Let the solution curve \( y = y(x) \) of the differential equation
$\left[ \frac{x}{\sqrt{x^2 - y^2}} + e^{\frac{y}{x}} \right] x \frac{dy}{dx} = x + \left[ \frac{x}{\sqrt{x^2 - y^2}} + e^{\frac{y}{x}} \right] y$
pass through the points \( (1,0) \) and \( (2\alpha,\alpha) \), \( \alpha > 0 \). Then \( \alpha \) is equal to:
