Differential Equations: JEE Main 2026 Mathematics question

Let $y=y(x)$ be the solution of the differential equation $(\tan x)^{1/2}dy=\left(\sec^3x-(\tan x)^{3/2}y\right)dx$, $0<x<\frac{\pi}{2}$, $y\left(\frac{\pi}{4}\right)=\frac{6\sqrt2}{5}$. If $y\left(\frac{\pi}{3}\right)=\frac45\alpha$, then $\alpha^4$ equals ______.