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Thread: Nonlinear ODE

  1. #1
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    Nonlinear ODE

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

    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:

    $\displaystyle y'\,y''=y\,y'''$, implies

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

    You can integrate then, to obtain

    $\displaystyle \ln|y|=\ln|y''|.$

    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

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

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

    so $\displaystyle y'' = c_1 y$ then look at cases when

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