Consider the first order nonlinear ODE problem: $$ y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0 $$ where $a, b>0$ are some constants. I would like to know if these kind of equations were studied somewhere else (i.e., existence of global solutions, uniqueness etc). Any references will be helpful.

Remarks:

Mathematica does not give me anything.

I have tried to use Maple and it shows me that $y$ is a solution of an equation involving some Whittaker functions but I am not sure about this.

If $a=b$ then $y=x+C$.

If $a=1, b=2$ then one can take $y(x) = \frac{1}{2}\left(x-\frac{1}{x} \right)$

inexactequation. Thus, we look for anintegration factor$\mu (x,y)$ such that $\partial_y (\mu A) = \partial_x (\mu B)$, which yields a PDE. $\endgroup$ – Rodrigo de Azevedo Jun 11 '16 at 17:26