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 :

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 :

$e^{\sqrt{2} e^{2}}$

$e^{2 e^{\sqrt{2}}}$

$e^{e^{2}}$

$e^{2 e^{2}}$

Solution

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