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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 257 21 "Forward Euler Method." }
{TEXT -1 47 "  Written for Math 251, Fall 2001 by J.Gerlach." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 48 "This is a routine which executes Euler's method." }}{PARA 0 "" 
0 "" {TEXT -1 21 "The input consists of" }}{PARA 0 "" 0 "" {TEXT -1 
36 "f          the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 
46 "a,b      the coordinates of the starting point" }}{PARA 0 "" 0 "" 
{TEXT -1 26 "L        the final x-value" }}{PARA 0 "" 0 "" {TEXT -1 
22 "h        the step size" }}{PARA 0 "" 0 "" {TEXT -1 83 "The output \+
is an array of points, containing the coordinates of the approximation
s." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "Euler := proc(f,a,b,L,h)\nlo
cal N, x,y,j,pt,dy, pts;\nN := round(L/h)+1;\nx := a;  y:= b;\npt(1) :
= [x,y];\nfor j from 2 to N do\ndy := h*f(x,y);\nx := x+h;  y := y + d
y;\npt(j) := [x,y];\nend do;\npts := [seq(pt(j),j=1..N)];\nend proc:\n
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 8 "Example:" }{TEXT -1 38 "  Sol
ve the initial value problem     " }{XPPEDIT 18 0 "y;" "6#%\"yG" }
{TEXT -1 4 "' = " }{XPPEDIT 18 0 "1-x*y^2;" "6#,&\"\"\"F$*&%\"xGF$*$%
\"yG\"\"#F$!\"\"" }{TEXT -1 3 "   " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%
\"yG6#\"\"!\"\"\"" }{TEXT -1 86 "     on the interval [0,2]  with step
size h=0.1, and generate a plot of the solution.." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 21 "f:= (x
,y) -> 1-x*y^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "pts := E
uler(f,0,1,2,.1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(p
ts,style=point);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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