Question

Let a curve $y=f\left(x\right)$ pass through the point $(2,({\log }_{e}2{)}^2)$ and have slope $\frac{2y}{x{\log }_{e}x}$ for all positive real value of $x.$ Then the value of $f\left(e\right)$ is equal to

Answer 1

Given: $\frac{dy}{dx}=\frac{2y}{x{\log }_{e}x}$
$\Rightarrow \frac{dy}{y}=\frac{2dx}{x\ln x}$
$\Rightarrow \ln \left(y\right)=2\ln \left(\ln x\right)+C$
$\because (2,(\ln 2{)}^2)$ lies on the curve
$\therefore \ln (\ln 2{)}^2=2\ln (\ln 2)+C$
$\Rightarrow C=0$
$\therefore \ln \left(y\right)=2\ln \left(\ln x\right)$
$\Rightarrow y=(\ln x{)}^2$
Now, at $x=e$
$y=(\ln e{)}^2$
$\therefore y=1=f\left(e\right)$