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