6#%\"tG-%\"$G6$F+\"\"#\"\"\"-%\"rGF*!\"\"F0*&* &-F&6$F(F+F0-F&6$F1F+F0F0*$)F1F/F0F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&*&-%$cosG6#-%&thetaG6#%\"tG\"\"\"-%%diffG6$F)F,F-F--%\"rGF+! \"\"F-*&*&-%$sinGF(F--F/6$F1F,F-F-*$)F1\"\"#F-F3F3" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*&-%%diffG6$-%&thetaG6#%\"tG-%\"$G6$F+\"\"#\"\"\"-% \"rGF*F0F0*&-F&6$F(F+F0-F&6$F1F+F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassig n('y');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 30;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 " Use a quadratic polynomial algorithm for generating the Maclaurin polynomials for the" }}{PARA 0 "" 0 "" {TEXT -1 25 " solut ion to the system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Thi s worksheet determines Maclaurin polynomial solutions to the " }} {PARA 0 "" 0 "" {TEXT -1 65 "initial value ordinary differential equat ion (IVODE) and uses the" }}{PARA 0 "" 0 "" {TEXT -1 7 "results" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " \+ y'(t) = " }{XPPEDIT 18 0 "Diff(y(t),t) = G(y);" "6#/-%%DiffG6$-%\"yG 6#%\"tGF*-%\"GG6#F(" }{TEXT -1 3 " ; " }{XPPEDIT 18 0 "y(0) = u;" "6#/ -%\"yG6#\"\"!%\"uG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "for \+ " }{XPPEDIT 18 0 "y*epsilon*R^n;" "6#*(%\"yG\"\"\"%(epsilonGF%)%\"RG% \"nGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "G;" "6#%\"GG" }{TEXT -1 4 " = (" }{XPPEDIT 18 0 "g[1];" "6#&%\"gG6#\"\"\"" }{TEXT -1 5 ",...," } {XPPEDIT 18 0 "g[n];" "6#&%\"gG6#%\"nG" }{TEXT -1 4 ") : " }{XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" }{TEXT -1 4 " -> " }{XPPEDIT 18 0 "R^n;" " 6#)%\"RG%\"nG" }{TEXT -1 33 " a second degree polynomial in ( " } {XPPEDIT 18 0 "y[1];" "6#&%\"yG6#\"\"\"" }{TEXT -1 5 ",...," } {XPPEDIT 18 0 "y[n];" "6#&%\"yG6#%\"nG" }{TEXT -1 10 "), that is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 80 "g[j] := a[j]+Sum(b[j,m]*y[m],m=1..n)+Sum(Sum(d[j,m, i]*y[m]*y[i],i=1..n),m=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\" gG6#%\"jG,(&%\"aGF&\"\"\"-%$SumG6$*&&%\"bG6$F'%\"mGF+&%\"yG6#F3F+/F3;F +%\"nGF+-F-6$-F-6$*(&%\"dG6%F'F3%\"iGF+F4F+&F56#FBF+/FBF8F7F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 " for j = 1,..,n and uses the results numerically." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 " The terms " }{XPPEDIT 18 0 "a[j];" "6#&%\"aG6#%\"jG" } {TEXT -1 26 " are the constant term in " }{XPPEDIT 18 0 "g[j];" "6#&% \"gG6#%\"jG" }{TEXT -1 12 ". The terms " }{XPPEDIT 18 0 "b[j,m];" "6#& %\"bG6$%\"jG%\"mG" }{TEXT -1 45 " are the coefficients of the linear t erms in " }{XPPEDIT 18 0 "g[j];" "6#&%\"gG6#%\"jG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 13 " The terms " }{XPPEDIT 18 0 "c[j,m];" " 6#&%\"cG6$%\"jG%\"mG" }{TEXT -1 48 " are the coefficients of the quadr atic terms in " }{XPPEDIT 18 0 "g[j];" "6#&%\"gG6#%\"jG" }{TEXT -1 12 ". The terms " }{XPPEDIT 18 0 "d[j,m,i];" "6#&%\"dG6%%\"jG%\"mG%\"iG" }{TEXT -1 24 " are the coefficients of" }}{PARA 0 "" 0 "" {TEXT -1 28 " the mixed product terms " }{XPPEDIT 18 0 "y[m]*y[i];" "6#*&&%\"yG 6#%\"mG\"\"\"&F%6#%\"iGF(" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "g[j];" " 6#&%\"gG6#%\"jG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "The solution to the I VODE is determined using Picard iteration and truncation on the Picard iterates to the " }}{PARA 0 "" 0 "" {TEXT -1 92 "Maclaurin polynomial . The computation is all algebraic since we are integrating polynomial s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 42 "Here is an example of the IVODE for n = 4." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "for j from 1 to 4 by 1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 " Diff(y[j],t) = a[j]+sum(b[j,m]*y[m],m=1..4)+sum(sum (d[j,m,i]*y[m]*y[i],i=1..4),m=1..4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$&%\"yG6#\"\"\"% \"tG,L&%\"aGF)F**&&%\"bG6$F*F*F*F'F*F**&&F16$F*\"\"#F*&F(6#F6F*F**&&F1 6$F*\"\"$F*&F(6#F " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 " Give the number of equations in your IVODE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "number_of_equations := 11;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%4number_of_equationsG\"#6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 " All the coefficients composing " }{XPPEDIT 18 0 "G;" "6 #%\"GG" }{TEXT -1 15 " are set to 0. " }}{PARA 0 "" 0 "" {TEXT -1 64 " That way you only have to set the non-zero ones for your IVODE" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "for j from 1 by 1 to number_of_equations