Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + \frac{\sqrt{2}y}{2\cos^4x - \cos 2x} = xe^{\tan^{-1}(\sqrt{2}\cot 2x)}$$, $$0 < x < \frac{\pi}{2}$$ with $$y\left(\frac{\pi}{4}\right) = \frac{\pi^2}{32}$$. If $$y\left(\frac{\pi}{3}\right) = \frac{\pi^2}{18}e^{-\tan^{-1}(\alpha)}$$, then the value of $$3\alpha^2$$ is equal to ______.