Solve Wave Equation using Separation of Variables

Question

Use separation of variables to solve the wave equation:

$$u_{tt}=a^2u_{xx},$$

on the domain $0<x<1$,$t>0$, with initial conditions

$$u(x,0)=0$$

$$u_t(x,0)=\cos(\pi x)$$

and boundary conditions

$$u_x(0,t)=u_x(1,t)=0$$

So I have rewrote this as

$$T''X=a^2X''T$$

then I set

$$T''/a^2T=X'/X=-\mu$$

Then I got loss.

Answer 1

Hint: Just solve the equation using the standard separate of variable approach with initial condition as a sinusoidal function. Be sure you can generalize this approach to dealing with any other initial conditions using Fourier series.