Maybe I'm missing something here. But your initial condition $y(x_0) = 0$ implies that $a = 0$. If
$y \frac{dy}{dx} = by - \frac{a}{x} $
then why not substitute $x = x_0$ to give $0 = -a/x_0$? Since $x_0 > 0$ it follows that $a=0$.
$ y \frac{dy}{dx} = by $
has a very simple solution: either $y \equiv 0$ or $y = bx + k$ for any $k \in \mathbb{R}.$ Imposing the condition that $y(x_0) = 0$ means that $k = -bx_0$ and so $y \equiv 0$ and $y(x) = b(x - x_0)$ are the two solutions. You'll need to change your initial conditions from $a,b > 0$ to $a,b \ge 0.$