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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 38 "The Language of Differen
tial Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 130 "The cell below illustrates how Maple can estimate the so
lution to a first order differential equation and the plot the estimat
e.  " }{TEXT 256 16 "Evaluate it now." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "with(plots):\n\nEndTim
e := 60:\nInitialT := 180:\n\nIVP := \{diff(T(t), t) = 0.05 * (65 - T(
t)),\n        T(0) = InitialT\}:\n\nsolution := dsolve(IVP, \{T(t)\}, \+
numeric):\n\nplt := display(\{odeplot(solution, [t, T(t)], 0..EndTime,
 color=blue)\}):\n\ngraf := plot(65, 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 257 18 "T' = 0.
05 (65 - T)" }}{PARA 0 "" 0 "" {TEXT -1 29 "\n\nwith the initial condi
tion " }{TEXT 258 10 "T(0) = 180" }{TEXT -1 15 " and the range " }
{TEXT 259 5 "0..60" }{TEXT -1 17 " for the variable" }{TEXT 260 2 " t
" }{TEXT -1 24 ".  The initial value for" }{TEXT 261 2 " T" }{TEXT -1 
11 " is called " }{TEXT 262 9 "InitialT " }{TEXT -1 42 "and the end of
 the range for the variable " }{TEXT 263 1 "t" }{TEXT -1 12 " is denot
ed " }{TEXT 264 7 "EndTime" }{TEXT -1 121 ".  You can change these val
ues by changing the obvious lines above.   \n\nThe cell below looks at
 the differential equation" }}{PARA 258 "" 0 "" {TEXT -1 2 "\n\n" }
{TEXT 265 9 "y'' = -3y" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 
"  \n\nwith the initial conditions  " }{TEXT 266 9 "y(0) = 0 " }{TEXT 
-1 6 " and  " }{TEXT 267 9 "y'(0) = 1" }{TEXT -1 17 "  and the range  \+
" }{TEXT 268 4 "0..1" }{TEXT -1 25 "0  for the time variable " }{TEXT 
269 1 "t" }{TEXT -1 28 ".   The initial values for  " }{TEXT 270 1 "y
" }{TEXT -1 7 "  and  " }{TEXT 271 1 "y" }{TEXT -1 14 "'  are called \+
" }{TEXT 272 9 " Initialy" }{TEXT -1 7 "  and  " }{TEXT 273 9 "Initial
yp" }{TEXT -1 167 "  respectively and you can change these values by c
hanging the obvious lines above. .  \n\nSpecifying the differential eq
uation is a bit tricky.  We use a new variable  " }{TEXT 274 1 "v" }
{TEXT -1 30 "  to represent the derivative " }{TEXT 275 3 " y'" }
{TEXT -1 13 "     Thus,   " }{TEXT 276 1 "v" }{TEXT -1 8 "'   is  " }
{TEXT 277 2 " y" }{TEXT -1 119 "''  and the original (second order) di
fferential equation is written as a pair of  (first order) differentia
l equations" }}{PARA 259 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 278 15 "y' =
 v\nv' = -3y" }{TEXT -1 2 "\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 280 "with(plots):\n\nEndTime  := 10:\nInitialy  := 0:\nInitialyp :
= 1:\n\nIVP := \{diff(y(t), t) = v(t),\n        diff(v(t), t) = -3 * y
(t),\n        y(0) = Initialy,\n        v(0) = Initialyp\}:\n\nsolutio
n := 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 
279 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 280 22 "y' \+
= v\nv' = -3y - 0.1v" }}{PARA 0 "" 0 "" {TEXT -1 29 "\n\nas shown in t
he 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\nIVP := \{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\nsolution := dsolve(IVP, \{y(t), v(t)\}, numeric)
:\n\ndisplay(odeplot(solution, [t, y(t)], 0..EndTime));" }}}}{MARK "0 \+
0 0" 38 }{VIEWOPTS 1 1 0 1 1 1803 }