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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "This Maple Procedure imple
ments the Euler Approximation to a differential equation of the type \+
" }{XPPEDIT 18 0 "dy/dx = f(x,y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$
%\"xG%\"yG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
467 "Euler:= proc(f,xstart, ystart, xend, n)\nlocal h, i, k, y, x;\nh \+
:= evalf((xend - xstart)/n);\ny:=ystart:   x:=xstart:\nprintf(\"%s  %s
          %s %f \\n\",\"Step\",\"x\",\"y with h =\",h);\nfor i from 1 \+
to n+1 do\n    printf(\"%d  %f    %f \\n\",i-1,x,y); # display current
 values    \n    k := f(x,y): \011\011                    # the left-h
and slope \n\011   y := y + h*k:  \011                   # Euler step \+
to update y\n\011   x := x + h:     \011                  # update x\n
\011   od:\nprint(f);\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Th
e Syntax is as follows: Euler(function, initial x, initial y, final x,
 number of steps);" }}{PARA 0 "" 0 "" {TEXT -1 44 "Below we compute th
e Euler Approximation to " }{XPPEDIT 18 0 "dy/dx = x+y;" "6#/*&%#dyG\"
\"\"%#dxG!\"\",&%\"xGF&%\"yGF&" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0) \+
= 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 16 ", with 10 steps." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=(x,y) -> x + y;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowG
F),&9$\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E
uler(f,0,1,1,10);" }}{PARA 6 "" 1 "" {TEXT -1 36 "Step  x          y w
ith h = .100000 " }}{PARA 6 "" 1 "" {TEXT -1 24 "0  0.000000    1.0000
00 " }}{PARA 6 "" 1 "" {TEXT -1 23 "1  .100000    1.100000 " }}{PARA 
6 "" 1 "" {TEXT -1 23 "2  .200000    1.220000 " }}{PARA 6 "" 1 "" 
{TEXT -1 23 "3  .300000    1.362000 " }}{PARA 6 "" 1 "" {TEXT -1 23 "4
  .400000    1.528200 " }}{PARA 6 "" 1 "" {TEXT -1 23 "5  .500000    1
.721020 " }}{PARA 6 "" 1 "" {TEXT -1 23 "6  .600000    1.943122 " }}
{PARA 6 "" 1 "" {TEXT -1 23 "7  .700000    2.197434 " }}{PARA 6 "" 1 "
" {TEXT -1 23 "8  .800000    2.487178 " }}{PARA 6 "" 1 "" {TEXT -1 23 
"9  .900000    2.815895 " }}{PARA 6 "" 1 "" {TEXT -1 25 "10  1.000000 \+
   3.187485 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6$%\"xG%\"yG6\"6$%)o
peratorG%&arrowGF',&9$\"\"\"9%F-F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
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