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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 20 "The Mid-Point Method" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots
):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 32 "Define the initial value problem" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=(t,x)->-2*t*x(t)^2;
x0:=1;\015" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "IVP:=\{diff(x
(t),t)=f(t,x),x(0)=x0\};\015" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "FCN:=\{x(t)\};\015" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 25 "Obtain the exact solution" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Exact:
=rhs(dsolve(IVP,FCN));\015" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 75 "Use the classical 4th order Runge-Kutta m
ethod to obtain the starting value" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "sv:=h->dsolve(IVP,FCN,type=
numeric,method=classical[rk4],stepsize=h);\015" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "SV:=(t,h)->rhs(sv(h)(t)[2]);\015" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Define a \+
procedure for the mid-point method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "midpt:=proc(n,h) option re
member;\nif n=1 then SV(h,h) elif n=2 then x0+2*h*f(h,SV(h,h)) \nelse \+
midpt(n-2,h)+2*h*f((n-1)*h,midpt(n-1,h))fi;\nend;\n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Define a proced
ure to plot the mid-point solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plotmidpt:=proc(N,h)\nloca
l l,i;\n l:=[];\n for i from 1 to N do\n l:=[op(l),i*h,midpt(i,h)];\n \+
od;\n pointplot(l,connect=true);\n end;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Plot the exact and numeri
cal solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "p1:=plotmidpt(50,0.1):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "p2:=plot(Exact,t=0..5):\015" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 19 "display(\{p1,p2\}); \015" }}}}{MARK "20" 0 }
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