do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " a[j] := 0; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for \+ i from 1 by 1 to number_of_equations do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for j from 1 by 1 to number_of_equations do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " b[i,j] := 0; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 43 "for i from 1 by 1 to number_of_equations do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for j from 1 by 1 to number_of_ equations do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for k from 1 by 1 t o number_of_equations do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " d[i,j ,k] := 0; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 " Put the coefficients o f the spring-pendulum into the quadratic polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "b[1,2] := 1; b[2,10] := -gra v; d[2,2,8] := -2; b[3,4] := 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"bG6$\"\"\"\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6$\"\"# \"#5,$%%gravG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"\"#F' \"\")!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6$\"\"$\"\"%\"\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "a[4] := K*re; b[4,3] := -K; b[4,6] := grav; d[4,2,11] := 1; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"%*&%\"KG\"\"\"%#reGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6$\"\"%\"\"$,$%\"KG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"bG6$\"\"%\"\"'%%gravG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6 %\"\"%\"\"#\"#6\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "d[ 5,2,6] := 1; d[6,2,5] := -1; d[7,7,8] := -1; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"\"&\"\"#\"\"'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"\"'\"\"#\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"\"(F'\"\")!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "b[8,7] := K*re; d[8,3,7] := -K; d[8,6,7] := grav; \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6$\"\")\"\"(*&%\"KG\"\"\"%# reGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"\")\"\"$\"\"(,$%\" KG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"\")\"\"'\"\"(%%g ravG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "d[8,8,8] := -1; d[8 ,9,11] := 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"\")F'F'!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"\")\"\"*\"#6\"\"\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "d[9,7,10] := -grav; d[9,8, 9] := -3; d[10,6,9] := 1; d[10,8,10] := -1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"\"*\"\"(\"#5,$%%gravG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"\"*\"\")F'!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"#5\"\"'\"\"*\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"#5\"\")F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "d[11,2,4] := 1; d[11,3,10] := -grav; d[11,8,11] := -2 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"#6\"\"#\"\"%\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"#6\"\"$\"#5,$%%gravG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6%\"#6\"\")F'!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 " Set up the initial conditions in u." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "al pha1 := evalf(Pi/4); beta1 := 0.0; l := grav/K; v := -1.1; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'alpha1G$\"??e%3m:'4$[uRj\")R&y!#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&beta1G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lG*&%%gravG\"\"\"%\"KG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG$!#6!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "K := 6; grav := 9.80; re := grav/K;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gravG $\"$!)*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#reG$\"?LLLLLLLLLLLLL L;!#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "u[1] := alpha1; u[ 2] := beta1; u[3] := l; u[4] := v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"uG6#\"\"\"$\"??e%3m:'4$[uRj\")R&y!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"#$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"$$\"?LLLLLLLLLLLLLL;!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"%$!#6!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "u[5] := sin(u[1]); u[6] := cos(u[1]); u[7] := 1/u[3]; u[8] := u[ 4]*u[7];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"&$\"?0@OW3SCv a'=\"y1rq!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"'$\"?0@OW 3SCva'=\"y1rq!