I was asked to solve the following $$ \begin{cases} \dfrac{dy}{dx} = \frac{3x^2}{6y^2-2} \\\\ y(2) = 0 \end{cases} $$
I solve it and I gave an answer as a function (which was correct), that is:
$$2y^3 - 2y = x^3 - 8$$
Yet I also have been asked about the domain of validity since this ODE is valid in a limited domain (with respect to $x$).
Can someone pleas explain it better to me?
You have in the end described the solution as the branch of a cubic curve that contains the initial point. The domain is now given by how far you can follow the branch according to the implicit function theorem. The critical values for the left side are $y=\pm1/\sqrt3$, inserting that gives an equation for $x$.