This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Systems of ﬁrstorder differential equations
found since
2e2t
2e2t
2
x
=
= 2t =
2
t )2
y
(3e
9e
9
Hence
y= 9x
2 Whether or not it is possible to readily ﬁnd a Cartesian representation of the
parametric curve, it is a simple matter to plot the parametric curve itself using
software packages.
Example 4.1 with Mathematica
The two commands used in this set of instructions, DSolve and ParametricPlot
are now both contained in the main package:11
Clear[x,y]
sol=DSolve[{x’
[t]==2x[t],y’
[t]==y[t],x[0]==2, y[0]==3},
{x[t],y[t]},t]
solx=sol[[1,1,2]]
soly=sol[[1,2,2]]
x[t]:=solx
y[t]:=soly
traj=ParametricPlot[{x[t],y[t]},{t,0,1}] If the equations for x(t) and y(t) are already known, then only the last instruction
need be given. For example, if it is known that x(t) = 2e2t and y(t) = 3et then all
that is required is
traj=ParametricPlot[{2e2t ,3et },{t,0,1}] Example 4.1 with Maple
To use Maple’s routine for plotting parametric equations that are solutions to
differential equations it is necessary to load the plots package ﬁrst. The following
input instructions will produce the trajectory for example 4.1:
restart;
with(plots):
sys:={diff(x(t),t)=2*x(t),diff(y(t),t)=y(t),
x(0)=2,y(0)=3}
vars:={x(t),y(t)}:
sol:=dsolve(sys,vars,numeric);
odeplot(sol,[x(t),y(t)],0..1,labels=[x,y]); 11 In earlier versions, DSolve and ParametricPlot needed to be loaded ﬁrst since these were contained in the additional packages. This is no longer necessary, since both are contained in the basic
built in functions. 195 ...
View Full
Document
 Fall '11
 Dr.Gwartney
 Economics

Click to edit the document details