Match List I with List II
| List I | List II | ||
| A. | The number of arbitrary constants in the particular solution of differential equation of order 2 | I. | 1 |
| B. | The number of arbitrary constants in the general solution of differential equation of order 2 | II. | 0 |
| C. | The integrating factor of differential equation \(\displaystyle\frac{dy}{dx} + \frac{1}{x}y = 3, x \gt 0\), is | III. | 2 |
| D. | For differential equation, \(x^2\displaystyle\frac{dy}{dx} + x = xy, x \gt 0, \lim_{x \to 0^+} y(x)\) is equal to | IV. | x |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A-II, B-III, C-IV, D-I
(A) The number of arbitrary constants in the particular solution of differential equation of order 2
This matches (II) 0.
A particular solution is obtained by giving definite values to all the constants, so it has none left.
(B) The number of arbitrary constants in the general solution of differential equation of order 2
This matches (III) 2.
The general solution of an equation of order n contains n arbitrary constants.
(C) The integrating factor of \(\frac{dy}{dx} + \frac{1}{x}y = 3\), x > 0
This matches (IV) x.
\( \displaystyle \text{I.F.} = e^{\int \frac{1}{x}\,dx} = e^{\log x} = x \quad (x \gt 0) \)
(D) \(x^2\frac{dy}{dx} + x = xy\), x > 0, \(\lim_{x \to 0^+} y(x)\)
This matches (I) 1.
Dividing by \(x^2\):
\( \displaystyle \frac{dy}{dx} = \frac{y - 1}{x} \)
\( \displaystyle \frac{dy}{y - 1} = \frac{dx}{x} \)
\(\log|y - 1| = \log x + \log|c|\)
\(y = 1 + cx\)
\( \displaystyle \lim_{x \to 0^+} (1 + cx) = 1 \)
So the matching is A-II, B-III, C-IV, D-I.