TITLE: Nonlinear ODE: $y'=(1+axy)/(1+bxy)$
QUESTION [5 upvotes]: 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 $y(x) = \frac{1}{2}\left(x-\frac{1}{x} \right)$

REPLY [2 votes]: Global existence (or not) can be clarified without any tools by just taking a good look at your equation. Clearly, the RHS $f(x,y)$ is bounded away from the curve $1+bxy=0$, where it is undefined. So global existence fails for a given solution precisely if this solution approaches $1+bxy=0$ in finite time (in particular, global existence for a solution $y(x)$ is guaranteed as soon as $y(x_0)\ge 0$ for some $x_0$).
This won't happen if $a\le b$, as we see by just checking what the ODE does close to our curve. For example, if $1+bx_0y(x_0)=\epsilon>0$, then $y'(x_0)>0$, so we're moving away from the curve.
If $a>b$, then, by the same argument, there are solutions that reach $1+bxy=0$ in finite time (those with sufficiently negative initial value $y(0)$).