The general solution of the differential equation \(y\log y\,dx-x\,dy=0\) is (where C is arbitrary constant):
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(y=C^{x}\)
The equation is variables separable:
\(y\log y\,dx = x\,dy\)
\( \displaystyle \frac{dy}{y\log y} = \frac{dx}{x} \)
On the left, put \(t = \log y\), so \(dt = \frac{dy}{y}\) and the left side becomes \(\frac{dt}{t}\). Integrating both sides:
\(\log|\log y| = \log|x|+\log|c|\)
\(\log y = cx\)
\(y = e^{cx} = \left(e^{c}\right)^{x}\)
Writing the constant \(e^{c}\) as C:
\(y = C^{x}\)
NCERT (2026-27 edition):
Differential Equations with Variables Separable
