Newton's law of cooling with multiple containers DE's

[asantas93]

I've been trying to figure out how to solve the following problem:

"A metal bar is placed in a container (call it container A) which is inside of a much larger container (call it container B), whose temperature can be assumed to be constant. Find the function for the temperature of the metal bar at time t."

First, I represent the temperature of container A as x(t) and the metal bar as y(t). The constant temperature of container B is T_B. I assume that the temperature of the bar is proportional to the temperature of container A but the temperature of container A is proportional to both T_B and the temperature of the bar. Then

dx/dt = k₁(T_B - x) + k₂(y - x) and
dy/dt = k₃(x - y)

I was told that I need to make a substitution to solve this, but I can't seem to take this DE down to two variables. Any help would be appreciated.

 

[gtoguy97]

Hint: Try solving for the derivative of (y - x) and substituting it into one of the equations. Substituting dy/dt = k3(x - y) into dx/dt = k1(TB - x) + k2(y - x) gives us: dx/dt = k1(TB - x) + k2(dy/dt - k3(x - y))Rearranging this and solving for dy/dt yields: dy/dt = (k1/k2)*(TB - x) + (k3/k2)*(y - x)Now we have two equations with two unknown variables, x and y. We can now proceed to solve the system of differential equations.