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Math Help - Separation of variables

  1. #1
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    Separation of variables

    Hi, I have this problem, but I am stuck in one thing.

    Assuming that solutions to y_{tt}=g(zy_z)_z are of the form y(z,t)=Z(z)T(t), and that the separation constant is negative, written as -v^2 with v>0, I need to show that Z(z) and T(t) must satisfy the differential equations

    T''+v^2gT=0, and
    zZ''+Z'+v^2Z=0

    So well, I kind know what to do once I have substituted y(z,t) into y_{tt}=g(zy_z)_z... However, I keep on getting the wrong results:

    I get this:

    Z(z)T''(t)=g(Z(z)Z'(z)T(t))_z
    Z(z)T''(t)=g_z(T(t)(Z'(z)Z'(z)+Z(z)Z''(z)))

    But this is clearly not what should be on both sides :/

    I would really appreciate it if any of you could give me a hand. Once I get both results on each side, I will be able to separate the variables.

    Thanks a lot!
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  2. #2
    MHF Contributor
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    Re: Separation of variables

    zy_z \neq Z(z)Z'(z)T(t) unless Z(z) = z.

    I think you should have:

    \begin{align*}Z(z)T''(t) & = g\left(zZ'(z)T(t)\right)_z \\ & = g'\left(zZ'(z)T(t)\right) \left[zZ''(t)T(t) + Z'(t)T(t)\right]\end{align*}

    Also, I am assuming that g is a function. If it is a constant, that changes things significantly.
    Last edited by SlipEternal; December 1st 2013 at 03:03 PM.
    Thanks from mathstudent1992
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