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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 = delta**_{pi}(t), y(0) = 1, y'(0) = 0
You may use the table of Laplace transforms.

4. Consider the function **F(x) = x**^{2} + 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.

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