Let the solution curve x=x(y), 0<y<(π)/2, of the differential equation ( log_e ( cos y))^2 cos y d x-(1+3 x log_e( cos y)) sin y d y=0 satisfy…

Let the solution curve $\displaystyle x=x(y), 0<y<\frac{\pi}{2}$, of the differential equation $\displaystyle \left(\log _{\mathrm{e}}(\cos y)\right)^2 \cos y \mathrm{~d} x-\left(1+3 x \log _e(\cos y)\right) \sin y d y=0$ satisfy $\displaystyle x\left(\frac{\pi}{3}\right)=\frac{1}{2 \log _e 2}$. If $\displaystyle x\left(\frac{\pi}{6}\right)=\frac{1}{\log _e m-\log _e n}$, where $\displaystyle m$ and $\displaystyle n$ are coprime, then $\displaystyle m n$ is equal to