Let's say I solve the exact differential equation
$(3x^2y + 3y^2 - 1) \ dx + (x^2 + 6xy) \ dy = 0$
and find the implicit general solution to be
$x^3y + 3xy^2 - x = C$.
Obviously, if I tried to solve this for $y$, we would have $2$ values due to the $y^2$ ($\pm$). In these cases, my instructor says that we cannot find an explicit solution. Is this equivalent to saying that we cannot find a unique solution? It seems to me like it would be, since an explicit solution would need to be of the form $y = \dots$, but in this case we have $2$ values -- $y = \pm \dots$ So it seems to me like we wouldn't have an explicit solution, per say, but explicit solutions, which, in other words, is saying that we cannot find an explicit solution, since there is no unique explicit solution?
Am I correct here, or am I getting this wrong?
I would greatly appreciate it if people could please take the time to clarify this.