Solution of the differential equation $y \log_e y\, dx-x\, dy = 0$ is (Where c is an arbitrary constant)

$|y|=|c\log_e(xy)|$

The correct answer is Option (1) → $|y|=|c\log_e(xy)|$ **

Form the differential equation:

$y\ln y\,dx - x\,dy = 0$

Separate variables:

$\frac{dx}{x} = \frac{dy}{y\ln y}$

Integrate:

$\displaystyle \int \frac{dx}{x} = \int \frac{dy}{y\ln y}$

$\ln|x| = \ln|\ln y| + C$

Rewrite:

$\ln|x| - \ln|\ln y| = C$

$\ln \left|\frac{x}{\ln y}\right| = C$

Exponentiate:

$\left|\frac{x}{\ln y}\right| = e^{C} = K$

$|x| = K |\ln y|$

or equivalently

$|y| = |C \ln (xy)|$