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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "  This example can be used
 as a template for Problem 9, p 118" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 297 "In this \+
example, we will plot numerically generated solutions to a system of d
ifferential equations.  In this example, we  have t, x(t), and y(t).  \+
We can either view the solution as a parametrized curve, (x(t), y(t)),
 or view two separate curves (t, x(t)) and (t, y(t)).  We look at both
 below.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
86 "Before we start, we have to define the system of equations and the
 initial conditions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 66 "Deqns:=[diff(x(t),t)=x(t)*(1-y(t)),diff(y
(t),t)=.3*y(t)*(x(t)-1)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "initconds:=[[x(0)=1.2,y(0)=1.2],[x(0)=1,y(0)=.7]];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "DEplot(Deqns,[x(t),y(t)],t=0..7,ini
tconds);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 36 "This plot isn't very good.  We will:" }}{PARA 0 "" 0 "" 
{TEXT -1 67 "  (a)  Change the step size to a smaller value.  Use:  st
epsize=0.1" }}{PARA 0 "" 0 "" {TEXT -1 73 "  (b)  Color the solution c
urves black.             Use:  linecolor=black" }}{PARA 0 "" 0 "" 
{TEXT -1 75 "  (c)  Expand the time interval                         U
se:   time = 0..15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 75 "DEplot(Deqns,[x(t),y(t)],t=0..15,initconds,
 stepsize=0.1, linecolor=black);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 431 "We will now construct a plot wher
e t is the independent variable, and x(t), y(t) are the two dependent \+
variables.  To do this in Maple, first construct two separate plots, t
hen display them together.  We will use an extra line:  scene=[t,x(t)]
 or scene=[t, y(t)] to get the right plots.  We'll only use the first \+
initial condition for this plot (otherwise we'll have too many curves)
, and we'll plot it over a longer time interval." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "initconds:=
[[x(0)=1.2,y(0)=1.2]];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "A
:=DEplot(Deqns,[x(t),y(t)],t=0..25,initconds,stepsize=0.1,linecolor=re
d,scene=[t,x(t)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "B:=DE
plot(Deqns,[x(t),y(t)],t=0..25,initconds,stepsize=0.1,linecolor=green,
scene=[t,y(t)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl
ots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(\{A,B\});
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 239 " By zooming in on the graph,
 and clicking on the first intersection of the curves, we see that the
 first time the populations are equal again is about t=11.57.  In this
 case, this will also be the approximate period for both x(t) and y(t)
." }}}}{MARK "15" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 
1 2 33 1 1 }