MA 226: Ordinary Differential Equations

Sample Exam 1

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. dy/dt = y(1-y)

2. Find the general solution of

dy/dt = 3y/t^2

At which points does this equation fail to have unique solutions? Sketch a number of the solution curves.

3. Sketch the slope field for

dy/dt = y^3 - y.

Then sketch the phase line, indicating all equilibrium solutions. Finally sketch various solution curves that illustrate which equilibrium points are sinks, sources, or nodes.

4. Here is a portion of the direction field for

dy/dt = sin(y) + exp(y).

In a two page essay, describe as much as possible of what the solution curves look like. Your essay should be in proper English. Faulty grammar, spelling, punctuation will result in points taken off. If I cannot read your writing, you also lose points. Include sketches of solution curves, equilibrium points, and the phase line. Show (approximate) locations of equilibrium points. Indicate whether they are sinks, sources, or nodes.

5. Give an example of a first order autonomous differential equation whose phase line looks like:

6. Use Euler's method with step size 0.1 to approximate the following initial value problem

dy/dt= t^2y^2, y(0)=1.

Starting with y_0 = 1, compute the values of y_1 and y_2 using Euler's method.

7. A new disease has been discovered. It is called "Engineer's Disease" 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 number of individuals currently infected and to the square of 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 1,234. It is also known that exactly 100 people have Engineer's disease 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. Draw a picture of someone infected with Engineer's Disease. Your drawing should not resemble any instructor at BU.

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