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Math Help - Nonlinear second order differential equation

  1. #1
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    Nonlinear second order differential equation

    What is the solution of the follwoing differential equation

    \frac{\partial^{2}y}{\partial x^{2}}-ay^{-1}\frac{dy}{dx}=0

    where a is a constant.
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Nonlinear second order differential equation

    Ok, This method works for all equations of the form  y'' = f(y', y) .
    Now Assume a new variable, v = y'. So  \frac{\mathrm{d}v}{\mathrm{d}x} = \frac{\mathrm{d}v}{\mathrm{d}y} \frac{\mathrm{d}y}{\mathrm{d}x}

    Since \frac{\mathrm{d}y}{\mathrm{d}x}} = v This now becomes  = \frac{\mathrm{d}v}{\mathrm{d}x} = \frac{\mathrm{d}v}{\mathrm{d}y} v
    Now you have a Differential equation of the first order,  v \frac{\mathrm{d}v}{\mathrm{d}y} = g(v, y) . For your specific equations turns into a seperable ODE,
     v \frac{\mathrm{d}v}{\mathrm{d}y} = \frac{a}{y} * v which turns into

     \mathrm{d}v} = \frac{a}{y} \mathrm{d}y

    So v =  v = ln(y) + k_0

    Now again Since  v = \frac{\mathrm{d}y}{\mathrm{d}x} which means  \frac{1}{ln(y)+k_0} \mathrm{d}y = \mathrm{d}x
    and you get some super ugly integral.
    Last edited by jakncoke; December 22nd 2012 at 08:54 AM.
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  3. #3
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    Re: Nonlinear second order differential equation

    Thank you
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  4. #4
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    Re: Nonlinear second order differential equation

    Hi !

    The obvious solutions are y(x)=constant.
    The other solutions cannot be expressed as a combination of a finit number of elementary functions.
    They involves some special functions such as li(x) or Ei(x) :
    Logarithmic Integral -- from Wolfram MathWorld
    Exponential Integral -- from Wolfram MathWorld
    About special functions, pp.18-36 of the paper "Safari on the contry of special Functions" :
    JJacquelin's Documents | Scribd
    Attached Thumbnails Attached Thumbnails Nonlinear second order differential equation-edo2.jpg  
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  5. #5
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    Re: Nonlinear second order differential equation

    Can someone please confirm that the above solution is OK? I tried a lot but I could not find the result. Is there any other way to solve this problem? I could not find any divisor for R[x].


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