Differential Equations

If the solution curve \(f(x, y)=0\) of the differential equation


\(\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x > 0\),


passes through the points \((1,0)\) and \((\alpha, 2)\), then \(\alpha^{\alpha}\) is equal to :