The integrating factor of the differential equation \((x + 1)\displaystyle\frac{dy}{dx} - y = (x + 1)^3\), \(x \gt -1\), is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{1}{x + 1}\)
Dividing by \(x + 1\) puts the equation in the linear form \(\frac{dy}{dx} + Py = Q\):
\( \displaystyle \frac{dy}{dx} - \frac{1}{x + 1}\,y = (x + 1)^2 \)
So
\( \displaystyle P = -\frac{1}{x + 1} \)
\( \displaystyle \int P\,dx = -\log (x + 1) \)
The integrating factor is
\( \displaystyle e^{\int P\,dx} = e^{-\log (x + 1)} = \frac{1}{x + 1} \)
Why the other options are incorrect
Option (1) \(x + 1\) — Incorrect. This takes \(P = +\frac{1}{x + 1}\); the minus sign in front of y makes P negative.
Option (3) \(\log (x + 1)\) — Incorrect. This is only \(\int P\,dx\) without the minus sign; the integrating factor is e raised to \(\int P\,dx\).
Option (4) \((x + 1)^2\) — Incorrect. This is the right-hand side Q after division, not the integrating factor.