Solution of the differential equation \(y \log_e y\, dx-x\, dy = 0\) is (Where c is an arbitrary constant)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(|x| = |c\log_e y|\)
The equation is variables separable:
\( y\log_e y\,dx = x\,dy \)
\( \displaystyle \frac{dx}{x} = \frac{dy}{y\log_e y} \)
Integrating both sides (on the right put \(u = \log_e y\), so \(du = \frac{dy}{y}\) and the integral is \(\log_e|u|\)):
\( \log_e|x| = \log_e|\log_e y| + \log_e|c| \)
\( |x| = |c\log_e y| \)
NCERT (2026-27 edition):
Differential Equations with Variables Separable
