 # I am curious if there are any examples of functions that are solutions to second order differential Elle Weber 2022-05-13 Answered
I am curious if there are any examples of functions that are solutions to second order differential equations, that also parametrize an algebraic curve.
I am aware that the Weierstrass $\mathrm{\wp }$ - Elliptic function satisfies a differential equation. We can then interpret this Differential Equation as an algebraic equation, with solutions found on elliptic curves. However this differential equations is of the first order.
So, are there periodic function(s) $F\left(x\right)$ that satisfy a second order differential equation, such that we can say these parametrize an algebraic curve?
Could a Bessel function be one such solution?
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The functions $\mathrm{sin}\left(x\right)$ and $\mathrm{cos}\left(x\right)$ solve a second-order differential equation, namely
${u}^{″}=-u,$
and $\left(\mathrm{cos}\left(x\right),\mathrm{sin}\left(x\right)\right)$ parametrizes the algebraic curve ${x}^{2}+{y}^{2}=1$

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