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.