Suppose you are considering the following system of two differential equations,

dx/dt = x - xy,
dy/dt = -y + xy,

which appear on page 570 of your text (x=w, y=r).

To see the phase plane (slope field) on your calculator, go to the "Y=" screen and set

Y1 = (-Y + XY)/(X - XY), ---Here Y1 represents dy/dx.

Then use the SLOPES program. Your calculator should give you a picture like the one on page 570.

To see a phase trajectory for the above system (i.e. the solution to the above equation which passes through some point), go back to the "Y=" screen and set

Y2 = X - XY, --- Here Y2 represents dx/dt.
Y3 = -Y + XY, --- Here Y3 represents dy/dt.

and then use the TRAJECT program.

TRAJECT will ask you for an inital point and for a step size. It will then use a version of Euler's method to graph an approximation of the solution which passes through the point you specified. The graph will be superimposed on top of the slope field you drew with SLOPE.

Note that the trajectory is approximate. That shows up in this example, when the curve does not quite meet itself after travelling around a path that should close exactly.

Short Cut

Instead of typing so much in for Y1, you can set Y2 and Y3 according to the given system, and then set Y1=Y3/Y2 (using the VARS button and YVARS option on the TI-83). It is more efficient and there is less chance of an error.

Warning

The program can hang up where the denominator (=dx/dt) vanishes. To avoid this you can:

• Change the window so that the graphed slope elements no longer hit a point where dx/dt vanishes, or...
• Change the problem by adding a very small quantity to dx/dt (e.g. 0.0001) which to the eye does not change any slopes but can avoid the vanishing denominator problem.