Looks good so far. Now expand x as a cosine series. Since cosine is even, you will really need to find the Fourier series for |x| on [-1, 1].
THis is my first of this type.
given and .
Using separation of variables with u(x,t)=F(x)G(t)
General solution for and
For non trivial solution let where a>0. Let therefore
The general solution for the BC's is
How am I doing so far?
With what I said above we take take derivatives inside the sum this gives
Now if we take one more derivative with respect to x we get
Now if you do the same thing with respect to t we get
Now if you plug these into the ode you get
The key idea is we need to exchange limits eg bring the derivative inside the infinite sum.
We actually need more than just uniform convergence of the function, they must also be continuous and their derivatives must be uniformly convergent.