Question:
Let a curve $y=f(x)$ pass through the point $\left(2,\left(\log _{e} 2\right)^{2}\right)$ and have slope $\frac{2 y}{x \log _{e} x}$ for all positive real value of $x$. Then the value of $f(e)$ is equal to
Solution:
$y^{\prime}=\frac{2 y}{x \ell n x}$
$\Rightarrow \frac{\mathrm{dy}}{\mathrm{y}}=\frac{2 \mathrm{dx}}{\mathrm{x} \ell \mathrm{nx}}$
$\Rightarrow \ell \mathrm{n}|\mathrm{y}|=2 \ell \mathrm{n}|\ell \mathrm{nx}|+\mathrm{C}$
put $x=2, y=(\ell n 2)^{2}$
$\Rightarrow \mathrm{c}=0$
$\Rightarrow \mathrm{y}=(\ell \mathrm{nx})^{2}$
$\Rightarrow f(\mathrm{e})=1$
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