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Non-linear second order equation

  1. Aug 14, 2009 #1
    I am not able to find the general integral of the following non-linear 2nd order equation:

    y^2 y'' + a y^3 - b = 0

    in which:

    y = f(x)

    0 < a <= 1, is a constant

    b > 0 , is a constant.
     
  2. jcsd
  3. Aug 14, 2009 #2
    Hello Leo Klem,

    Not an easy one. The only method I know that can be used to solve this equation is Lie point transformations. By setting:

    [tex]u=y[/tex]
    [tex]v=y'[/tex]
    and
    [tex]w=\frac{dv}{du}=\frac{y''}{y'}[/tex]

    You can transform the equation into:

    [tex]v\frac{dv}{du}=\frac{b}{u^2}-au[/tex]

    Which has as solution:

    [tex]v^2=-\frac{2b}{u}-au^2+K_1[/tex]

    Taking the root and inverting the substitution you end up with:

    [tex]\pm x+K_2=\int \frac{\sqrt{y}dy}{\sqrt{K_1y-2b-ay^3}}[/tex]

    Which is not an easy integral. Maybe by trying to find the roots you can solve it, I did not try this. (check the calculations for errors to be sure)

    Hope this helps so far,

    coomast
     
  4. Aug 15, 2009 #3
    It's an elliptic integral, not elementary.
    Maple evaluates it in terms of the elliptic integrals [itex]F[/itex] and [itex]\Pi[/itex], but too complicated to copy here.
     
  5. Aug 16, 2009 #4
    16 August 2009

    Many thanks for the attention paid both by "coomast" and by "g_edgar".
    The solution for x given by "ccomast" is just the point where I too had to stop. Unfortunately, I am not in condition to calculate the integral indicated.
    Leo Klem
     
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