Match List-I with List-II
| List-I | List-II | ||
| Differential equation | Integrating factor | ||
| (A) | \( \displaystyle\frac{dy}{dx}+y\cot x=4x\,\mathrm{cosec}\,x \) | (I) | \( x \) |
| (B) | \( \displaystyle\frac{dy}{dx}+2y=\sin x \) | (II) | \( \displaystyle\frac{1}{x} \) |
| (C) | \( \displaystyle\frac{dy}{dx}+\frac{y}{x}=x^2 \) | (III) | \( \sin x \) |
| (D) | \( x\displaystyle\frac{dy}{dx}-y=2x^2 \) | (IV) | \( e^{2x} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
For a linear equation in the standard form \( \frac{dy}{dx}+Py=Q \), the integrating factor is \( e^{\int P\,dx} \).
(A) \( \frac{dy}{dx}+y\cot x=4x\,\mathrm{cosec}\,x \)
This matches (III) sin x.
\( \displaystyle P=\cot x \)
\( \displaystyle \text{I.F.}=e^{\int\cot x\,dx}=e^{\log\sin x}=\sin x \)
(B) \( \frac{dy}{dx}+2y=\sin x \)
This matches (IV) e2x.
\( \displaystyle P=2 \)
\( \displaystyle \text{I.F.}=e^{\int 2\,dx}=e^{2x} \)
(C) \( \frac{dy}{dx}+\frac{y}{x}=x^2 \)
This matches (I) x.
\( \displaystyle P=\frac{1}{x} \)
\( \displaystyle \text{I.F.}=e^{\int\frac{1}{x}dx}=e^{\log x}=x \)
(D) \( x\frac{dy}{dx}-y=2x^2 \)
This matches (II) 1/x.
Dividing by x gives the standard form:
\( \displaystyle \frac{dy}{dx}-\frac{y}{x}=2x \)
\( \displaystyle P=-\frac{1}{x} \)
\( \displaystyle \text{I.F.}=e^{-\int\frac{1}{x}dx}=e^{-\log x}=\frac{1}{x} \)
So the matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).