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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 + x^{2}
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

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