Linear Differential Equations with Integrating Factor

Let $y = y(x)$ be the solution of the differential equation $(\tan x)^{1/2} dy = (\sec^3 x - (\tan x)^{3/2} y) dx$, $0 < x < \dfrac{\pi}{2}, y\left(\dfrac{\pi}{4}\right) = \dfrac{6\sqrt{2}}{5}$. If $y\left(\dfrac{\pi}{3}\right) = \dfrac{4}{5} \alpha$, then $\alpha^4$ equals _______.