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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Euler's method - Example 1
" }}{PARA 0 "" 0 "" {TEXT -1 36 "------------------------------------
" }}{PARA 0 "" 0 "" {TEXT -1 32 "Solve the initial value problem:" }}
{PARA 0 "" 0 "" {TEXT -1 52 "                         dy/dx = 2y    y(
0) = 1     " }}{PARA 0 "" 0 "" {TEXT -1 125 "in the domain [ 0 , 2 ] u
sing various values of step size h. Compare the approximations with th
e true solution  y = e ^ (2x)." }}{PARA 0 "" 0 "" {TEXT -1 19 "       \+
            " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ExactSolution := (x) -> exp(
2*x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%.ExactSolutionG:6#%\"xG6\"6$%)operatorG%&arrowGF(-%$e
xpG6#,$9$\"\"#F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The exact s
olution is indicated by the blue curve in the graph." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "picture1 \+
:= plot(ExactSolution(x), x=0..2, color=COLOR(RGB,0,0,1)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "h:=0.5;  # the step size" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 10 "oldx:=0;  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%o
ldxG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "oldy:=1;  # th
e initial condition" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG\"\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "lastx:=2;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&lastxG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 6 "L:=[];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"LG7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "L:=[op(L),[oldx,
oldy]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7#7$\"\"!\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "slope:=(x,y) -> 2*y;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&slopeG:6$%\"xG%\"yG6\"6$%)operatorG
%&arrowGF),$9%\"\"#F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "
while oldx < lastx do oldy:=oldy+h*slope(oldx,oldy);oldx:=oldx+h;L:=[o
p(L),[oldx,oldy]];od;" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%oldyG$\"#?!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%oldxG$\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7$7
$\"\"!\"\"\"7$$\"\"&!\"\"$\"#?F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
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{XPPMATH 20 "6#>%%oldyG$\"'++;!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
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\"\"\"7$$\"\"&!\"\"$\"#?F,7$$\"#5F,$\"$+%!\"#7$$\"#:F,$\"%+!)!\"$7$F-$
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%)picture2G-%/INTERFACE_PLOTG6%-%'CURVESG6$7'7$\"\"!$\"\"\"F-7$$\"1+++
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qoF%Q!GFfv7$$\"1LL$3N1#4<F7$\"1'QBH2$4_IFfv7$$\"1nm\"HYt7v\"F7$\"1.'=d
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F(F(F*-%+AXESLABELSG6$%\"xG%!G-%%VIEWG6$;F(F/%(DEFAULTG" 2 304 304 
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{TEXT -1 108 "You can copy from the above all the text in red and chan
ge the value of step size h for your own experiment." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "We now
 decrese the step size h and re-run the programme. It is necessary to \+
reset the values of oldx and oldy to the initial condition: " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "h:=0.1;  # the step size" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 10 "oldx:=0;  " }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%%oldxG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "oldy:=1
;  # the initial condition" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldyG
\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "L:=[];" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "L:=[op(L),[oldx,oldy]];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"LG7#7$\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "slope:=(x,y) -> 2*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%&slopeG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$9%\"\"#F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "while oldx < lastx do oldy:=
oldy+h*slope(oldx,oldy);oldx:=oldx+h;L:=[op(L),[oldx,oldy]];od;" }}
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\"\"'F+$\"(%)f)H!\"'7$$\"\"(F+$\")3=$e$!\"(7$$\"\")F+$\"*'p\")*H%!\")7
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{XPPMATH 20 "6#>%%oldyG$\"+ILLiE!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%oldxG$\"#=!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"LG757$\"\"
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20 "6#>%%oldyG$\"+'***z%>$!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%o
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" 1 "" {XPPMATH 20 "6#>%%oldyG$\"+&**fP$Q!\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%oldxG$\"#?!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%\"LG777$\"\"!\"\"\"7$$F(!\"\"$\"#7F+7$$\"\"#F+$\"$W\"!\"#7$$\"\"$F+$
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H!\"'7$$\"\"(F+$\")3=$e$!\"(7$$\"\")F+$\"*'p\")*H%!\")7$$\"\"*F+$\"+_.
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he array L gives the solution obtained by Euler's method in the form:
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }
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{PARA 0 "" 0 "" {TEXT -1 191 "The circlces give the approximation with
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) compared to the the diamonds, which is the approximation with step s
ize h=0.25." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 
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