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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "# Improved Euler me
thod \n" }{MPLTEXT 1 0 48 "restart:Digits:=5:  with(plots):with(linalg
):  \n" }{MPLTEXT 1 0 63 "  f:=(x,y)->-y; x0:=0.0; y0:=2.0;    # f(x,y
) in DE dy/dx=-y  \n" }{MPLTEXT 1 0 7 "     # " }{MPLTEXT 1 0 4 "and "
 }{MPLTEXT 1 0 52 "initial condition -  exact solution is y=2*exp(-x) \+
\n" }{MPLTEXT 1 0 56 "n:=10; xn:=1.0; h:=(xn-x0)/n;        #  make  n=
10  now\n" }{MPLTEXT 1 0 63 "    xx:=vector(n+1):  yy:=vector(n+1):  x
x[1]:=x0:  yy[1]:=y0:\n" }{MPLTEXT 1 0 22 "for i from 1 to n do \n" }
{MPLTEXT 1 0 23 "  x:=xx[i]: y:=yy[i]: \n" }{MPLTEXT 1 0 21 "  xx[i+1]
:=x+h:  k1:=" }{MPLTEXT 1 0 7 "f(x,y):" }{MPLTEXT 1 0 8 "\n  u:=y+" }
{MPLTEXT 1 0 43 "h*k1: \n  k2:=f(xx[i+1],u):  k:=0.5*(k1+k2):" }
{MPLTEXT 1 0 19 "\n  yy[i+1]:=y+h*k:\n" }{MPLTEXT 1 0 5 "od: \n" }
{MPLTEXT 1 0 36 "# now plot computed and exact solns\n" }{MPLTEXT 1 0 
44 "  points:= \{seq([xx[i],yy[i]],i=1..n+1) \}: \n" }{MPLTEXT 1 0 36 
"  pointplot(points,symbol=asterisk,\n" }{MPLTEXT 1 0 62 "          ti
tle=\"Approximate soln by improved Euler method\");\n" }{MPLTEXT 1 0 
51 " display(plot(2*exp(-t),t=0..1),pointplot(points),\n" }{MPLTEXT 1 
0 56 "                    title=\"Exact and computed solns\" ) ;" }
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