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+x2)
dy/dt = x -xy/(1+x2).

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) = Ax2 + x3 + y2.

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?