## MA 226: Ordinary Differential Equations Sample Exam 3

Do all problems and show all work.

1. In an essay, describe the trace-determinant plane for linear systems. Be sure to tell where this picture comes from, and what it means. Discuss what the different regions in this picture mean. Sketch the typical phase planes for a variety of different regions in your essay.

2. Consider the linear system of differential equations:

x' = -5x + y
y' = -x - 3y

First find the general solution of this system. Then sketch its phase plane.

3. Consider the harmonic oscillator

y" + ky' +3y = 0

where the damping constant k is non-negative. For which values of k is this system undamped? Underdamped? Critically Damped? Overdamped?

4. Consider the damped and periodically forced harmonic oscillator:

y" + 3y' +3y = sin(2t).

Solve the initial value problem y(0) = 0, y'(0) = 0. What is the ultimate behavior of this system?

5. In a brief essay, discuss what will happen to solutions of the periodically forced system

y" +3y = sin(At)

for various values of the constant A.

6. Consider the system of equations:

x' = -2x + y
y' = -y + x2

First find all equilibrium points for this system. Classify each of them as either a sink, source, or saddle. Then sketch the phase plane near each of these points.

7. Consider the system of equations:

x' = x(-x -3y + 150)
y' = y(-2x -y + 100)

Sketch the x(t) and y(t) graphs of the solution satisfying the initial condition x(0) =30, y(0) = 40.

8. Quickies. Answers only; no partial credit.

A. My favorite differential equations teacher is: _____________

B. The differential equation for a nonlinear pendulum is:

C. The system of differential equations corresponding to the equation

y" + 3y' + 6y = cos(2t)

is:

D. I am going to get a (an)______ on exam 3.

9. Use the method of Laplace Transforms to solve the following initial value problem:

dy/dt + y = 0 if 0 < t < 1;
dy/dt + y = 5 if 1 <= t:
y(0) = 2