Originally Given 1994 B Term

**(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
*T*(0) = 32. **(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.

**(a)**-
Write an expression for the
*rate*at which TCE is added to the tank. Write an expression for the*rate*at 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.

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