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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
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