# PDE and more boundary conditions

#### Markov

##### Member
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

##### Well-known member
MHB Math Helper
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?

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#### Markov

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