Cylinder with heat generation, Separation of variables

I'm having some difficulty setting up a problem. I'm trying to model the temperature of a thermistor connected to a constant current source. The thermistor's resistance varies with temperature, so with a fixed current, I would expect to see the thermistor's temperature to oscillate with time. For simplicity, I'm only interested in the surface temperature ##T_s## as a function of time. I'm trying to derive the result analytically. This is how I am setting it up:



Suppose you have a cylinder with some radius ##a##, and length ##L##. It has internal heat generation per unit volume ##Q(T) = \frac{I^2 R(T)}{\pi a^2 L}##, where ##I## is a fixed current, and ##R## is a resistance that varies with temperature. Assume ##R≈-mT+b## where ##T## is the temperature of the cylinder and ##c## is a constant of proportionality. Assume that heat only flows through the cylinder's side surface (not through it's ends).



At ##t=0##, ##T = T_0## everywhere.



Using the heat equation:



$$\rho C \frac{\partial T}{\partial T} = \kappa \nabla^2 T + Q(T)$$



where ##\rho## is the density, ##C## is the heat capacity, ##\kappa## is the thermal conductivity. For simplicity, assume these are all constant.



If we replace ##Q(T)## with ##\left(\frac{I^2}{\pi a^2 L}\right) (-mT+b) = -\mu T + \beta##, then:



$$\rho C \frac{\partial T}{\partial T} = \kappa \nabla^2 T - \mu T + \beta$$



My questions are:



1. Have I set this up correctly?



2. What is the appropriate way to solve this for ##T(T_s,t)##?



Using separation of variables, you would normally let ##T(r,t) = R(r) \Gamma (t)##, but I cannot solve it this way since the ##\beta## term makes it difficult to separate variables.



Any help on this would be much appreciated. Thanks in advance!