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 $y(x) = \frac{1}{2}\left(x-\frac{1}{x} \right)$



Nonlinear ODE: $y'=(1+axy)/(1+bxy)$
You might find it useful to make a change of variables to reduce the equation to a more familiar form. For example, if we assume, as we may, that $a$ and $b$ are not equal, then we can substitute $y = (z+ax)/b$, where $z$ is a new unknown, and then the equation can be written in the form
$$
\frac{dx}{dz} = \frac{(1 + xz + a x^2)}{b-a},
$$ 
which is a Riccati equation for $x$ as a function of $z$. Consequently, we understand all of the blow-up conditions since, by the usual linearization of Ricatti equations, we can express the solutions as the ratio of linear combinations of two solutions of a linear equation. Moreover, if we are given a particular, solution, we can now get the general solution by quadrature.


