# Solving the wave equation with Neumann conditions

I have some difficulties solving this Wave Equation under Neumann BC. Here is what I have so far.

$u_{tt} = 4u_{xx}$ for $0<x<\pi, t>0$

$u_x(0,t) = u_x(\pi$, t) = 0 for t>0

$u(x,0) = 0, u_t(x,0) = \sin(x), 0 <= x <= \pi$

I used the separation of variables method to solve this problem.

$u(x,t) = X(x)T(t)$

$\frac{-X''(x)}{X(x)} = \frac{-1}{4}\frac{T''(t)}{T(t)} = \lambda$

BC: $X'(0) = 0$ and $X'(\pi) = 0$

IC: $T(0) = 0$ and $T'(0) = \sin (x)$

I get $u(x,t) = \frac{1}{2}A_0t + \frac{1}{2}B_0$ + $\sum_{n=1}^\infty(A_n \cos (4nt) + B_n \sin (4nt))\cos(nx)$

I think everything here so far is correct.

Here is where I am stuck: