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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 28 "Section 8.1 - Euler's Method" }
}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Here i
s a procedure for using the Euler method.  It takes as input a functio
n " }{XPPEDIT 18 0 "f(t,y);" "6#-%\"fG6$%\"tG%\"yG" }{TEXT -1 0 "" }
{TEXT -1 19 ", an initial value " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6#\"
\"!" }{TEXT -1 0 "" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "y;" "6#%\"yG" 
}{TEXT -1 0 "" }{TEXT -1 18 ", an initial time " }{XPPEDIT 18 0 "t[0];
" "6#&%\"tG6#\"\"!" }{TEXT -1 1 " " }{TEXT -1 4 "(so " }{XPPEDIT 18 0 
"y(t[0]) = y[0];" "6#/-%\"yG6#&%\"tG6#\"\"!&F%6#F*" }{TEXT -1 0 "" }
{TEXT -1 15 "), a step size " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 
0 "" }{TEXT -1 20 ", a number of steps " }{XPPEDIT 18 0 "n;" "6#%\"nG
" }{TEXT -1 0 "" }{TEXT -1 15 ", and a number " }{XPPEDIT 18 0 "m;" "6
#%\"mG" }{TEXT -1 0 "" }{TEXT -1 35 " so that results are printed ever
y " }{TEXT -1 0 "" }{TEXT -1 93 " steps.  This procedure is a bit diff
erent from the one in Differential Equations with Maple." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " eule
r_method := proc(f,y0,t0,h,n,m)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 248 
"    y(t0) := evalf(y0);\n    for i from 1 to n\n       do\n          \+
 y(t0+i*h) := y(t0+(i-1)*h)+ h*f(t0+(i-1)*h,y(t0+(i-1)*h));           \+
                                    if type(i/m,integer)='true'  then \+
print(t0+i*h,y(t0+i*h)) fi;\n      od;\nend;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Let's try it on the \+
function in Problem 1, p. 392." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := (t,y) -> 2*y-1;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "euler_method(f,1,0,.1,4,1);
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " euler_method(f,1,0,.
05,8,2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "euler_method(
f,1,0,.025,16,4);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 131 "Here is the exact solution. I gave the solution a new \+
name so Maple wouldn't confuse it with the result of the previous calc
ulation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "dsolve(\{diff(z(t),t)=2*z(t)-1,z(0)=1\},z(t));\n" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " I'll ren
ame the solution " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 1 " " }
{TEXT -1 12 "(instead of " }{XPPEDIT 18 0 "z(t);" "6#-%\"zG6#%\"tG" }
{TEXT -1 0 "" }{TEXT -1 39 ")--this reduces problems evaluating it." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "z := rhs(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 111 " Here is the exact solution evaluated at several point
s. Notice that we have to substitute the desired value of" }{XPPEDIT 
18 0 "t;" "6#%\"tG" }{TEXT -1 1 " " }{TEXT -1 5 "into " }{XPPEDIT 18 
0 "z;" "6#%\"zG" }{TEXT -1 0 "" }{TEXT -1 60 " before evaluating.  Als
o, we have to use evalf to evaluate " }{XPPEDIT 18 0 "z(t);" "6#-%\"zG
6#%\"tG" }{TEXT -1 1 " " }{TEXT -1 28 "as a floating point number.\n" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "evalf(subs(t=.1,z));\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "evalf(subs(t=.2,z)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "evalf(subs(t=.3,z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 
"evalf(subs(t=.4,z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
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