## 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 = x + y
dy/dt = x

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

2. Consider the system of differential equations:

dx/dt = 1
dy/dt = x.

• Sketch the vector field for this system.
• Find the general solution.
• Determine the solution satisfying x(0)=y(0) =1.
• Sketch the solutions in the phase plane.

3. Consider the linear system Y' = AY and suppose that Y1(t) = (x1(t), y1(t)) and Y2(t) = (x2(t), y2(t)). First define the Wronskian W(t) of these solutions. Then prove that Y1 and Y2 are independent if W(0) not = 0. Then prove that W(t) not = 0 for any t if W(0) not = 0.

4. Consider the first order differential equation

dy/dt = y2 + By +C

where B and C are parameters.

• Fix B=2. Sketch the bifurcation diagram for this family of differential equations (i.e., for the equation y' = y2 + 2y +C.
• Now fix instead C=1 and B vary. Sketch the bifurcation diagram for this family.
• Finally provide a sketch in the B,C-plane of the various equilibrium point configurations for this system. Explain what your diagram means in a paragraph or two.
5. List all possible 2 by 2 matrices with real entries which have repeated 0 eigenvalues, i.e., list all possible entries a, b, c, d for which the matrix
|a b|
|c d|

has repeated 0 eigenvalues. Be careful--- there are several different types.