JEE 2024 — Mathematics PYQ

If $\sin \left(\frac{y}{x}\right)=\log |x|+\frac{\alpha }{2}$ is the solution of the differential equation $x\cos \left(\frac{y}{x}\right)\frac{dy}{dx}=y\cos \left(\frac{y}{x}\right)+x$ and $y(1)=\frac{\pi }{3}$, then ${\alpha }^2$ is equal to