Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | Integrating factor of \(xdy-(y + x^2)dx = 0\) | (I) | \(x^2\) |
| (B) | Integrating factor of \(xdy + (2y + x^2)dx = 0\) | (II) | \(x^3\) |
| (C) | Integrating factor of \((3y-x^2)dx + xdy = 0\) | (III) | \(x\) |
| (D) | Integrating factor of \((y + 3x^2)dx + xdy = 0\) | (IV) | \(\displaystyle\frac{1}{x}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(IV), (B)-(I), (C)-(II), (D)-(III)
Each equation is put in the linear form \(\frac{dy}{dx} + Py = Q\). The integrating factor is then \(e^{\int P\,dx}\) (taking x > 0).
(A) Integrating factor of \(xdy-(y + x^2)dx = 0\)
This matches (IV) \(\frac{1}{x}\).
\(\displaystyle \frac{dy}{dx} = \frac{y + x^2}{x}\)
\(\displaystyle \frac{dy}{dx} - \frac{1}{x}\,y = x\)
\(\displaystyle \text{IF} = e^{-\int \frac{1}{x}dx} = e^{-\log x} = \frac{1}{x}\)
(B) Integrating factor of \(xdy + (2y + x^2)dx = 0\)
This matches (I) \(x^2\).
\(\displaystyle \frac{dy}{dx} = -\frac{2y + x^2}{x}\)
\(\displaystyle \frac{dy}{dx} + \frac{2}{x}\,y = -x\)
\(\displaystyle \text{IF} = e^{\int \frac{2}{x}dx} = e^{2\log x} = x^2\)
(C) Integrating factor of \((3y-x^2)dx + xdy = 0\)
This matches (II) \(x^3\).
\(\displaystyle \frac{dy}{dx} = \frac{x^2 - 3y}{x}\)
\(\displaystyle \frac{dy}{dx} + \frac{3}{x}\,y = x\)
\(\displaystyle \text{IF} = e^{\int \frac{3}{x}dx} = e^{3\log x} = x^3\)
(D) Integrating factor of \((y + 3x^2)dx + xdy = 0\)
This matches (III) \(x\).
\(\displaystyle \frac{dy}{dx} = -\frac{y + 3x^2}{x}\)
\(\displaystyle \frac{dy}{dx} + \frac{1}{x}\,y = -3x\)
\(\displaystyle \text{IF} = e^{\int \frac{1}{x}dx} = e^{\log x} = x\)
So the matching is (A)-(IV), (B)-(I), (C)-(II), (D)-(III).