I encountered this differential equation $x'' = x^3$ during one of my work and couldn't find an analytical solution to the above. I've used numerical methods to solve the equation in the end.
I was just wondering if there is any way to find an analytical solution or show that an elementary solution doesn't exist.
WolframAlpha expresses the answer with Jacobi theta functions, and I was wondering if that's the only way to express the answer.
If you use $${\dfrac{d^2x}{dt^2}}=-\frac{\dfrac{d^2t}{dx^2}}{\left(\dfrac{dt}{dx}\right)^3}$$ the equation becomes $$t''+(t')^3 x^3=0$$ Let $t'=y$ to get $$y'+y^3x^3=0$$ Now $y=\frac 1 {\sqrt z}$ to get $$z'=2x^3 \implies z=c_1+\frac{x^4}{2}\implies y=\frac{1}{\sqrt{c_1+\frac{x^4}{2}}}$$ and finally $$t+c_2=\int\frac{dx}{\sqrt{c_1+\frac{x^4}{2}}}$$ leading to elliptic integrals.
I suppose that back to $x$, we should get some Jacobi elliptic function.