Solving a PDE with Non-homogenous Boundary Conditions

[kgal]

Homework Statement

If u_tt - u_xx= 1-x for 0<x<1, t>0
u(x,0) = x²(1-x) for 0≤x≤1
u_t(x,)=0 for 0≤x≤1
u_x(x,)=0
u(1,t)=0

find u(1/4,2)

Homework Equations

The Attempt at a Solution

I was thinking to make a judicious change of variables that not only converts the PDE to a homogenous PDE, but also makes the boundary conditions homogenous.
I am quite unsure how to even start this problem...

 

[kostas230]

Consider that the solution has the form: u(x,t)=v(x,t)+w(x). Can you find the boundary and initial conditions to make this problem simpler?