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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 41 "Differential Equations -
- Guess-and-Check" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 118 "The cell below illustrates how Maple can estimate the so
lution to an initial valueproblem and the plot the estimate.  " }
{TEXT 256 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) = InitialT\}:\n\nsolution := dsolve(IVP, \{T(t)\}, numeric):\n\np
lt := display(\{odeplot(solution, [t, T(t)], 0..EndTime, color=blue)\}
):\n\ngraf := plot(50, 0..EndTime, 0..200, color=black):\n\ndisplay([p
lt, 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 11 "T' = 50 - T" }}
{PARA 0 "" 0 "" {TEXT -1 29 "\n\nwith the initial condition " }{TEXT 
258 10 "T(0) = 100" }{TEXT -1 15 " and the range " }{TEXT 259 5 "0..10
" }{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 denoted " }{TEXT 264 7 
"EndTime" }{TEXT -1 121 ".  You can change these values by changing th
e obvious lines above.   \n\nThe cell below looks at the differential \+
equation" }}{PARA 258 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 265 9 "y'' = -3
y" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "  \n\nwith the initi
al 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 " Initi
aly" }{TEXT -1 7 "  and  " }{TEXT 273 9 "Initialyp" }{TEXT -1 167 "  r
espectively and you can change these values by changing the obvious li
nes above. .  \n\nSpecifying the differential equation is a bit tricky
.  We use a new variable  " }{TEXT 274 1 "v" }{TEXT -1 30 "  to repres
ent 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) differential equation is writ
ten as a pair of  (first order) differential 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\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 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 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" 41 }{VIEWOPTS 1 1 0 
1 1 1803 }