The integrating factor of the differential equation \(x\displaystyle\frac{dy}{dx}+y-x+xy\cot x=0\), (x ≠ 0) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → x sin x
Dividing by x puts the equation in the linear form \(\frac{dy}{dx} + Py = Q\):
\( \displaystyle \frac{dy}{dx} + \left(\frac{1}{x} + \cot x\right)y = 1 \)
So \(P = \frac{1}{x} + \cot x\), and
\( \displaystyle \int P\,dx = \log x + \log \sin x = \log(x\sin x) \)
The integrating factor is
\( \displaystyle e^{\int P\,dx} = e^{\log(x\sin x)} = x\sin x \)