**MA2051--C97****Name**

**TEST I****Section**
**Instructions:** Do your work on the paper provided. Put your
name and section number on the cover and on each page. Work neatly
and show your work.
Remember, *your work and explanations are graded, not just the
final answer.*
**1.** (35 points)
In this problem, you will analyze the following linear
differential equation:

**(a)**- Find a nontrivial solution for the corresponding homogeneous equation.
**(b)**- Find a particular solution for the original differential equation.
**(c)**-
Find the solution for the differential equation which
satisfies the initial condition
*y*(0) = 5. **(d)**-
Can you choose the initial condition so that the solution to the
initial value problem contains only the particular solution?
Explain your answer clearly.

Let *A*(*t*) denote the *amount* of sugar in the tank at time
*t*. Let *C*(*t*) denote the *concentration* of sugar in the
tank at time *t*.

**(a)**-
Write an expression for the
*rate*at which sugar is entering the tank. **(b)**-
Write an expression for the
*rate*at which sugar is leaving the tank.Why is it important to assume that the liquid in the tank is ``thoroughly mixed'' before the mixture is removed from the room?

**(c)**-
Write a balance equation that relates the
*net change*in the amount of sugar in the tank over a time with the*rate*at which it is added and lost. Use this equation to derive a differential equation for the amount of sugar in the tank at time*t*. (Do not solve the differential equation.) **(d)**-
If you wait a long time, the concentration in the mixture flowing out
of the tank will be constant. Use your model to predict this constant
level.

**3.** (15 points) The following equation models a
population that grows faster than the standard exponential model:

Solve the equation and
show that the population will become infinite in finite time.
**4.** (15 points)
Work with the following differential equation:

**(a)**- Find all of the steady states for this equation.
**(b)**-
Determine the regions in the (
*t*,*y*) plane where the solution curves are increasing and where the solution curves are decreasing. **(c)**-
Sketch solution curves starting at three different initial points:
*y*(0) = 0,*y*(0) = 8.9, and*y*(0) = 12.

is a nonlinear equation. Show that a new variable defined by

satisfies a linear equation. Use this linear equation to solve the original logistic equation. Explain your answer completely.

Thu Feb 6 16:50:22 EST 1997