The only thing you need to know here is that if you are solving an equation with non-homogenous boundary conditions and you seperate it into a sum of two solutions one of which is a steady-state solution then the full solution to the original problem would be the sum of the steady-state function and the solution to the equation with homogenous boundary conditions. In this case the steady-state function (similar to first part) is

. Then you need to solve the equation for

. Where

with

and

. But this is a homogoneous heat equation. Its solutions is

. And the

are determined by integral forumals while solving this problem.