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Solving a Wave Equation

  1. Oct 30, 2013 #1
    The problem statement, all variables and given/known data

    2nrq1x2.png


    The attempt at a solution

    I'm using the method of separation of variables by first defining the solution as [itex]u(x,t) =X(x)T(t)[/itex]

    Putting this back into the PDE I get: [itex]T''X = x^{2}X''T + xX'T[/itex]

    which is simplified to [tex]\frac{T''}{T} = \frac{x^{2}X'' + xX'}{X} = -\lambda^{2}[/tex]

    The spatial problem is then: [itex]x^{2}X'' + xX' = -X\lambda^{2}[/itex]

    Is this correct so far? How do I continue?
     
  2. jcsd
  3. Oct 31, 2013 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Yes, that is correct. You proceed, of course, by solving that equation, together with the boundary conditions X(1)= 0, X(e)= 0.

    (Those boundary conditions follow from u(t, 1)= T(t)X(1)= 0 and u(t, e)= T(t)X(e)= 0. If T(t) is not identically 0, which would not satisfy the initial conditions, then X(1)= X(e)= 0.)

    (An obvious solution is the trivial X(x)= 0. But then you could not satisfy the initial conditions. [itex]\lambda[/itex] must be such that the equation has non-trivial solutions- i.e. eigenvalues.)
     
  4. Oct 31, 2013 #3

    pasmith

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    Homework Helper

    Looking at the boundary conditions, the substitution [itex]X(x) = Z(\log (x))[/itex] looks helpful.
     
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