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