Match List-I with List-II, where \(0 \lt x \lt \displaystyle\frac{\pi}{2}\).
| List-I | List-II | ||
|---|---|---|---|
| Differential equation | Integrating factor | ||
| (A) | \(\displaystyle\frac{dy}{dx} + y\tan x = x\) | (I) | \(\sin x\) |
| (B) | \(\displaystyle\frac{dy}{dx} - y\tan x = x\) | (II) | \(\mathrm{cosec}\,x\) |
| (C) | \(\displaystyle\frac{dy}{dx} + y\cot x = x\) | (III) | \(\sec x\) |
| (D) | \(\displaystyle\frac{dy}{dx} - y\cot x = x\) | (IV) | \(\cos x\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
The integrating factor of \(\frac{dy}{dx} + Py = Q\) is \(e^{\int P\,dx}\). On \(0 \lt x \lt \frac{\pi}{2}\) all the trigonometric values are positive, and
\( \displaystyle \int \tan x\,dx = \log \sec x, \qquad \int \cot x\,dx = \log \sin x \)
(A) \(\frac{dy}{dx} + y\tan x = x\)
This matches (III) \(\sec x\): \(e^{\log \sec x} = \sec x\).
(B) \(\frac{dy}{dx} - y\tan x = x\)
This matches (IV) \(\cos x\): \(e^{-\log \sec x} = \frac{1}{\sec x} = \cos x\).
(C) \(\frac{dy}{dx} + y\cot x = x\)
This matches (I) \(\sin x\): \(e^{\log \sin x} = \sin x\).
(D) \(\frac{dy}{dx} - y\cot x = x\)
This matches (II) \(\mathrm{cosec}\,x\): \(e^{-\log \sin x} = \frac{1}{\sin x} = \mathrm{cosec}\,x\).