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

#### josh146

Wave equation with inhomogeneous boundary conditions

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

1. Homework Statement

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. Homework 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:
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