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Math Help - Nonlinear ODE

  1. #1
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    Nonlinear ODE

    Here is the equation \dfrac{dy}{dx}\dfrac{d^2 y}{dx^2}=y\dfrac{d^3 y}{dx^3}

    Any ideas?
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  2. #2
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    Two thoughts:

    1. Any affine function y = mx + b solves the equation trivially.
    2. You could try this:

    y'\,y''=y\,y''', implies

    \displaystyle{\frac{y'}{y}=\frac{y'''}{y''}}.

    You can integrate then, to obtain

    \ln|y|=\ln|y''|.

    You can go from there further, I think.
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  3. #3
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    There's also a constant of integration to think about.
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  4. #4
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    Indeed. As always. So you've really got

    \ln|y|=\ln|y''|+C_{1} so far.
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  5. #5
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    Might be a little easier to write ( y \ne 0)

    y y''' - y' y'' = 0 as \dfrac{d}{dx} \left(\dfrac{y''}{y}\right) = 0

    so y'' = c_1 y then look at cases when

    (i) c_1 = \omega^2,
    (ii) c_1 = 0,
    (iii) c_1 = - \omega^2.
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