The particular solution of the differential equation \(\displaystyle\frac{dy}{dx} = -6xy^2\) satisfying \(y(1) = \displaystyle\frac{1}{4}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(y = \displaystyle\frac{1}{3x^2 + 1}\)
The equation is variables separable (y is not zero, since y(1) = 1/4):
\( \displaystyle \frac{dy}{y^2} = -6x\,dx \)
Integrating both sides:
\( \displaystyle -\frac{1}{y} = -3x^2 + C \)
Putting x = 1, y = 1/4:
\(-4 = -3 + C\)
\(C = -1\)
\( \displaystyle \frac{1}{y} = 3x^2 + 1 \)
\( \displaystyle y = \frac{1}{3x^2 + 1} \)
Why the other options are incorrect
Every option gives y(1) = 1/4, so the condition alone cannot decide; each must also satisfy the equation.
Option (1) \(y = \displaystyle\frac{1}{x^2 + 3}\) — Incorrect. Here \(\frac{dy}{dx} = \frac{-2x}{(x^2 + 3)^2} = -2xy^2\), not \(-6xy^2\).
Option (2) \(y = \displaystyle\frac{1}{6x^2 - 2}\) — Incorrect. Here \(\frac{dy}{dx} = \frac{-12x}{(6x^2 - 2)^2} = -12xy^2\).
Option (3) \(y = \displaystyle\frac{4}{3x^2 + 13}\) — Incorrect. Here \(\frac{dy}{dx} = \frac{-24x}{(3x^2 + 13)^2} = -\frac{3}{2}xy^2\).