Solution of the differential equation \( x\log_{e}x\,dy=y\,dx \) is: ( where \( c \) is an arbitrary constant)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( y=c\log_{e}x \)
The variables can be separated:
\( \displaystyle \frac{dy}{y}=\frac{dx}{x\log_{e}x} \)
On the right put \( \log_{e}x=t \), so that \( \frac{dx}{x}=dt \). Integrating both sides,
\( \log_{e}|y|=\log_{e}|\log_{e}x|+\log_{e}|c| \)
Hence
\( y=c\log_{e}x \)