Match List-I with List-II
| List-I | List-II | ||
| Differential equation | Integrating factor | ||
| (A) | \( \displaystyle\frac{dy}{dx}+2y=x \) | (I) | \( e^{x} \) |
| (B) | \( \displaystyle\frac{dy}{dx}+\left(\displaystyle\frac{2}{x}\right)y=x^2 \) | (II) | \( e^{2x} \) |
| (C) | \( \displaystyle\frac{dy}{dx}-\left(\displaystyle\frac{2}{x}\right)y=\sin x \) | (III) | \( \displaystyle\frac{1}{x^2} \) |
| (D) | \( \displaystyle\frac{dy}{dx}+y=e^{-x} \) | (IV) | \( x^2 \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (II), (B) - (IV), (C) - (III), (D) - (I)
For \( \frac{dy}{dx}+Py=Q \), the integrating factor is \( e^{\int P\,dx} \).
(A) \( \frac{dy}{dx}+2y=x \)
This matches (II) \( e^{2x} \). Here P = 2:
\( \displaystyle e^{\int 2\,dx}=e^{2x} \)
(B) \( \frac{dy}{dx}+\left(\frac{2}{x}\right)y=x^2 \)
This matches (IV) \( x^2 \). Here \( P=\frac{2}{x} \):
\( \displaystyle e^{\int\frac{2}{x}dx}=e^{2\log x} \)
\( \displaystyle =e^{\log x^2} \)
\( \displaystyle =x^2 \)
(C) \( \frac{dy}{dx}-\left(\frac{2}{x}\right)y=\sin x \)
This matches (III) \( \frac{1}{x^2} \). Here \( P=-\frac{2}{x} \):
\( \displaystyle e^{\int-\frac{2}{x}dx}=e^{-2\log x} \)
\( \displaystyle =e^{\log x^{-2}} \)
\( \displaystyle =\frac{1}{x^2} \)
(D) \( \frac{dy}{dx}+y=e^{-x} \)
This matches (I) \( e^{x} \). Here P = 1:
\( \displaystyle e^{\int 1\,dx}=e^{x} \)
So the matching is (A)-(II), (B)-(IV), (C)-(III), (D)-(I).