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PDE: Wave equation with first order derivative

  1. Aug 6, 2013 #1
    1. The problem statement, all variables and given/known data

    Solve using separation of variables utt = uxx+aux
    u(0,t)=u(1,t)=0
    u(x,0)=f(x)
    ut=g(x)



    3. The attempt at a solution
    if not for the ux I'd set
    U=XT
    such that X''T=TX'' and using initial conditions get a solution.

    In my case I get T''X=T(aX'+X'') which is still solvable but highly unpleasant. The question comes from an old exam and I wonder if their is something I'm not seeing?

    Thanks!!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 6, 2013 #2

    pasmith

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    Consider the substitution [itex]u(x,t) = e^{\beta x}v(x,t)[/itex], for a suitable value of [itex]\beta[/itex].

    EDIT: Forget that, it doesn't simplify as I thought it would. But if you separate the variables as is, you end up with a perfectly normal 2nd order linear ODE with constant coefficients for [itex]X(x)[/itex].
     
    Last edited: Aug 6, 2013
  4. Aug 6, 2013 #3

    haruspex

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    Is it stated anywhere that the solution will be pleasant?
    Post your subsequent working down to the point where you find it suspiciously messy.
     
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