If \(\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}\) is the solution of the differential equation \(x \cos \…

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

[JEE Main 2024, 29 Jan (Shift 2)]