[Solved] PDE problem - Separation of variables - Dirichlet condition

Hi all

I'm in quite some trouble with this PDE problem, that is part of my assignment for wednesday.

First of all I got the PDE problem:

**PDE: u_t - 2*u_xx = 0 , 0<x<1, t>0**

BC: u(0,t) = t, u(1,t)=0

IC: u(x,0) = 0

I then have to show, that **v(x,t)=u(x,t)-t(1-x)** satisfies

**PDE: v_t - 2*v_xx = -(1-x) , 0<x<1, t>0**

BC: v(0,t) = 0, v(1,t)=0

IC: u(x,0) = 0

That is fairly easy.

Then I have to find a solution for X(x) to the ODE problem:

ODE: -2*X''(x) = -(1-x)

BC: X(0) = X(1) = 0

Again, this is simple, and my result is:

**X(x) = x^2/4 - x^3/12 -x/6**

But then i get in some trouble!

First I must derive a PDE problem for w(x,t)=v(x,t)-X(x), and then solve it. And after that, I have to find u(x,t)

To derive the PDE problem, it is just like I did before, and I end up with:

**PDE: w_t - 2*w_xx = 0**

BC: w(0,t) = 0, w(1,t) = 0

IC: w(x,0) + X(x) = 0

But how do I solve this? I know, that I should know, but I'm confused, and missed class when this was explaned.

I'm in desperate need of help.

Thanks!