Is there a general theory for equations of the type $ f_y^2 = A(x,y) f_x$? where one first derivative is expressed as a multiple of the other one.
Concretely, I'm interested in the equation
$$ ( x+ y^2 + c) f_x = f_y^2$$
where $f_x = \frac{\partial f}{\partial x}$, $f_y = \frac{\partial f}{\partial y}$.
I tried $f(x,y) = g( z)$ with $z=x+ y^2 + c$ so far, and some variants of this.
I would take a look at the method of characteristics. This equation is fully nonlinear, so $\S$2.4 would be of interest. If this equation is paired with data to form a Cauchy problem, then the solution is determined at least locally.