1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Mar 26, 2009 #1
    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) [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: Mar 26, 2009
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted