This is from Stewart - Calculus - Early Transcedentals
Writing the equation in differential form and integrating both sides, we have $$ \begin{aligned} (2 y+\cos y) d y &=6 x^{2} d x \\ \int(2 y+\cos y) d y &=\int 6 x^{2} d x \\ y^{2}+\sin y &=2 x^{3}+C \end{aligned} $$ where $C$ is a constant. This equation gives the general solution implicitly. In this case it's impossible to solve the equation to express $y$ explicitly as a function of $x$.
I would like to hear a very technical answer on why it is impossible to solve the equation to express y explicity as a function of x. Right now I know that I do not know how to solve it for $y$. How can I logically conclude that, it is in fact impossible to solve and not me who ignores some special technique.