##
MA 231:
Honors Differential Equations
Exam #2

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 solution of the following second
order differential equation that satisfies **y(0)=0, y'(0)=2**

**
y'' + 4y = cos(2t).
**

Then sketch the **y(t)**-graph for this solution. Describe in a
sentence or two what happens to the corresponding forced harmonic
oscillator.
2. Consider the system of differential equations that
describe a chemical reaction involving iodide (whose concentration is
given by **x(t)**) and chlorine dioxide (**y(t)**).

**
dx/dt = -x + 10 - 4xy/(1+x**^{2})

dy/dt = x -xy/(1+x^{2}).

- Find all equilibrium points for this system.
- Determine the linearizations at these equilibria.
- Sketch the approximate locations of the
nullclines for this system and the
corresponding phase plane.
- Describe briefly in words what happens to typical
solutions for this system in terms of the behavior of the chemicals.

3. Define a Hamiltonian system. What are the only types of
equilibrium points possible for a Hamiltonian system? Prove that this
is true.

4. Consider the gradient system whose gradient function is given by

**G(x, y) = Ax**^{2} + x^{3} +
y^{2}.

- Find all equilibrium points for this system and their
types.
- Where does this system undergo a bifurcation?
- Describe how the phase plane of this system changes as you
pass through the bifurcation.

5. In an essay of no more than two pages,
discuss the behavior of the nonlinear
pendulum. What are the equations of motion? Equilibrium points?
Behavior in the damped and undamped cases?