Heat Transfer Problem

1. The problem statement, all variables and given/known data
Two bodies of masses [itex]m[/itex][itex]_{1}[/itex] and [itex]m[/itex][itex]_{2}[/itex] and specific heat capacities [itex]s[/itex][itex]_{1}[/itex] and [itex]s[/itex][itex]_{2}[/itex] , are connected by a rod of length [itex]l[/itex] and cross-sectional area [itex]A[/itex], thermal conductivity [itex]K[/itex] and negligible heat capacity. The whole system is thermally insulated. At time [itex]t=0[/itex], the temperature of the first body is [itex]T[/itex][itex]_{1}[/itex] and the temperature of the second body is [itex]T[/itex][itex]_{2}[/itex] ([itex]T[/itex][itex]_{2}[/itex] [itex]>[/itex] [itex]T[/itex][itex]_{1}[/itex] ). Find the temperature difference between the bodies at time [itex]t[/itex].

2. Relevant equations
[itex] dQ/dt = KAdT/dx[/itex]

[itex] dQ = msdθ [/itex]

3. The attempt at a solution
I was able to set up [itex]2[/itex] equations relating the amount of heat transferred through the rod in a time [itex]dt[/itex] to the rise and fall of temperatures of the masses [itex]m[/itex][itex]_{2}[/itex] and [itex]m[/itex][itex]_{1}[/itex] respectively. I don’t know how to proceed after this ?


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