# I'm not sure how to transform this into two ODEs

1. Mar 26, 2009

### josh146

Wave equation with inhomogeneous boundary conditions

Sorry about the thread title, I've tried changing it but it won't work.

1. The problem statement, all variables and given/known data

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.

2. Relevant equations
(1) $\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}$

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

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

3. The attempt at a solution

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

The PDE is:

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

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

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

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

Last edited: Mar 26, 2009