Let y=y(x) be the solution of the differential equation ( tan x)^1 / 2 d y=( sec^3 x-( tan x)^3 / 2 y) d x, 0<x<(π)/2, y((π)/4)=(6 √ 2)/5. If…

Let $\displaystyle y=y(x)$ be the solution of the differential equation $\displaystyle (\tan x)^{1 / 2} \mathrm{~d} y=\left(\sec ^3 x-(\tan x)^{3 / 2} y\right) \mathrm{d} x, 0<x<\frac{\pi}{2}, y\left(\frac{\pi}{4}\right)=\frac{6 \sqrt{2}}{5}$. If $\displaystyle y\left(\frac{\pi}{3}\right)=\frac{4}{5} \alpha$, then $\displaystyle \alpha^4$ equals $\displaystyle \_\_\_\_$.