I've been trying to figure out how to solve the following problem:(adsbygoogle = window.adsbygoogle || []).push({});

"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 = k_{1}(T_{B}- x) + k_{2}(y - x) and

dy/dt = k_{3}(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.

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

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