Using separation of variables, construct the general solution of the wave equation $$u_{tt}=u_{xx},$$ for $0<x<1, \ t>0$ with boundary conditions $u(0,t)=0$ and $u_x(1,t)=0$ for $t>0$ and initial condition $u(x,0)=0.$
My attempt:
We assume the solution takes the form $u(x,t)=X(x)T(t).$ Hence, $$\frac{T''}{T}=\frac{X''}{X}=\lambda \ \ \ (\text{$\lambda$ is a separation constant}).$$ First consider $X''-\lambda X=0$. The cases where $\lambda=0$ and $\lambda>0$ lead to the trivial solution. I am having trouble with the case where $\lambda<0$. In this case, $$X(x)=A\cos(wx)+B\sin(wx) \ \ A,B\in\mathbb{R}.$$ The boundary condition $$u(0,t)=0\implies X(0)=0\implies A=0.$$
My question is, how do I deal with $u_x(1,t)=0$? I know that this implies $X'(1)=0$, so does this mean $Bw\cos(w)=0?$ If so, $B\neq 0$ and $w\neq 0$ (as the solution would be trivial). So does this mean $$w=\frac{\pi}{2}+n\pi, \ n=0,1,2,3...?$$
Thank you in advance!