## MA 226: Ordinary Differential Equations Sample Final Exam

Do all problems and show all work.

1. Find the general solutions of each of the following differential equations:

A. dy/dt = (y^2 + 1)t^2

B. y" + 3y' +3y = 0

2. Find the general solution of

x' = 2x+y
y' = x + y

Also sketch the phase plane for this system. Draw the x(t) and y(t) graphs for the solution of this system starting at (1,1)

3. Use the method of Laplace transforms to solve the following initial value problem

y" + y = deltapi(t), y(0) = 1, y'(0) = 0

You may use the table of Laplace transforms.

4. Consider the function F(x) = x2 + 0.25. First find all fixed points and classify them as attracting, repelling, or neutral. Then use graphical analysis to determine the fate of all other orbits under iteration.

5. In a two page essay, describe as much as possible what bifurcation means. You should give at least one example of a bifurcation. Provide graphs as necessary.

6. Find all equilibrium points for the following system and determine their type. Then sketch the phase plane for this system, indicating the locations of all nullclines.

x ' = y
y' = -y - sin(x)

7. Another new disease has been discovered. It is called "Engineer's Disease #2" and it is quite deadly. Unhappily, this disease affects people who are not necessarily engineers. It is known that the spread of this disease is governed by the following facts. The rate of change of the population infected by this disease is directly proportional to both the square of the number of individuals currently infected and also to the difference between the number of engineers in the universe and the infected population, i.e., is proportional to the product of these two quantities. It is known that the number of engineers in the universe is the (mercifully small and constant) number 1,002. It is also known that exactly 100 people have Engineer's disease #2 now. 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 2000 people were infected at this time instead of 100.

8. Write a brief essay about Euler's method. Explain the formula used in this method. Show a picture of what Euler's method is actually accomplishing.

9. 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.