So I'm not sure how to start on this problem.

$\displaystyle u_{t} = u_{xx} +1$

$\displaystyle u(0,t) = 1 ; u_{x}(1,t) = 0$

$\displaystyle u(x,0)=1$

$\displaystyle t>0; 0<x<1$

Where im having my difficulty is finding the steady state solution.

$\displaystyle v''(x) = -\frac{1}{\beta}P(x)$ with P(x) =1

Where do I go from here?