Prof. Golé, Spring 99
MTH 222: Ordinary Differential Equations

Final Exam

Return your exam by Friday May 7 at 4 PM at the Science Center Office, ground floor of Burton. Give it to any secretary there: they will have you sign your name. You've been a fun class to work with, have a good summer!

1. (6, 6 pts) Find the general solutions of each of the following differential equations:

A. dy/dt - y3 sint=0

B. y" + 6y' +13y = e-2t

2. (6, 4, 4, 6 pts) A. Find the general solution of

x' = -y
y' = -13x -6y

B. Sketch the phase plane for this system.

C. Draw the x(t) and y(t) graphs for the solution of this system starting at (1,1).

D. Use Euler's method in five steps to approximate the solution starting at (1,1) up to time t=1. What is the error you make with this approximation?

3. (6, 6, + 6 Xtra credit pts) Use the method of Laplace transforms to solve the following initial value problems

A. y' + 2y = e3t, y(0) = 1

B. y' + 2y =u3(t), y(0) = 0

(Xtra credit): C. y'' + 4y = u4(t)e2(t-4), y(0) = 0, y'(0) = 0

You may use the table of Laplace transforms and Mathematica (in particular the command Apart to split a fraction into partial ones).

4. (8 pts) Find the bifurcation point(s) for the one-parameter family of ODE

y'=fa(y)=y3-ay

and draw the phase line for values of the parameter slightly smaller and slightly larger than, and at the bifurcation value.

5. (8 pts) A new disease has been discovered, which is transmitted by rare marsupials. It is known that the spread of this disease is governed by the following facts. There are 31 families having one marsupial each. 10 people have been infected so far. It is estimated that about 50 people are immune to the disease. The rate of change of the population infected by this disease is directly proportional to the following quantities: the number of individuals currently infected; the number of marsupials; the difference between the number of marsupials and the number of infected; the difference between the number of people infected and those that are immune to the disease. Write a differential equation that describes the spread of this disease in the future.

Without solving your ODE, or determining the constants exactly, tell what happens to the infected population in the future. Also predict what would happen if 40 people were infected at this time instead of 10. What about if 60 people have been infected?

6. (20 pts) One of the nonlinear systems of differential equations govening a laser is given by

x' = 1 -x -3xy
y' = -y + 3xy

Using any and all means possible, describe as much of the phase plane for this system as you can. Some features to not miss: find all equilibria and their type, the nullclines and the behavior of the vector field on the nullclines as well as in the regions they separate, connecting orbits between equilibria. Note that there is no restriction on x and y here: they can be negative. Don't be shy and draw BIG and neat...