Consider the ODE $$(4x^3 y^3 + 1/x)dx + (3x^4 y^2 - 1/y)dy = 0.$$
I can easily be shown that when $x\not =0$ and $y \not = 0$, this ODE is an exact DE, hence by doing the computation we get that any solution $y=y(x)$ satisfies $$F(x,y) = x^4 y^3 + \ln|x/y| = c,$$ for some $c \in \mathbb{R}.$
My question is that since in order to for $F(x,y)$ to be defined we need $x \not = 0$ and $y \not = 0$, when we solve $F(x,y) = c$, are the values $x_0$ s.t $y(x_0) = 0$ also a singular point of this ODE ? I mean I'm confused with the domain of the solutions in this particular ODE because by existance theorem, to have a solution, we need $y \not = 0$, but as can be seen in this question, there is the concept of "Analytic continuation" of the solution, so I don't know whether should I consider the point where $y = 0$ as in the solution or not.