This is $y(x)$ expressed in an implicit form because a closed form solution doesn't exist. For a given $C$ and for all $x$ in the domain of $y(x)$ you can solve that equation, somehow, to obtain the value of $y$ corresponding to that $x$ i.e. $y(x)$

Look at this

It makes it a bit more clear what $\Psi(x,y)$ is all about.