Match List-I with List-II (C is an arbitrary constant, \(x \ne 0\), \(y \ne 0\))
| List-I | List-II | ||
|---|---|---|---|
| Differential equation | General solution | ||
| (A) | \(\displaystyle\frac{dy}{dx} = 3y\) | (I) | \(y = Cx^3\) |
| (B) | \(\displaystyle\frac{dy}{dx} = 3x^2y\) | (II) | \(y^3 - x^3 = C\) |
| (C) | \(\displaystyle\frac{dy}{dx} = \frac{x^2}{y^2}\) | (III) | \(y = Ce^{3x}\) |
| (D) | \(\displaystyle\frac{dy}{dx} = \frac{3y}{x}\) | (IV) | \(y = Ce^{x^3}\) |
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)-(II), (D)-(I)
Each equation is solved by separating the variables and integrating both sides.
(A) \(\frac{dy}{dx} = 3y\)
This matches (III) \(y = Ce^{3x}\):
\( \displaystyle \frac{dy}{y} = 3\,dx \)
\(\log|y| = 3x + C_1\)
\(y = Ce^{3x}\)
(B) \(\frac{dy}{dx} = 3x^2y\)
This matches (IV) \(y = Ce^{x^3}\):
\( \displaystyle \frac{dy}{y} = 3x^2\,dx \)
\(\log|y| = x^3 + C_1\)
\(y = Ce^{x^3}\)
(C) \(\frac{dy}{dx} = \frac{x^2}{y^2}\)
This matches (II) \(y^3 - x^3 = C\):
\(y^2\,dy = x^2\,dx\)
\( \displaystyle \frac{y^3}{3} = \frac{x^3}{3} + C_1 \)
\(y^3 - x^3 = 3C_1 = C\)
(D) \(\frac{dy}{dx} = \frac{3y}{x}\)
This matches (I) \(y = Cx^3\):
\( \displaystyle \frac{dy}{y} = \frac{3\,dx}{x} \)
\(\log|y| = 3\log|x| + \log|C|\)
\(y = Cx^3\)