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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 258 32 "Euler Method for First \+
Order ODE" }}{PARA 257 "" 0 "" {TEXT 256 47 "Date: Jan 21, 2002\nLast \+
Revision: Jan 21, 2002\n" }{TEXT 267 7 "Maple 6" }}{PARA 259 "" 0 "" 
{TEXT 259 16 "Bent E. Petersen" }}{PARA 258 "" 0 "" {TEXT 260 17 "bent
@alum.mit.edu" }}{PARA 258 "" 0 "" {TEXT 261 22 "petersen@math.orst.ed
u" }}{PARA 0 "" 0 "" {TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT 263 15 "Co
urse: Mth 256" }}{PARA 0 "" 0 "" {TEXT 264 17 "Term: Winter 2002" }}
{PARA 0 "" 0 "" {TEXT 265 11 "File name: " }{TEXT 257 25 "256w2002-eul
er-method.mws" }{TEXT 266 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 327 "Maple has a number of numeric solvers fo
r differential equations built-in. Normally Maple chooses a numeric me
thod when we request a numeric answer, but we can specify the method. \+
In particular we can specify the simple Euler method. Our goal here th
ough is to study the Euler method and therefore we roll our own Euler \+
method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 333 "Here is a simple Euler procedure for sol
ving  dy/dx=f(x,y),  y(x0)=y0  on a specified interval with a specifie
d number of steps. We do no error checking - be careful. We return the
 approximate solution as a list of lists - the first is the list of ab
scissae, the second is the list of ordinates. This form is convenient \+
for plotting." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "First here is our Eul
er routine, eul(). Note we force Maple to use floating point arithmeti
c. Otherwise Maple will attempt exact evaluation, which will waste imm
ense amounts of time and RAM." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eul:=proc(f,x0,y0,x1,N)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  local h,XX,YY,k,x,y;" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "x:=evalf(x0): XX:=[x];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "y:=evalf(y0): YY:=[y];" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "h:=(x1-x0)/N;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from
 1 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "  y:=y+h*f(x,y);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  x:=x+h;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "  XX:=[op(XX),x];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "  YY:=[op(YY),y];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "[XX,YY];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 106 "Here is a routine to convert the data returned
 by eul() into a piecewise linear (or polygonal) expression:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "e
ulpoly:=proc(EUL,x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "  spline(EU
L[1],EUL[2],x,1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Le
t's try an example. Let" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f01:=(x,y)->x^2+y^2;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$f01GR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&
*$)9$\"\"#\"\"\"F2*$)9%F1F2F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "init01:=y(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
'init01G/-%\"yG6#\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 
"ode01:=diff(y(x),x)=f01(x,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%&ode01G/-%%diffG6$-%\"yG6#%\"xGF,,&*$)F,\"\"#\"\"\"F1*$)F)F0F1F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Ma
ple can solve this initial value problem." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "soln01:=dsolve(\{ode
01,init01\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'soln01G/-%\"y
G6#%\"xG,$*&*&F)\"\"\",&-%(BesselJG6$#!\"$\"\"%,$*$)F)\"\"#F-#F-F8!\"
\"-%(BesselYGF1F-F-F-,&-F06$#F-F4F5F:-F<F?F-F:F:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Let's plot this so
lution on  [0,1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 61 "plot(rhs(soln01),x=0..1,title=\"Actual solut
ion\",thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 334 251 251 
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45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 46 "Now let's see how well our Euler routine \+
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{MPLTEXT 1 0 38 "esoln01:=eulpoly(eul(f01,0,0,1,20),x);" }}{PARA 11 "
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h(*F=F5F,F5F5%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
61 "plot(esoln01,x=0..1,title=\"Euler approximation\",thickness=3);" }
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "yy:=subs(soln02,y(x));" }}
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