# Solving the heat equation using Laplace Transforms

I am trying to solve the 1-D heat equation using Laplace Transform theory. The equation is as follows. I don’t have the capability to write the symbols so I will write it out.

``````                     partial u/partial t = 2(partial squared u/ partial x squared) -x
boundary conditions are partial u/partial x(0,t)=1, partial u/partial x(2,t)=beta.
``````

The problem asks the following:
(a). For what value of beta does there exist a steady-state solution?
(b). if the initial temperature is uniform such that u(x,0)=5 and beta takes the value suggested by the answer to part (a), derive the equilibrium temperature distribution.

I was able to get an equation that looks like U(x,s)=c e^(s/2)^1/2 -(1/s)((x/s)-u(x,0)). But I am not sure how to go from here to solve for beta using the boundary conditions. I need some assistance from someone.