Let a curve $y=y(x)$ be given by the solution of the differential equation $$ \cos \left(\frac{1}{2} \cos ^{-1}\left(e^{-x}\right)\right) d x=\sqrt{e^{2 x}-1} d y $$ If it intersects $y$-axis at $y=-1$, and the intersection point of the curve with $x$-axis is $(\alpha, 0)$, then $e^\alpha$ is equal to