##
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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