Problems on Equilibrium Solutions to Nonhomogeneous Systems of DE's

1. Consider the two room apartment shown below.

   

   Let T₁(t) and T₂(t) denote the temperatures in rooms one and two at time t, and let Q₁(t) and Q₂(t) denote the heat energy in rooms one and two. The rate of change of heat energy in each room is equal to the rate at which heat enters the room minus the rate at which it leaves the room, as shown by the arrows. The rate at which heat flows across each arrow is proportional to the difference in the temperature between the two areas, with the proportionality constant being the heat transfer constant marked on the arrow. For instance,

   Q₁'(t) = 300 BTU/hr ^{. o}F(T₂(t) - T₁(t)) - 100 BTU/hr ^{. o}F(T₁(t) - 10^oF).
   (Don't forget the heat pump attached to room 2.)

   The rate at which the temperature changes in each room is directly proportional to the rate at which the heat energy in each room changes, with proportionality constant begin the heat capacity marked in the middle of the room. For instance,

   Q₁'(t) = 250 BTU/^oF T₁'(t).

   (a)

   Use these facts to write a system of differential equations satisfied by T₁(t) and T₂(t). (Eliminate Q₁(t) and Q₂(t).)

   (b)

   Find the equilibrium solution of the system. (That is, the solution when the temperatures are constant.)

   (c)

   Find the general solution of the system, using the eigenvalue method and the equilibrium solution.

   (d)

   Assume the inital temperatures are T₁(0) = T₂(0) = 80^oF. How cold can it get in the unheated room?

   (e)

   Find the size of the heat pump such that the equilibrium temperature in room two is 80^o F. What is the corresponding temperature in room one?

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Problems on Equilibrium Solutions to Nonhomogeneous Systems of DE's

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