$u''+\cosh u=0$ or $u''+\sinh u=0$

Question

I would like to know a physical or chemical background of the ODE of the second order $u''(x)+\cosh u(x)=0$ or $u''(x)+\sinh u(x)=0$.

These two ODEs have beautiful structures. Hence, I am wondering whether these ODEs have some backgrounds or not.

In the case of $u''+u=0$ it appears in an oscillatory behavior of a spring motion.

Answer 1

$$u''(x)+\cosh (u(x))=0$$ switch variables and write it $$\frac{x''(u) }{[x'(u)]^3}=\cosh(u)$$ Use reduction of order $p(u)=x'(u)$ $$\frac{p'(u) }{[p(u)]^3}=\cosh(u)\quad\implies\quad p(u)=\pm \frac{1}{ \sqrt{c_1-2\sinh (u)}}=x'(u)$$ $$x(u)+c_2=\pm \frac{2 i}{\sqrt{c+2 i}} F\left(\frac{\pi -2 i u}{4} |\frac{4 i}{c_1+2 i}\right)$$ where appears the elliptic integral of the first kind.

Inversing will give

$$u(x)=-\frac{1}{2} i \left(\pi \pm 4 \text{am}\left(\frac{1}{2} \sqrt{-\left((c_1+2 i) (x+c_2){}^2\right)}|\frac{4 i}{c_1+2 i}\right)\right)$$ where appears the amplitude for Jacobi elliptic functions.