{VERSION 5 0 "IBM INTEL LINUX" "5.0" }
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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "#This worksheet does a simple appr
oximation of solutions of a first-order linear\n#differential equation
 of the for y'=f(x,y) using Euler's method" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "#Enter function below:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "f:=(x,y)->x+y;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 289 "#The following procedure does the actual approximati
on. To find an approximate \n#solution of the differential equation\n#
                                y' = f(x,y)\n#                        \+
        y(x0) = y0\n#with stepsize h on the interval [x0, x0+n*h], ent
er eulermethod(f, x0, y0, h, n)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 230 "eulermethod:=proc(f, x0, y0, h, n)\n  local x, y, i, dydx;\n \+
 x:=x0: y:=y0;\n  for i from 1 to n do\n    dydx:=f(x,y);\n    printf(
\"Step=%d, x=%6.2f, y=%6.2f, y'=%6.2f\\n\", i-1, x, y, dydx);\n    x:=
x+h: y:=y+dydx*h;\n  od;\n  return;\nend;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "eulermethod(f,0,1,0.2,16);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "#F
or your interest, here is the exact solution of the above differential
 equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dsolve([D(y)(
x)=f(x,y(x)), y(0)=1], y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "g:=unapply(op(2,%),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 104 "#And for comparison's sake, here is a procedure to print the va
lue of this #solution at the same points:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 141 "valuetable:=proc(g, x0, h, n)\n  local i;\n    for
 i from 0 to n-1 do\n    printf(\"x=%6.2f, y=%6.2f\\n\", x0+i*h, g(x0+
i*h));\n  od;\n  return;\nend;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "valuetable(g,0,0.2,15);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "#Experiment wit
h different functions and stepsizes. Try an initial value which leads \+
to an equilibrium solution. If you are having fun, try a few of the pa
thological examples in the exercises in FHMW 1.4." }}}}{MARK "16" 0 }
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