The integrating factor of the differential equation \(\displaystyle\frac{dy}{dx}=x+xy\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(e^{-x^2/2}\)
In the linear form \(\frac{dy}{dx} + Py = Q\), the equation reads
\( \displaystyle \frac{dy}{dx} - xy = x \)
so \(P = -x\) and \(Q = x\).
\( \displaystyle \text{I.F.} = e^{\int P\,dx} = e^{\int -x\,dx} \)
\(= e^{-x^2/2}\)
Why the other options are incorrect
Option (1) \(e^{-x/2}\) — Incorrect. It would need \(\int P\,dx = -\frac{x}{2}\), that is \(P = -\frac{1}{2}\), but \(P = -x\).
Option (3) \(e^{x^2/2}\) — Incorrect. This takes P = x. When xy is moved to the left side it becomes −xy, so P = −x.
Option (4) \(e^{x/2}\) — Incorrect. It would need \(P = \frac{1}{2}\), but \(P = -x\).