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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# Runge-Kutta metho
d  for " }{MPLTEXT 1 0 15 "dy/dx=-y, with " }{MPLTEXT 1 0 20 " x0:=0.0
; y0:=2.0; \n" }{MPLTEXT 1 0 24 "#   on interval [0,1]   " }{MPLTEXT 
1 0 33 "->  exact solution is y=2*exp(-x)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "restart:Digits:=5:  with(plots):with(linalg):  \n" }
{MPLTEXT 1 0 50 "  f:=(x,y)->-y; x0:=0.0; y0:=2.0;  # f(x,y) in DE " }
{MPLTEXT 1 0 9 "dy/dx=-y " }{MPLTEXT 1 0 5 "and \n" }{MPLTEXT 1 0 57 "
                                   # initial condition  \n" }{MPLTEXT 
1 0 50 "n:=5; xn:=1.0; h:=(xn-x0)/n;       #  make  n=5  \n" }{MPLTEXT
 1 0 63 "    xx:=vector(n+1):  yy:=vector(n+1):  xx[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 70 "  k1:=f(x,y); k2:=f(x+0.5*h,y+
0.5*h*k1); k3:=f(x+0.5*h,y+0.5*h*k2);  \n" }{MPLTEXT 1 0 70 "   k4:=f(
x+h,y+h*k3); xx[i+1]:=x+h; yy[i+1]:=y+h/6*(k1+2*k2+2*k3+k4);\n" }
{MPLTEXT 1 0 5 "od: \n" }{MPLTEXT 1 0 36 "# now plot computed and exac
t 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 72 "              title=\"Approximate soln computed by Ru
nge-Kutta method\");\n" }{MPLTEXT 1 0 51 " display(plot(2*exp(-t),t=0.
.1),pointplot(points),\n" }{MPLTEXT 1 0 66 "                          \+
    title=\"Exact and computed solns\" ) ;" }}{PARA 0 "> " 0 "" 
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