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

A

$$3$$

B

$$12$$

C

$$4$$

D

$$9$$