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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "                          \+
                    Math 2280: Computer Project 1" }}{PARA 0 "" 0 "" 
{TEXT -1 70 "                                                        E
uler's Method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 106 "In this project you will compute some numerical solution
s to initial value problems using Euler's method. " }}{PARA 0 "" 0 "" 
{TEXT -1 60 "First let's restart Maple and load the appropriate packag
es." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "r
estart: with (DEtools): with (plots): with (linalg):" }}{PARA 7 "" 1 "
" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}
{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the name adjoint has been redefi
ned\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names nor
m and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 116 "We will consider the differential equation y' = 2
y. Here's a plot of the slope field for this differential equation." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f := y \+
-> 2*y;\nfieldplot ([1,f(y)], x = -2..2, y = -2..2);" }}{PARA 11 "" 1 
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FduFbft7$$\"3%z[uS*)Ro9\"F*Fgft7%7$F[vF_ft7$F^vFbft7$$\"3hhR')4iOd8F*F
gft7%7$FevF_ft7$FhvFbft7$$\"3GNMlDD*yc\"F*Fgft7%7$F_wF_ft7$FbwFbft7$$
\"3'*3HWT)=%y<F*Fgft7%7$FiwF_ft7$F\\xFbft7$$\"3&GQKs:X*))>F*Fgft-%+AXE
SLABELSG6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "We're
 going to solve this differential equation for 0 < x < 1, with a steps
ize of 1/100 = .01 (or, 100 steps)." }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "n:= 100; h := .01;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"nG\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"hG$\"\"\"!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Next we set up
 the data array for our solution and initialize it." }{MPLTEXT 1 0 0 "
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "y:= \+
vector[n+1];\nx := vector[n+1];\nx[1] := 0;\ny[1] := 1;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"yG&%'vectorG6#\"$,\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"xG&%'vectorG6#\"$,\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%\"xG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6
#\"\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Now we fill in the d
ata array with the iteration procedure" }}{PARA 0 "" 0 "" {TEXT -1 59 
"              x[i+1] = x[i] + h,  y[i+1] = y[i] + hf(y[i])." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The basic for
mat of a do loop in Maple is the following: " }}{PARA 0 "" 0 "" {TEXT 
-1 52 " for i from 1 to n do <do stuff at the ith step> od:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 73 "Next we will plot the numerical solution and the real sol
ution together. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "numerical := pointplot(\{seq([x[i],y[i]], i = 1..n+1)
\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "actual := plot(exp(
2*x), x = 0..1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "display
 (\{numerical,actual\}, title = \"numerical solutiuon vs. actual solut
ion\");" }}{PARA 13 "" 1 "" {GLPLOT2D 381 381 381 {PLOTDATA 2 "6'-%'PO
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ULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 240 "You have to look \+
fairly closely to tell that the numerical solution lies below the actu
al solution, but it does. This is easier to see if you look on a large
r interval, or at a differential equation with a steeper slope field. \+
Play around!." }}}}{MARK "15 0 0" 240 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }