{VERSION 2 3 "APPLE_68K_MAC" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" 
-1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 
-1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 
0 0 1 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 
257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 
-1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 
0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 
259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 
-1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 
0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 45 "Numerical Solutions of I
nitial Value Problems" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 335 "This module has two parts.  The first part looks at t
he computational details of Euler's Method, a straightforward but not \+
very efficient method of estimating the solution of an initial value p
rblem.  The second part illustrates how the much more efficient algori
thms built into Maple are used.\n\nThe cell below defines two procedur
es  " }{TEXT 256 11 "printeuler " }{TEXT -1 6 " and  " }{TEXT 257 9 "p
loteuler" }{TEXT -1 45 "  that you can use to study Euler's Method.  \+
" }{TEXT 258 27 "Evaluate the next cell now." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 982 "with(plottools):
\nwith(plots):\n\nprinteuler :=\n    proc(fcn, a, b, n, start)\n    lo
cal h, nx, ny, w, x, y, i;\n    \n    x := a:\n    y := start:\n    h \+
:= (b - a)/n:\n    printf(`%s\\n`, \n       `    i     t start     y s
tart     f start       t end       y end`):\n    printf(`%s\\n`, ` `):
\n    for i from 1 to n do\n        nx := x + h:\n        w  := evalf(
fcn(x, y)):\n        ny := y + h * w:\n        printf(`%5d %11.4f %11.
4f %11.4f %11.4f %11.4f\\n`,\n                i, x, y, w, nx, ny):\n  \+
      x := nx:\n        y := ny:\n    od:\n  end:\n \n  ploteuler :=\n
    proc(fcn, a, b, n, start)\n    local h, lst, nx, ny, w, x, y, i, p
lt;\n    x := a:\n    y := start:\n    h := (b - a)/n:\n    lst := \{
\}:\n    for i from 1 to n do\n        nx := x + h:\n        w  := eva
lf(fcn(x, y)):\n        ny := y + h * w:\n        lst := lst union \{l
ine([x, y], [nx, ny])\}:\n        x := nx:\n        y := ny:\n    od:
\n\n    printf(`%s %11.4f`, `Final value of y:`, y):\n\n    plt := dis
play(lst):\n\n    display(plt);\n\n  end:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "The  " }{TEXT 259 10 "prin
teuler" }{TEXT -1 136 "  procedure takes five arguments.\n\n\0111.  Th
e name of the function appearing on the right hand side of the differe
ntial equation\n\n        " }{TEXT 260 15 "dy/dt = f(t, y)" }{TEXT -1 
60 "\n\n\011\011  This function must be defined in a line like\n\n    \+
    " }{TEXT 261 29 "MyFcn := (t, y) -> t * sin(y)" }{TEXT -1 14 "\n\n
\011\011  with the" }{TEXT 262 2 " t" }{TEXT -1 24 " variable first an
d the " }{TEXT 263 1 "y" }{TEXT -1 127 " variable second.  It must be \+
a function of two \011variables even if only one is used.\n\n\0112.
\011The lower limit for the range of the " }{TEXT 264 1 "t" }{TEXT -1 
53 " variable.\n\n\0113.\011The upper limit for the range of the " }
{TEXT 265 1 "t" }{TEXT -1 66 " variable.\n\n\0114.\011The number of st
eps.\n\n\0115.\011The initial value of the" }{TEXT 266 2 " y" }{TEXT 
-1 18 " variable.\n\n\011The  " }{TEXT 267 10 "ploteuler " }{TEXT -1 
39 " procedure takes the same arguments as " }{TEXT 268 10 "printeuler
" }{TEXT -1 11 " procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 74 "The cell below illustrates the way in which the
se two procedures are used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "MyFcn := (t, y) -> y:\n\npri
nteuler(MyFcn, 0, 1, 4, 1);\n\nploteuler(MyFcn, 0, 1, 4, 1);\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 251 "U
sing Euler's method and the more sophisticated methods built in to Map
