##
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 **Y**_{1}(t) = (x_{1}(t),
y_{1}(t)) and **Y**_{2}(t) =
(x_{2}(t), y_{2}(t)). First define the Wronskian
**W(t)** of these
solutions. Then prove that **Y**_{1} and **Y**_{2} 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 = y**^{2} + 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' =
y**^{2} + 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.