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"($\"? .^v(QpMn$=fz*[C7'!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\") $!?81`Ej\"3/-^v(QpMn!#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " u[9] := u[2]*u[7]; u[10] := u[5]*u[7]; u[11] := u[2]*u[3]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"*$\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"#5$\"?t/;Fgx1Y!Q4>D#HV!#I" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"uG6#\"#6$\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 " You can now check to see if this is the " }{XPPEDIT 18 0 "G;" "6#%\"GG" }{TEXT -1 11 " you wanted" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "unassign('m,i');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for j from 1 to number_of_equations by 1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 136 " g[j] := a[j]+sum(b[j ,m]*y[m],m=1..number_of_equations)+sum(sum(d[j,m,i]*y[m]*y[i],i=1..num ber_of_equations),m=1..number_of_equations);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " G[j] := g[j];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " for k from 1 to number_of_equations by 1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " G[j] := subs(\{y[k]=u[k]\},G[j]);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " p rint(g[j],G[j]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " unassign('m', 'i');" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"yG6#\"\"#$\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$,&&%\"yG6#\"#5$!$!)*!\"#*(\"\"#\"\"\"&F%6#F,F-&F%6#\"\")F-!\"\"$!?ks h10k9&G>roSEC%!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"yG6#\"\"%$!#6 !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,*$\"?)********************** ****z*!#H\"\"\"*&\"\"'F'&%\"yG6#\"\"$F'!\"\"*&$\"$!)*!\"#F'&F+6#F)F'F' *&&F+6#\"\"#F'&F+6#\"#6F'F'$\"?j[ZFG\"Rd;GcXY'HpF&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$*&&%\"yG6#\"\"#\"\"\"&F%6#\"\"'F($\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&&%\"yG6#\"\"#\"\"\"&F&6#\"\"&F)!\"\"$!\" !\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&&%\"yG6#\"\"(\"\"\"&F&6# \"\")F)!\"\"$\"?-n\"**>?:NcZe'>GBT!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,,&%\"yG6#\"\"($\"?)**************************z*!#H*(\"\"'\"\"\"&F% 6#\"\"$F-F$F-!\"\"*($\"$!)*!\"#F-&F%6#F,F-F$F-F-*$)&F%6#\"\")\"\"#F-F1 *&&F%6#\"\"*F-&F%6#\"#6F-F-$\"?Fki%GtfJ0w3nz!*y$F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$,&*&&%\"yG6#\"\"(\"\"\"&F&6#\"#5F)$!$!)*!\"#*(\"\"$F) &F&6#\"\")F)&F&6#\"\"*F)!\"\"$!?%G'H;c1kFGc9^`(f#!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&&%\"yG6#\"\"'\"\"\"&F&6#\"\"*F)F)*&&F&6#\"\")F)& F&6#\"#5F)!\"\"$\"?Lg+/jqh5]zQ1g:H!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,(*&&%\"yG6#\"\"#\"\"\"&F&6#\"\"%F)F)*($\"$!)*!\"#F)&F&6#\"\"$F)&F& 6#\"#5F)!\"\"*(F(F)&F&6#\"\")F)&F&6#\"#6F)F7$!?h[ZFG\"Rd;GcXY'Hp!#H" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 54 " We now find the Maclaurin polynomials algebraically! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 " Give the degree of the Macl aurin polynomials you want for the approximation to the solution to yo ur IVODE." }}{PARA 0 "" 0 "" {TEXT -1 46 " The time step h and the num ber of time steps." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "degree_of_maclaurin_polynomi al := 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%?degree_of_maclaurin_pol ynomialG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "h := 1./2^ 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"?+++++++++++++]i!#J" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "num_time_steps := 100;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/num_time_stepsG\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 " The 0th and 1st Maclaurin and Picard polynomials are the same ! The 1st " }}{PARA 0 "" 0 "" {TEXT -1 55 " Maclaurin polynomial for \+ each component is outputted." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for i from 1 to nu mber_of_equations by 1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " alph a[i,0] := u[i];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%&alphaG6$\"\"\"\"\"!$\"??e%3m:'4$[uRj\")R&y!# I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6$\"\"#\"\"!$F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6$\"\"$\"\"!$\"?LLLLLLLLLLLL LL;!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6$\"\"%\"\"!$!#6! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6$\"\"&\"\"!$\"?0@OW 3SCva'=\"y1rq!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6$\"\"' \"\"!