Problems on Nonhomogeneous Systems of DE's

1. Earlier we looked at salt tank problems such as the one shown in the following picture.

   

   This is a homogeneous system, since there is no salt entering the system. Now we are going to make it a non-homogeneous system and use Laplace transforms to solve it.

   (a)

   Instead of fresh water coming in, assume there is an exponentially decaying concentration of salt f (t) = 3e^-5t lbs/gal entering the system. The initial amount of salt in the left tank is 100 lbs and there is no salt initially in the right tank. Use Laplace transforms to solve the system. What is the maximum amount of salt in the right tank, and when?

   (b)

   Now suppose there is an oscillating concentration of salt f (t) = 3 sin($\pi$t) entering the system. (The initial amounts are the same as in part (a).) Use Laplace transforms to solve the system. What does the solution look like? (Exponential functions, oscillations, damped oscillations, something else?)

2. Now consider the two room apartment shown below. (We dealt with this situation in an earlier assignment.)

   

   (a)

   Suppose the heat pump is failing over the course of the day, so that its output is f (t) = 10000e^-3t BTU/hr. Assume the initial temperatures are T₁(0) = T₂(0) = 80^oF. How long will it take the unheated room to drop to 60^oF? (Use Laplace transforms to solve the system.)

   (b)

   Now suppose the heat pump is working, but it is on a timer to keep it warmer during the day and cooler at night. Its output is now f (t) = 10000 + 1000 sin($\pi$t/12) BTU/hr. Assume the inital temperatures are T₁(0) = T₂(0) = 80^oF. How cold can it get in the unheated room? (Use Laplace transforms to solve the system.)

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Problems on Nonhomogeneous Systems of DE's

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