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:
