The solution of the differential equation \(y\,dx + (x-y^2)\,dy = 0\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(3xy-y^3 = C\): C is an arbitrary constant
The equation is linear in x when y is taken as the independent variable:
\( \displaystyle y\,\frac{dx}{dy} + x - y^2 = 0 \)
\( \displaystyle \frac{dx}{dy} + \frac{1}{y}\,x = y \)
The integrating factor is
\( \displaystyle e^{\int \frac{1}{y}\,dy} = e^{\log y} = y \)
Multiplying by y:
\( \displaystyle y\,\frac{dx}{dy} + x = y^2 \)
\( \displaystyle \frac{d}{dy}(xy) = y^2 \)
Integrating with respect to y:
\( \displaystyle xy = \frac{y^3}{3} + c \)
Multiplying by 3 and writing C = 3c:
\(3xy - y^3 = C\)
Shortcut: Differentiate Option (3):
\(3y\,dx + 3x\,dy - 3y^2\,dy = 0\)
Dividing by 3 gives \(y\,dx + (x - y^2)\,dy = 0\), the given equation.
NCERT (2026-27 edition):
Linear Differential Equations
