Question: Solve the wave equation (1) on the region 0<x<2 subject to the boundary conditions (2) and the initial condition (3) by separation of variables.

(1)

(2) ;

(3)

I've defined where u_st is the steady state solution (the solution to subject to (2)). I've used this to create a new PDE with homogeneous boundary conditions.

The PDE is:

.

By subbing in I get:

I'm not sure how to separate this into two ODEs. Can someone help?