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I am interested in the solution to the following 1-parameter family of 2nd order ODE: $$(\lambda (xH' - H)^3 + 4(H')^2 (1-H') x^2)H'' - 4 H'(1-H')(xH'-H)(x+H')=0,$$

with $H(0) = 1$ and $H(1) = 0$. $\lambda < 0$ is the parameter.

I am interested in either analytic solution or proof that no analytic solutions exists. Uniqueness of solutions is also appreciated. Thanks!

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  • $\begingroup$ Have you tried simplifying it? $\endgroup$ – mathreadler May 10 '17 at 20:05
  • $\begingroup$ I have tried solving it using mathematica. $\endgroup$ – John Jiang May 10 '17 at 21:18
  • $\begingroup$ Some things which seem to be recurring all over the place: $(1-H')$, $(x+H')$, $(xH'-H)$, maybe we can find some function to substitute which makes all of those simpler. $\endgroup$ – mathreadler May 10 '17 at 21:29
  • $\begingroup$ Then you can break down into an equation system with writing the new functions in relations to each other. Also what do you mean by "analytic" solution? Expressibility in some specific class of functions? $\endgroup$ – mathreadler May 10 '17 at 21:33
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    $\begingroup$ Maybe you can stress the uniqueness by adding some cost term / constraint to the numerical solver. For example trying to add $H'(0.5) = \lambda$ and see if it manages to find different solutions for some different $\lambda$. You can start with sampling $\lambda$ from the solution found and doing small displacements of it : $\pm \epsilon$. $\endgroup$ – mathreadler May 10 '17 at 22:36

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