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