le you can investigate complicated situations as easily as the simple \+
one above.  The cells below illustrate some of the ways in which the p
rocedures built in to Maple can be used." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 119 "The cell below illustrates how Maple can estimate the solution
 to an initial value problem and the plot the estimate.  " }{TEXT 269 
16 "Evaluate it now." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 298 "with(plots):\n\nEndTime := 10:\nInitialT
 := 100:\n\nIVP := \{diff(T(t), t) = 50 - T(t),\n        T(0) = Initia
lT\}:\n\nsolution := dsolve(IVP, \{T(t)\}, numeric):\n\nplt := display
(\{odeplot(solution, [t, T(t)], 0..EndTime, color=blue)\}):\n\ngraf :=
 plot(50, 0..EndTime, 0..200, color=black):\n\ndisplay([plt, graf]);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 
"The cell above looks at the differential equation" }}{PARA 257 "" 0 "
" {TEXT -1 2 "\n\n" }{TEXT 270 11 "T' = 50 - T" }}{PARA 0 "" 0 "" 
{TEXT -1 29 "\n\nwith the initial condition " }{TEXT 271 10 "T(0) = 10
0" }{TEXT -1 15 " and the range " }{TEXT 272 5 "0..10" }{TEXT -1 17 " \+
for the variable" }{TEXT 273 2 " t" }{TEXT -1 24 ".  The initial value
 for" }{TEXT 274 2 " T" }{TEXT -1 11 " is called " }{TEXT 275 9 "Initi
alT " }{TEXT -1 42 "and the end of the range for the variable " }
{TEXT 276 1 "t" }{TEXT -1 12 " is denoted " }{TEXT 277 7 "EndTime" }
{TEXT -1 121 ".  You can change these values by changing the obvious l
ines above.   \n\nThe cell below looks at the differential equation" }
}{PARA 258 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 278 9 "y'' = -3y" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "  \n\nwith the initial conditi
ons  " }{TEXT 279 9 "y(0) = 0 " }{TEXT -1 6 " and  " }{TEXT 280 9 "y'(
0) = 1" }{TEXT -1 17 "  and the range  " }{TEXT 281 4 "0..1" }{TEXT 
-1 25 "0  for the time variable " }{TEXT 282 1 "t" }{TEXT -1 28 ".   T
he initial values for  " }{TEXT 283 1 "y" }{TEXT -1 7 "  and  " }
{TEXT 284 1 "y" }{TEXT -1 14 "'  are called " }{TEXT 285 9 " Initialy
" }{TEXT -1 7 "  and  " }{TEXT 286 9 "Initialyp" }{TEXT -1 167 "  resp
ectively and you can change these values by changing the obvious lines
 above. .  \n\nSpecifying the differential equation is a bit tricky.  \+
We use a new variable  " }{TEXT 287 1 "v" }{TEXT -1 30 "  to represent
 the derivative " }{TEXT 288 3 " y'" }{TEXT -1 13 "     Thus,   " }
{TEXT 289 1 "v" }{TEXT -1 8 "'   is  " }{TEXT 290 2 " y" }{TEXT -1 
119 "''  and the original (second order) differential equation is writ
ten as a pair of  (first order) differential equations" }}{PARA 259 "
" 0 "" {TEXT -1 2 "\n\n" }{TEXT 291 15 "y' = v\nv' = -3y" }{TEXT -1 2 
"\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 280 "with(plots):\n\nE
ndTime  := 10:\nInitialy  := 0:\nInitialyp := 1:\n\nIVP := \{diff(y(t)
, t) = v(t),\n        diff(v(t), t) = -3 * y(t),\n        y(0) = Initi
aly,\n        v(0) = Initialyp\}:\n\nsolution := dsolve(IVP, \{y(t), v
(t)\}, numeric):\n\ndisplay(odeplot(solution, [t, y(t)], 0..EndTime));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
62 "If you wanted, for example, to study the differential equation" }}
{PARA 260 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 292 16 "y'' = -3y -0.1y'" }
}{PARA 0 "" 0 "" {TEXT -1 25 "\n\nyou would rewrite it as" }}{PARA 0 "
" 0 "" {TEXT -1 2 "\n\n" }{TEXT 293 22 "y' = v\nv' = -3y - 0.1v" }}
{PARA 0 "" 0 "" {TEXT -1 29 "\n\nas shown in the cell below." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "
with(plots):\n\nEndTime  := 10:\nInitialy  := 0:\nInitialyp := 1:\n\nI
VP := \{diff(y(t), t) = v(t),\n        diff(v(t), t) = -3 * y(t) - 0.1
 * v(t),\n        y(0) = Initialy,\n        v(0) = Initialyp\}:\n\nsol
ution := dsolve(IVP, \{y(t), v(t)\}, numeric):\n\ndisplay(odeplot(solu
tion, [t, y(t)], 0..EndTime));" }}}}{MARK "0 0 0" 45 }{VIEWOPTS 1 1 0 
1 1 1803 }