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 TB. 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 TB and the temperature of the bar. Then
dx/dt = k1(TB - 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.