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Planar Systems of Differential Equations

MA 233-02

February 12, 1999

Richard Hitt

Introduction

This worksheet discusses some of the material in section 1.7 of the Borrelli and Coleman book.

Getting started

Every Maple worksheet should begin by re-initializing the Maple "kernel" and loading the additional packages that we are most likely to use.

> restart;

> with( plots ):

> with( DEtools ):

Example 1

Consider the following separable DE which we solved in the worksheet from section 1.7 on separable equations: . In order to get integral curves of this DE plotted we had to use the contourplot command in Maple. But we were not able to specify an initial condition and get an integral curve precisely through that point.

In this section, we learn how to use parametric systems of differential equations to do this.

Think of and as each being functions of time . Then set and . We can solve this system of differential equations using a numerical solver, say DEplot, and then graph the ordered pairs as varies to see the integral curve for the original DE. Here's one way to do this in Maple.

> DE1 := diff(y(t),t) = x(t)^2;
DE2 := diff(x(t),t) = 1-y(t)^2;

We can combine the vector field and the contours into a single graph as follows.

First we generate a list of initial conditions.

> IC := seq([0,0,i/4],i=-12..12):

> DEplot(
[DE1,DE2],
[x,y],
t=-5...5,
x=-3..3,
y=-3..3,
[IC],
linecolor=blue,
stepsize=.1
);

Question: What is happening near the points (0,1) and (0,-1)? There are curves that seem to stop near those points.

Exercise

For the separable differential equation , procede as above to plot the vector field together with several integral curves. Choose a reasonable viewing window for the curves.

Example 1.7.2 on page 58 of Borrelli/Coleman

The example mentioned in the section heading can be reproduced using these methods as well.

> DE3 := diff(x(t),t) = sin(y) - 2*sin(x^2)*sin(2*y);
DE4 := diff(y(t),t) = -cos(x) - 2*x*cos(x^2)*cos(2*y);

> IC := {seq([0,-2,i],i=-4..4)} union {seq([0,0,i],i=-4..4)}:

> DEplot(
[DE3,DE4],
[x,y],
t=-10...10,
x=-4..2,
y=-4..4,
IC,
linecolor=blue,
stepsize=.1,
arrows=NONE
);

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