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.

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 $y(0)$).

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 $y(0)$).