From your conditions it seems like separation of variables is probably not going to work. When you do separation of variables, your solution will be of the form u(x,t) = X(x)T(t). Now whatever T(t) is (in this case it will be an exponential), u(x1,t) and u(x2,t) are only going to differ by a multiplicative constant factor (i.e X(x2)/X(x1)). But you have 2 different functions of time for the 2 boundaries. So you can see that it can't work. I also have a feeling doing this in an analytical fashion might not be easy.