JEE Main 2024 — Differential Equations Question with Solution

$\begin{aligned} & a|x|=|y| e^{y x-\beta}, a, b \in N \\ & x d y-y d x+x y(x d y+y d x)=0 \\ & \frac{d y}{y}-\frac{d x}{x}+(x d y+y d x)=0 \\ & \ell n|y|-\ell n|x|+x y=c \\ & y(1)=2 \\ & \ell n|2|-0+2=c \\ & c=2+\ell n 2 \\ & \ell n|y|-\ell n|x|+x y=2+\ell n 2 \\ & \ell n|x|=\ell n\left|\frac{y}{2}\right|-2+x y \\ & |x|=\left|\frac{y}{2}\right| e^{x y-2} \\ & 2|x|=|y| e^{x y-2} \\ & \alpha=2 \quad \beta=2 \quad \alpha+\beta=4 \end{aligned}$