Consider the following non-homogenous heat equation on $\displaystyle 0 \leq x \leq \pi$

$\displaystyle u_t = k u_{xx} - 1$ with $\displaystyle u(x,0) = 0, u(0,t) = 0, u(\pi, t) = 0$

Find a solution of the form

$\displaystyle \displaystyle \sum_1^{\infty} b_n(t) \phi_n (x)$

where $\displaystyle \phi_n(x)$ are the eigenfunctions of an appropriate homogenous problem, and find explicit expressions for $\displaystyle b_n(t)$