Heat Equation Time dependant heat loss

PAR

1. The problem statement, all variables and given/known data
Solve the heat equation: [tex]u_{t} = u_{xx} - u - x*e^{-t}[/tex]
BC: u(0,t) = 0, u(1,t) = 0
IC: u(x,0) = x


2. Relevant equations



3. The attempt at a solution

The only progress I've made so far is figuring out that the steady state solution is zero. Other than that I don't know where to start with the time dependent solution. Basically I could use some help starting this problem, thanks!

gabbagabbahey

Try finding functions [itex]f(t)[/itex] and [itex]g(x)[/itex] such that [itex]w(x,t) \equiv f(t)u(x,t)+g(x)[/itex] satisfies the homogeneous heat equation [itex]w_t=w_{xx}[/itex]...then use your boundary conditions for u(x,t) to find corresponding BCs for w(x,t) and solve for w(x,t) and use that to find u(x,t).