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A Help solving non-linear ODE analytically

  1. Sep 8, 2016 #1

    joshmccraney

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    Hi PF!

    Anyone have any ideas for a solution to this $$0 = F F''+\left.2F'\right.^2+ xF' + F$$ where primes denote derivatives with respect to ##x##.
    So far I have tried this $$0=\left( F F'\right)'+\left({xF}\right)'+\left.F'\right.^2$$

    Which obviously failed. I also thought of this $$0 = F^2 F''+2F\left.F'\right.^2+ xFF' + F^2\\
    = (F'F^2)' + (xF^2)'+xFF'$$
    which also fails. Any ideas? I know an analytic solution exists, but how to derive it?
     
  2. jcsd
  3. Sep 9, 2016 #2

    Stephen Tashi

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    F = -x is a particular solution. Look up how to do the "reduction of order" of a differential equation. (I say look it up, because I'd have to look it up myself before attempting to explain it.)
     
  4. Sep 9, 2016 #3

    joshmccraney

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    That's an idea, which is all I'm asking for. But ##-x## doesn't solve this. Comes close though.
     
  5. Sep 9, 2016 #4

    pasmith

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    If [itex]F(x) = kx^2[/itex] then every term on the right is a constant times [itex]x[/itex]. You can then choose [itex]k[/itex] so that those constants sum to zero.
     
  6. Sep 9, 2016 #5

    joshmccraney

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    Thanks! I do know the exact solution is ##3(x^{1/3}-x^2)/10## but was wondering how to find this solution apart from guessing.
     
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