Let \( y=y(x) \) be the solutions of the differential equation \( (\tan x)^{\tfrac{1}{2}}\,dy=\left(\sec^3 x-(\tan x)^{\tfrac{3}{2}}\,y\right)dx \),\(0 < x < \frac{\pi}{2},\)\(y\left(\frac{\pi}{4}\right) = \frac{6\sqrt{2}}{5}.\) If \(y\left(\frac{\pi}{3}\right) = \frac{4}{5}\alpha,\) \(\text{then } \alpha^4\) equals_______________
