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I'm writing something about the double pendulum. I want to answer why the differential equation, which describes the movement of the double pendulum is analytically unsolvable. Is there a general theorem that describes which differential equations are solvable/unsolvable? Or are we just too dumb to find an analytical solution? Or is there a proof for some special cases?

Thanks for the answers. Remarks on where I can look the answers up for myselfe are also very helpful.

Arctic Char
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Leo Horber
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  • Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Jul 11 '24 at 15:50
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    @LeoHorber: https://math.stackexchange.com/questions/3782499/is-there-a-reason-it-is-so-rare-we-can-solve-differential-equations – Moo Jul 11 '24 at 15:50
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    Pretty much everything is unsolvable analytically really. Classroom examples of solvable systems are the exception rather than the rule. – Paul Jul 11 '24 at 16:37

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The simple pendulum is also unsolvable analytically, unless you allow an Elliptic Integral, a special function.

This is in direct relation to Liouville's theorem on analytical integration. ODEs are as difficult as straight antiderivation.

https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)

Ponder $y''+2xy'=0$ or $\log(x)\,y'+y=1$.