Is there a well-known type of differential equation consisting of $y$, $y'$, and $y''$ multiplied together? Is there some sort of well known type of differential equation consisting of first and second derivatives multiplied together? For example (I just made this up):
$$y''(y')^2-y-x^2=0$$
Edit: Are there any applications of these sort of differential equations?

The height profile of a viscous gravity currents on horizontal surfaces can be described using a similarity solution as
$$ (y^3y')' + 1/5(3\alpha+1)xy' - 1/5(2\alpha-1)y = 0, $$
on the domain $ 0 \leq x \leq 1$.  Analytic solutions exist for $\alpha = 0$.
Is this similar enough to what you're looking for?