$\"?0@OW3SCva'=\"y1rq!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&a lphaG6$\"\"(\"\"!$\"?.^v(QpMn$=fz*[C7'!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6$\"\")\"\"!$!?81`Ej\"3/-^v(QpMn!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6$\"\"*\"\"!$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6$\"#5\"\"!$\"?t/;Fgx1Y!Q4>D#HV!#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6$\"#6\"\"!$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "theta[0] := u[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%&thetaG6#\"\"!$\"??e%3m:'4$[uRj\")R&y!#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "r[0] := u[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%\"rG6#\"\"!$\"?LLLLLLLLLLLLLL;!#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 " Now we gener ate the rest of the story! Since we are squaring polynomials and" }} {PARA 0 "" 0 "" {TEXT -1 86 "then integrating polynomials in the Picar d iteration it is easy to determine what the " }}{PARA 0 "" 0 "" {TEXT -1 84 "next term in the Maclaurin polynomial is from the Picard \+ iterate. The coefficient of" }}{PARA 0 "" 0 "" {TEXT -1 59 "the next t erm in the Maclaurin polynomial for the solution " }{XPPEDIT 18 0 "y[j ];" "6#&%\"yG6#%\"jG" }{TEXT -1 13 " is given by " }{XPPEDIT 18 0 "alp ha[j,k];" "6#&%&alphaG6$%\"jG%\"kG" }}{PARA 0 "" 0 "" {TEXT -1 81 "whe re k is the degree of the next term in the Maclaurin polynomial. The M aclaurin" }}{PARA 0 "" 0 "" {TEXT -1 23 "polynomial solution to " } {XPPEDIT 18 0 "y[j];" "6#&%\"yG6#%\"jG" }{TEXT -1 13 " is given by " } {XPPEDIT 18 0 "U[j,k];" "6#&%\"UG6$%\"jG%\"kG" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "unassign('m','j','i');" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "ne := number_of_equations;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#neG\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "nd := degree_of_maclaurin_polynomial;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ndG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "for n from 1 by 1 to num_time_steps do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " print(step, n); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " for num from 1 by 1 to number_of_equations do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " sum1 := 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " sum2 := 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " for mm from 1 by 1 to ne do " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " sum1 := sum1 + b[num,mm]*alph a[mm,0];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " #print(sum1);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " for jj from 1 by 1 to ne do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " sum2 := sum2 + d[num,mm,j j]*alpha[mm,0]*alpha[jj,0]; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " \+ #print(sum2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " od:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " alpha[num,1] := a[num]+sum1+sum2;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " for k from 2 by 1 to degree_of_ maclaurin_polynomial do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " for \+ num from 1 by 1 to number_of_equations do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " #print(num);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " sum1 := 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " sum2 \+ := 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " for mm from 1 by 1 t o ne do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " sum1 := sum1 \+ + b[num,mm]*alpha[mm,k-1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " \+ for jj from 1 by 1 to ne do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " \+ sum3 := 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " fo r i from 0 by 1 to k-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " \+ sum3 := sum3 + alpha[mm,i]*alpha[jj,k-i-1];" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 13 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " sum2 := sum2 + d[num,mm,jj]*sum3; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " \+ od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " #print(eqn,num);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " alpha[num,k] := (s um1+sum2)/k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " #print(num,k, alpha[num,k]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " od :" }}{PARA 0 "> " 0 "" 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