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Nota Bene: These remarks are provided by Robert Lewis on Wednesday 15 July 2015 6:50 PM PST in effort to provide context and motivation for this engaging question. Our OP originally asked:

"How to prove that if $f$ is a non-trivial solution of the non-linear differential equation $x^{''}+\sin x=0$, then $f$ is periodic?"

Which cannot be done, since the equation has non-trivial, non-periodic solutions (see below), although most of the solutions are in fact periodic. Therefore we need to broaden the scope of inquiry as indicated in the (now edited) title. Essentially what we now seek is an answer to something more along the lines of

How may we rigorously describe the non-trivial solutions to

$$x'' + \sin x = 0;$$

are any or all of them periodic?

This, I think, captures much of the spirit of the original question whilst expanding the scope so that an answer is possible.

I have probed, though not to excessive depth, the question data base here and did not find this exact question; hopefully, it's not a duplicate. It is, to my mind, a good question since it illustrates, in a fundamental way, some essential basic principles of analyzing non-linear ODEs.

These things being said, let me add that attempts based upon linearization or power series analysis might prove problematic, since the questions asked are global in nature.

With these modifications and caveats on the table, I bid you have at it, Ladies and Gentlemen of Math Stack Exchange.

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    $\begingroup$ Welcome to MSE! Users are more likely to offer aid if you provide evidence that you have made an attempt in good faith to solve the problem by yourself. In addition, we can offer more targeted aid if you specify previous attempts you have made, as then we may be able to identify where you are struggling. For more on how to ask questions on this site, please visit math.stackexchange.com/help/asking. $\endgroup$ – Gyu Eun Lee Jul 15 '15 at 20:01
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    $\begingroup$ @zed111 Presumably he means $x(t)$ $\endgroup$ – user223391 Jul 15 '15 at 20:30
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    $\begingroup$ the main reason is a (total) energy conservation law $\endgroup$ – Will Jagy Jul 15 '15 at 20:41
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    $\begingroup$ Actually, it's not true. The title question, that is. The equation has non-periodic, heteroclinic trajectories "joining" phase points of the form $(x, x') = (n \pi, 0)$ with those of form $(x, x') = ((n \pm 1) \pi, 0)$ for all $n \in \Bbb Z$. For further explication, see my edits to the question and (hopefully) forthcoming answer. Cheers! $\endgroup$ – Robert Lewis Jul 15 '15 at 20:59
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    $\begingroup$ Please Note: My previous comment should have said that the heteroclinic trajectories join points of the form $((2n + 1) \pi, 0)$ with those of the form $(2(n + 3) \pi, 0)$ or $((2n -1)\pi, 0)$; the points $(2n\pi, 0)$ are centers. Sorry for any confusion. Cheers! $\endgroup$ – Robert Lewis Jul 16 '15 at 3:34
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Multiplying by $x' $ we get $$x'' x' +x' \sin x =0 $$ hence $$\left(\frac{1}{2} x'^2 \right)' -(\cos x )' =0$$ thus $$x'^2 -2\cos x =c$$ hence $$\pm\int \frac{dx}{\sqrt{2\cos x +c} }=t+c_1 $$ substituting $u =\cos x , $ $dx =\frac{- du}{\sqrt{1-u^2}}$ we get $$\pm \int \frac{du}{\sqrt{(1-u^2)(2u +c)}} =t+c_1 $$

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  • $\begingroup$ Beautiful elliptic integral :) $\endgroup$ – A.Γ. Jul 15 '15 at 21:09
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    $\begingroup$ And this answers the question how? Generally, the point of exercises "prove property X of solutions of this ODE" is not to solve the ODE and then look at property X. $\endgroup$ – user147263 Jul 15 '15 at 22:08

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