Heat Equation Time dependant heat loss

[PAR]

Homework Statement

Solve the heat equation: $$u_{t} = u_{xx} - u - x*e^{-t}$$
BC: u(0,t) = 0, u(1,t) = 0
IC: u(x,0) = x

Homework Equations

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 $f(t)$ and $g(x)$ such that $w(x,t) \equiv f(t)u(x,t)+g(x)$ satisfies the homogeneous heat equation $w_t=w_{xx}$...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).