A rod of length L with insulated lateral surface has thermal constants c, p, k. One end of the rod is kept at temperature zero. The other end is connected to a black box which manages to maintain that end at the instantaneous average temperature of the entire rod. Initially, the rod is at temperature U. Formulate the initial-boundary value problem for the temperature in the rod. Is it a linear problem?

The solution is

I don't understand how this part (see below) is obtained.