The general solution of the differential equation \(\displaystyle\frac{dy}{dx}=-4xy^2\) is given by
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(2x^2-\displaystyle\frac{1}{y}= C\): C is an arbitrary constant
The variables separate:
\( \displaystyle \frac{dy}{y^2} = -4x\,dx \)
\( \displaystyle \int y^{-2}\,dy = -4\int x\,dx \)
\( \displaystyle -\frac{1}{y} = -2x^2 + C \)
\( \displaystyle 2x^2 - \frac{1}{y} = C \)
Why the other options are incorrect
Option (1) \(2x^2-y= C\): C is an arbitrary constant — Incorrect. Differentiating gives \(\frac{dy}{dx} = 4x\), not \(-4xy^2\).
Option (3) \(2x^2-\displaystyle\frac{1}{y^2}= C\): C is an arbitrary constant — Incorrect. Differentiating gives \(4x + \frac{2}{y^3}\frac{dy}{dx} = 0\), so \(\frac{dy}{dx} = -2xy^3\).
Option (4) \(2x^2+\displaystyle\frac{1}{y}= C\): C is an arbitrary constant — Incorrect. This has the wrong sign: \(\int y^{-2}\,dy = -\frac{1}{y}\), not \(+\frac{1}{y}\). Differentiating it gives \(\frac{dy}{dx} = 4xy^2\).