Let a curve \( y=y(x) \) be given by the solution of the differential equation \( \cos\!\left(\dfrac{1}{2}\cos^{-1}\!\left(e^{-x}\right)\right)dx=\sqrt{e^{2x}-1}\,dy \). 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 ______.
