General solution of the differential equation \( (x+2y^3)dy=ydx \) is
(Where C is an arbitrary constant)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( x=y(y^2+\mathrm{C}) \)
The equation is linear in x when x is taken as the dependent variable:
\( \displaystyle \frac{dx}{dy}=\frac{x+2y^3}{y} \)
\( \displaystyle \frac{dx}{dy}-\frac{x}{y}=2y^2 \)
The integrating factor is
\( \displaystyle e^{-\int\frac{dy}{y}}=\frac{1}{y} \)
So
\( \displaystyle \frac{x}{y}=\int 2y^2\times\frac{1}{y}\,dy \)
\( \displaystyle \frac{x}{y}=y^2+C \)
\( x=y(y^2+C) \)
NCERT Exemplar:
Differential Equations
