## MA 231: Honors Differential Equations

Do all problems on the blank sheets in order. Please write the problem number in the upper right hand corner of each sheet. Be sure to show all work.

1. First find the general solution of the following system of differential equations:

dx/dt = -y
dy/dt = 3x.

Then sketch a representative sample of solutions in the phase plane.

2. Consider the system of differential equations depending on a parameter A:

dx/dt = Ax + y
dy/dt = 2y.

a. Find the general solution. Your solution will, of course, depend on A.
b. Determine which values of A lead to saddles, sinks, spiral sinks, etc.
c. Sketch all possible different phase portraits for this system.

3. Find all solutions of the differential equation

dy/dt = y2/3

satisfying y(0) = 0. In a paragraph, explain why you have so many solutions. (That's a little hint!)

4. Consider the first order differential equation

dy/dt = Ay + 1

where A is a parameter. Describe any bifurcations that occur in this family of differential equations and sketch the bifurcation diagram. You need not solve this equation, by the way. Just a sketch and a description is sufficient.

5. In an essay, discuss Euler's Method and how it is used to approximate solutions of differential equations. Be sure to include all relevant formulas and some pictures.