Question

If \(tanx\) is an integrating factor of the differential equation \(\frac{dy}{dx}+Py = Q\), then \(P\) can be

• A. \(2sec2x\)
• B. \(tan2x\)
• C. \(sin2x\)
• D. \(2csc2x\)

Hint:

The integrating factor is e^{integral of P}, so P is the log derivative of tan x.

Answer 1

The Correct Option is D

Step 1: Logarithmic derivative:
If $\mu = e^{\int P}$ then $P = \frac{\mu'}{\mu}$.

Step 2: Compute:
With $\mu = \tan x$: $\mu' = \sec^2x$, so $P = \frac{\sec^2 x}{\tan x} = \frac{1}{\cos^2x}\cdot\frac{\cos x}{\sin x} = \frac{1}{\sin x\cos x}$.

Step 3: Double angle:
$\sin x\cos x = \frac12\sin 2x$, so $P = \frac{2}{\sin 2x} = 2\csc 2x$.

Final Answer:
P is 2 cosec 2x, option (D). \[ \boxed{2\csc 2x} \]