PDE — lost on this separation of variables problem

[fahraynk]

<< Mentor Note -- thread moved from the technical math forums >>

I am getting stuck on this partial differential equation.
U_t = U_xx - U + x ; 0<x<1
U(0,t) = 0
U(1,t) = 1
U(x,0) = 0

Here is my work so far :
U = e^-tw + x
gives the new eq w_t=w_xx
to get rid of boundary conditions :
w=x+W
W_t=W_xx
W(0,t) = 0
W(1,t)=0
W(x,0)=-x

W=X(x)T(t)
Plug that in, and I get these :
T'=μT
X''=μx

w = e^-(nπ)2t[a_nsin(nπx)]
a_n = -2∫xsin(nπx) = 2cos(nπ)/nπ
w = x + W
w = x +(2/π)Σ(1/n)cos(nπ)sin(nπx)e^-(nπ)2t
u = e^-tw + x
u = x + e^-t(x +(2/π)Σ(1/n)cos(nπ)sin(nπx)e^-(nπ)2t)

But the books answer is :
u(x,t) = x - (2/π)e^-t* [ e^-π2tsin(πx) - (1/2) e^-2π2tsin(2πx)+...]

What am I doing wrong?

 

[fahraynk]

I figured out the answer. I was not plugging U = we^-t+x into the boundary conditions properly. I did not need the extra substitution. Cant find a delete button for the thread though !