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DiffEq question (solving damped wave equation)

  1. Mar 5, 2009 #1
    How do you use separation of variables to solve the damped wave equation
    y_tt + 2y_t = y_xx

    where y(0,t) = y(pi,t) = 0
    y(x,0) = f(x)
    y_t (x,0) = 0

    ---
    These are partial derivatives where y = X(x)T(t)

    So rewriting the equation I get

    X(x)T''(t) + 2X(x)T'(t) = X''(x)T(t)
    which results in the following differential equations (lambda a constant)

    X''(x) + lambda*X(x) = 0..........(1)
    T''(t) + 2T'(t) + lambda*T(t) = 0.......(2)

    I am supposed to use the boundary conditions somehow and find y(x,t) which will be a Fourier series but I am completely stuck and would appreciate some help.
     
  2. jcsd
  3. Mar 5, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You are told that y(0, t)= X(0)T(t)= 0 and that y(pi, t)= X(pi)T(t)= 0. In order that those be true for all t, you must have X(0)= 0 and X(pi)= 0.

    Start by solving the equation X"+ lambda X= 0, X(0)= 0, X(pi)= 0.

    The "type" of soltutions will (1) linear if lambda= 0, (2) exponential if lambda< 0, (3) sine and cosine if lambda> 0. But you know that X(0)= X(pi)= 0 so lambda must be what?
     
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