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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "John Cole" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 9 "Problem 1" }{TEXT -1 0 
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0 "" {TEXT 258 36 "A.  Picards successive approximation" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "y0:=0;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "y1 := y0 + int(2*(y+1), y=0..t)
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "y2 := y0 + int(2*(y+1)+subs(t=
y,pic1), y=0..t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "y3 := y0 + int
(2*(y+1)+subs(t=y,pic2), y=0..t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "y4 := y0 + int(2*(y+1)+subs(t=y,pic3), y=0..t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#y0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G
,&*$)%\"tG\"\"#\"\"\"F**&F)F*F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#y2G,(*$)%\"tG\"\"#\"\"\"F)*&F)F*F(F*F**&#F*\"\"$F*)F(F.F*F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G,**$)%\"tG\"\"#\"\"\"F)*&F)F*F(F
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{XPPMATH 20 "6#>%#y4G,,*$)%\"tG\"\"#\"\"\"F)*&F)F*F(F*F**&#F)\"\"$F*)F
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{PARA 0 "" 0 "" {TEXT -1 20 "Show that yn(t)=etc." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 54 "show := sum('(((2^k)*(t^k)))/factorial(k)', 'k'=1..
4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%showG,*%\"tG\"\"#*&F'\"\"\")
F&F'F)F)*&#\"\"%\"\"$F))F&F.F)F)*&#F'F.F))F&F-F)F)" }}}{EXCHG {PARA 
256 "" 0 "" {TEXT -1 229 "C.  Plot y1, y2, y3, y4, on the same graph f
or [-1.5,1.5] and hence show that iterates converge to a limit functio
n e^2t-1.  Visualize the convergence of the sequence of iterates more \+
clearly by plotting y(t)-yn(t) for n=1,2,3,4." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 48 "plot(\{(exp(2*t)-1), y1, y2, y3, y4\}, t=-0.4..1
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