MHB PDE and more boundary conditions

[Markov2]

Solve

$\begin{aligned} & {{u}_{tt}}={{u}_{xx}}+1+x,\text{ }0<x<1,\text{ }t>0 \\
& u(x,0)=\frac{1}{6}{{x}^{3}}-\frac{1}{2}{{x}^{2}}+\frac{1}{3},\text{ }{{u}_{t}}(x,0)=0,\text{ }0<x<1, \\
& {{u}_{x}}(0,t)=0=u(1,t),\text{ }t>0.
\end{aligned}
$

Here's something new for me, the boundary condition $u_x.$ I've always seen the $u_t$ condition, but what to do in this case?

 

[HallsofIvy]

Try a "Fourier series" solution of the form
$\sum_{n=0}^\infty A_n(t)cos(n\frac{\pi}{2}t)$
Do you see why that will work?

 

[Markov2]

Not actually. I thought this can be solved by using another function, etc, don't know how to make it yet. :(