Math 282 Name: Key

Quiz 4 October 22, 2009

You must SHOW ALL WORK to receive credit!

1. Consider the differential equation:

For what value(s) of a will this have more than one equilibrium solution? Where are these equilibria?

The determinant is a-6, so there are multiple equilibria if a-6=0, or a=6. These occur along the line 1x+2y=0, or y=-1/2 x

2. a. What are the straight line solutions for the differential equation: ?

. Setting this equal to 0, we get the eigenvalues are 5 and -1.

Solving , and

solving

b. Find the general solution to . Give your answer in vector form.

From above, the we know the eigenvalues are 5 and -1, and corresponding eigenvectors are and .

So the general solution is

Extra credit: On the back of this sheet, graph the phase plane for this system. You should clearly show the straight line solutions as well as other “typical” solutions, including the direction of motion.