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I am currently dealing with the following ODE as a stationary, special case version of a PDE model derived from Kuramoto-Sivashinsky.

$$ y'' y' = ay $$

Where $a$ is a real (constant) parameter.

I am going through different handbooks (even asked chatgpt haha) to try and classify this equation or even (best case scenario) find some insight on analytical solutions/solution properties. However, this search has not provided much yet and I'm currently working on the linearization.

Has anyone here stumbled across a similar ODE before? Any help would be greatly appreciated!

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    $\begingroup$ Multiply by $y'$ to get $$\frac{1}{3} (y'^{3})' = \frac{1}{2} a (y^{2})'$$ then integrate, then separate and integrate again. It might not be a nice integral though. $\endgroup$
    – Matthew Cassell
    Commented Apr 29 at 11:46
  • $\begingroup$ Yup, tried subbing m = y'(x) and then integrating once more but mathematica spit out a crazy special function solution. I guess that's just the way the solution is. I will give your idea a try in a moment as well. Thanks! $\endgroup$
    – Vasil
    Commented Apr 29 at 11:47
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    $\begingroup$ Ah, leads to the same spot with the sub :) Thanks again Matthew $\endgroup$
    – Vasil
    Commented Apr 29 at 12:18

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This type is classified as missing $x$. Let $p=y'=\frac{dy}{dx}$ and then $$ y''=p'=\frac{dp}{dx}=\frac{dp}{dy}\frac{dy}{dx}=\frac{dp}{dy}p. $$ So the equation becomes $$ \frac{dp}{dy}p^2=ay $$ or $$ p^2dp=aydy. $$ So $$ \frac13p^3=\frac12ay^2+C_1$$ and hence $$ (y')^3=\frac32(y^2+C_1)$$ or $$ y'=\sqrt[3]{\frac32}\sqrt[3]{y^2+C_1}$$ So $$ \frac{dy}{\sqrt[3]{y^2+C_1}}=\sqrt[3]{\frac32}dx$$ Integrating gives the solution $$ x=\sqrt[3]{\frac23}\int\frac{dy}{\sqrt[3]{y^2+C_1}}+C_2. $$

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  • $\begingroup$ Thanks for the in depth answer! I was getting stuck at the final integral and mathematica is spitting some special function mess. I guess it's just not a "nice" integral. Marking this as the final answer (and credit to Matthew Cassell for his initial answer as well) $\endgroup$
    – Vasil
    Commented May 4 at 12:33

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