The integrating factor of the differential equation \(\displaystyle\frac{dy}{dx} + y\sec^2 x = x\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(e^{\tan x}\)
Comparing with \(\frac{dy}{dx} + Py = Q\):
\( P = \sec^2 x \)
\( \displaystyle \int P\,dx = \int \sec^2 x\,dx = \tan x \)
The integrating factor is \(e^{\int P\,dx} = e^{\tan x}\).
Why the other options are incorrect
Option (1) \(e^{\sec x}\) — Incorrect. \(\sec x\) is not an antiderivative of \(\sec^2 x\); the derivative of \(\sec x\) is \(\sec x \tan x\).
Option (3) \(\tan x\) — Incorrect. This is only \(\int P\,dx\); the integrating factor is e raised to it.
Option (4) \(e^{\sec^2 x}\) — Incorrect. P itself is put in the power, without integrating it.