Let \( y = y(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{\sqrt{2}\,y}{2\cos^4 x - \cos 2x} = x e^{\tan^{-1}(\sqrt{2}\cot 2x)} \), \( 0 < x < \frac{\pi}{2} \) 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 ________.
