Solve,

$\displaystyle \frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}=F(x,t)$

$\displaystyle u(0,t)=0, \frac{\partial u}{\partial x}(\pi,t )=0, u(x,0)=0$

where $\displaystyle F(x,t)$ is a given function.

(a) Determine the x-dependent eigenfunctions appropriate to the solution of the equation with given boundary conditions.

(b) Solve the method of eigenfunction expansions to obtain a series solution for the the problem in terms of the eigenfunctions determined in (a)

(c) Simplify for the particular case $\displaystyle F(x,t)=1$

Any help appreciated. Thanks