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Second order nonlinear ODE - not autonomous

  1. May 6, 2010 #1
    second order nonlinear ODE -- not autonomous

    This equation has arisen from a steady state problem of diffusion with nonlinear reaction:

    (dependent var=c; independent variable=x)

    c'' = ko + k1 c + k2 c^2 + k3 c x

    ko, etc are constants.

    I can obtain a solution if I drop the last term which involves the product of the dependent and independent variable. Is there any way to get a solution with this term?
  2. jcsd
  3. May 7, 2010 #2
    Re: second order nonlinear ODE -- not autonomous

    Hello Prof. Mind if I write it as:

    [itex]y''-k_0-(k_1+k_3 x)y-k_2 y^2=0[/itex]

    and if you wish an analytical expression for the solution, I suggest two options unless someone comes up with a better way:

    (1) Compute an IVP numerically and fit the data to a high-degree polynomial within your range of acceptable error.

    (2) Compute the power series [itex]y=\sum_{n=0}^{\infty} a_n x^n[/itex]

    I realize there is a square in the power series approach but I would just take the Cauchy product. Would need to figure out the convergence radius of course which I suspect is no more than one. I recommend this approach for the following reasons: It's not in the Handbook of DEs, not in another book on non-linear ones, and Mathematica's DSolve can't solve it. Of course, that doesn't mean a nice way doesn't exists but rather I'm just out of bullets. :)
    Last edited: May 7, 2010
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