Let $y=y\left(x\right)$ be the solution of the differential equation ${cosec}^{2}xdy+2dx=\left(1+y\cos 2x\right){cosec}^{2}xdx,$ with $y\left(\frac{\pi }{4}\right)=0.$ Then, the value of ${\left(y\left(0\right)+1\right)}^2$ is equal to:

Let $y=y\left(x\right)$ be solution of the following differential equation 

$e^y\frac{dy}{dx}-2e^y\sin x+\sin x{\cos }^{2}x=0, y\left(\frac{\pi }{2}\right)=0$.

If $y\left(0\right)={\log }_{\mathrm{e}}\left(\alpha +\beta {\mathrm{e}}^{-2}\right)$, then $4(\alpha +\beta )$ is equal to            .