Question

The integrating factor of the linear DE \(\dfrac{dy}{dx}-y\sin x=\cot x\) is

• A. \(\sin x\)
• B. \(e^{-\sin x}\)
• C. \(e^{\sin x}\)
• D. \(e^{\cos x}\)

Hint:

Linear DE \(y'+P y=Q\) \(\Rightarrow\) IF \(=\exp(\int P\,dx)\). Watch the sign when moving terms.

Answer 1

The Correct Option is D

Idea. Standard linear form is \(y'+P(x)y=Q(x)\). The integrating factor is \(e^{\int P(x)\,dx}\).
Step 1. Identify \(P(x)\). Here \(y'-y\sin x=\cot x\Rightarrow P(x)=-\sin x\).
Step 2. Compute IF. \[ \text{IF}=e^{\int -\sin x\,dx}=e^{\cos x}. \] That's the required integrating factor (no need to solve for \(y\) here).