If the solution curve $f(x, y)=0$ of the differential equation $\left(1+\log _e x\right) \frac{d x}{d y}-x \log _e x=e^y, x0$, passes through the points $(1,0)$ and $(\alpha, 2)$ then $\alpha^\alpha$ is equal to [JEE Main 2023, 6 Apr (Shift 2)]

Question

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 [JEE Main 2023, 6 Apr (Shift 2)]

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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=…

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