Match each family of curves in List-I with the order of its differential equation in List-II (a, b, c, d and m are arbitrary constants).
| List-I | List-II | ||
|---|---|---|---|
| Family of curves | Order | ||
| (A) | \(x^2 + y^2 = a^2\) | (I) | 3 |
| (B) | \(y = mx + c\) | (II) | 1 |
| (C) | \(y = ae^{x} + be^{2x} + ce^{3x}\) | (III) | 4 |
| (D) | \(y = ax^3 + bx^2 + cx + d\) | (IV) | 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)-(I), (D)-(III)
The order of the differential equation of a family equals the number of arbitrary constants in it, since each differentiation removes one constant.
(A) \(x^2 + y^2 = a^2\)
This matches (II) 1. There is one constant, a.
(B) \(y = mx + c\)
This matches (IV) 2. There are two constants, m and c.
(C) \(y = ae^{x} + be^{2x} + ce^{3x}\)
This matches (I) 3. There are three constants, a, b and c, attached to three different functions.
(D) \(y = ax^3 + bx^2 + cx + d\)
This matches (III) 4. There are four constants, and \(\frac{d^4y}{dx^4} = 0\) is the differential equation.