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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "This Maple Procedure impl
ements the Runge-Kutta Approximation to a differential equation of the
 type " }{XPPEDIT 18 0 "dy/dx = f(x,y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%
\"fG6$%\"xG%\"yG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 647 "RungeKutta:= proc(f,xstart, ystart, xend, n)\nlocal \+
h, i, k, y, x, k1, k2, k3, k4;\nh := evalf((xend - xstart)/n);\ny:=yst
art:   x:=xstart:\nprintf(\"%s  %s          %s %f \\n\",\"Step\",\"x\"
,\"y with h =\",h);\nfor i from 1 to n+1 do\n   printf(\"%d  %f    %f \+
\\n\",i-1,x,y); # display current values\n   k1 := f(x,y): \011\011
\011         # the left-hand slope\n\011  k2 := f(x+h/2,y+h*k1/2):\011
 # 1st midpoint slope\n\011  k3 := f(x+h/2,y+h*k2/2):\011 # 2nd midpoi
nt slope\n\011  k4 := f(x+h,y+h*k3): \011\011   # the right-hand slope
\n   k := (k1+2*k2+2*k3+k4)/6:\011# the average slope\n\011  y := y + \+
h*k:  \011\011         # R-K step to update y\n\011  x := x + h:     \+
\011\011        # update x\nod:\nprint(f);\nend:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 94 "The Syntax is as follows: RungeKuttafunction, init
ial x, initial y, final x, number of steps);" }}{PARA 0 "" 0 "" {TEXT 
-1 50 "Below we compute the Runge-Kutta Approximation to " }{XPPEDIT 
18 0 "dy/dx = x+y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&%\"xGF&%\"yGF&" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }
{TEXT -1 16 ", with 10 steps." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "f:=(x,y) -> x + y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$
%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&9$\"\"\"9%F/F)F)F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "RungeKutta(f,0,1,1,10);" }}{PARA 6 
"" 1 "" {TEXT -1 36 "Step  x          y with h = .100000 " }}{PARA 6 "
" 1 "" {TEXT -1 24 "0  0.000000    1.000000 " }}{PARA 6 "" 1 "" {TEXT 
-1 23 "1  .100000    1.110342 " }}{PARA 6 "" 1 "" {TEXT -1 23 "2  .200
000    1.242805 " }}{PARA 6 "" 1 "" {TEXT -1 23 "3  .300000    1.39971
7 " }}{PARA 6 "" 1 "" {TEXT -1 23 "4  .400000    1.583648 " }}{PARA 6 
"" 1 "" {TEXT -1 23 "5  .500000    1.797441 " }}{PARA 6 "" 1 "" {TEXT 
-1 23 "6  .600000    2.044236 " }}{PARA 6 "" 1 "" {TEXT -1 23 "7  .700
000    2.327503 " }}{PARA 6 "" 1 "" {TEXT -1 23 "8  .800000    2.65107
9 " }}{PARA 6 "" 1 "" {TEXT -1 23 "9  .900000    3.019203 " }}{PARA 6 
"" 1 "" {TEXT -1 25 "10  1.000000    3.436559 " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF',&9$\"\"\"9%F-F
'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }