Separable ODE - Integral involving Elliptic integral

Question

I am trying to solve the equation $$ \dot y(t) = -\sqrt{ y(t)^6 + y(t)^2 + a } $$ with $a \leq y(t)^6+y(t)^2$ for all $t\geq 0$. This is a first order separable ODE. According to Wolfram Alpha the integral involves elliptic integrals of the first kind. For $a=0$, the integral involves hyperbolic trigonometric functions. Can this integral be solved avoiding elliptic integrals?. Else, I don't see any hope for getting an explicit solution for the ODE.

Answer 1

Notice that if you replace $\,y(t)^2\,$ with $\,y(t)\,$, then the ODE becomes $$ \dot y(t) = -\sqrt{ y(t)^3 + y(t) + a }. $$ This has a similar solution depending of the elliptic integral $\,F(\cdot|m).\,$ You asked

Can this integral be solved avoiding elliptic integrals?

The answer is no. It is not avoidable. You wrote

Else, I don't see any hope for getting an explicit solution for the ODE.

That depends on what you consider an "explicit" solution. The Legendre elliptic integral of the first kind is a well studied special function and there is no reason to avoid it if it is available for computations.

However, you are asking for an explicit solution to an ODE and in general, there is no way to get a "closed form" solution that you want. This case is no exception .