• Support PF! Buy your school textbooks, materials and every day products via PF Here!

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

  • Thread starter josh146
  • Start date
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) [itex]\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}[/itex]

(2) [itex]\frac{\partial u}{\partial x}(0,t)=1[/itex] ; [itex]\frac{\partial u }{\partial x}(2,t)=1[/itex]

(3) [itex]\frac{\partial u}{\partial t}(x,0)=0[/itex]

3. The Attempt at a Solution

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

The PDE is:

[itex]\frac{\partial^2 \theta}{\partial t^2} + h''(t)=c^2 \frac{\partial^2 \theta}{\partial x^2}[/itex].

By subbing in [itex]\theta=f(t)g(x)[/itex] I get:

[itex]f''(t)g(x)+h''(t)=c^2 f(t) g''(x)[/itex]

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

Want to reply to this thread?

"I'm not sure how to transform this into two ODEs" You must log in or register to reply here.

Related Threads for: I'm not sure how to transform this into two ODEs

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving