# MA2051 - Ordinary Differential EquationsOld Exam 1

Instructions:  Do your work on the paper provided.  Put your
name and section number on the cover and on each page.  Work neatly

1. (40 points) Work with 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 general solution as well as the solution satisfying the initial condition .
(d)
Is it possible to choose the initial value so that the solution to the initial value problem is exactly equal to the particular solution found in part (b)? If it is not possible, explain why not. If it is possible, explain how to do it.

2. (25 points) Solve each of the following initial value problems:

3. (35 points) Water containing 2 oz of TCE (a toxic waste) per gallon flows into a treatment tank at a rate of 500 gallons per minute. The volume of the tank is fixed at 10,000 gallons (so water must flow out at the same rate that it flows in). The treatment process is able to remove 12% of the current amount of TCE in the tank per minute. Assume that the water is thoroughly stirred and that the tank is initially filled with clean water.

(a)
Write an expression for the rateat which TCE is added to the tank. Write an expression for the rateat which TCE is removed from the tank. Define all of the variables used in your expressions. (Note: TCE is removed by treatment as well as the out-flow.)
(b)
Write a balance equation reflecting the fact that the net change in the amount of TCE in the tank over a time is equal to the amount of TCE added minus the amount of TCE removed. Use this law and the results from part (a) to derive a differential equation model for the amount of TCE in the tank at time t. (Do not solve the initial value problem.)
(c)
If you wait a long time (like 24 hours), the concentration of TCE in the water flowing out of the tank approaches a constant level. Without solving the differential equation derived in part (b), determine this constant level.
(d)
Use your model (and the results from part (c)) to determine ways to reduce the concentration of TCE in the water flowing out of the tank. Justify your answer.

4: BONUS (5 points) Government regulations specify that the maximum allowed TCE concentration released from the tank is 0.05 oz per gallon. Change the parameters in the model if you need to, but find a way to satisfy the government regulations (or go out of business).

Solutions To This Exam
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