MA 226: Ordinary Differential Equations

Sample Exam 2

Do all problems and show all work.

1. Consider the predator prey system

R' = 2R(1 - 0.5R) -1.2 RF

F' = -F + 0.9 RF.

Here is the phase plane for this system with two solution curves sketched. First indicate the approximate locations of all equilibrium solutions on this picture. Then, for each of the solution curves A and B, draw the corresponding x(t) and y(t) graphs.

2. IMPORTANT: We have not covered all of the material relevant to this question yet. Consider the system of differential equations

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

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

3. Consider the linear system

x' = -4x + y

y' = 2x - 3y

4. In an essay, discuss the different kind of phase plane pictures that can occur for constant coefficient linear systems of the form

x' = ax + by

y' = cx + dy

when the eigenvalues of the system are real (and distinct). Be sure to include pictures of typical types of systems and discuss the eigenvalues corresponding to each type.

5. Consider the second order equation for a harmonic oscillator

y'' + y' + 4y = 0.

6. Here are four phase planes and four systems. Match them.

A. x' = 2y, y' = -2x

B. x' = 2y, y' = 2x

C. x' = x+2y, y' = 2x-y

D. x' = x+2y, y' = -2x + y

7. Show that, for the linear system Y' = AY, if AV = kV for some nonzero vector V, then

Y(t) = ektV

is a solution of the system.

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