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) $\displaystyle \frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}$

(2) $\displaystyle \frac{\partial u}{\partial x}(0,t)=1$ ; $\displaystyle \frac{\partial u }{\partial x}(2,t)=1$

(3) $\displaystyle \frac{\partial u}{\partial t}(x,0)=0$

I've defined $\displaystyle \theta(x,t)=u(x,t)-u_{st}(x) = u(x,t)-x-h(t)$ where u_st is the steady state solution (the solution to $\displaystyle \frac{\partial^2 u}{\partial x^2} = 0$ subject to (2)). I've used this to create a new PDE with homogeneous boundary conditions.

The PDE is:

$\displaystyle \frac{\partial^2 \theta}{\partial t^2} + h''(t)=c^2 \frac{\partial^2 \theta}{\partial x^2}$.

By subbing in $\displaystyle \theta=f(t)g(x)$ I get:

$\displaystyle f''(t)g(x)+h''(t)=c^2 f(t) g''(x)$

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