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Math Help - Solving a wave equation with inhomogeneous boundary conditions

  1. #1
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    Solving a wave equation with inhomogeneous boundary conditions

    Question: 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.

    (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

    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 (the solution to \frac{\partial^2 u}{\partial x^2} = 0 subject to (2)). 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 separate this into two ODEs. Can someone help?
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  2. #2
    Super Member Rebesques's Avatar
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    Errrr... That trick you tried does not apply here. For the method of separation of variables, set u(x,t)=X(x)T(t) to construct two ODEs, then add up all the solutions. It must be in your notes.
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  3. #3
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    Thanks but I already realised my mistake:

    h is a constant, not a function of t.
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