IIT JAM 2023 Question 16: Difference and differential equations

Let $y=y(x)$ solve $x\dfrac{dy}{dx}=y\ln\!\left(\dfrac{y}{x}\right)$, $y>x>0$. If $y(1)=e^2$ and $y(2)=\alpha$, then $\dfrac{dy}{dx}$ at $(2,\alpha)$ equals