## 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).

• 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.