Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (Differential equation) | (Order and Degree) | ||
| (A) | \(\displaystyle\frac{d^3y}{dx^3} + y^2 + e^{dy/dx} = 0\) | (I) | order = 3, degree = 1 |
| (B) | \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^3 + \left(\displaystyle\frac{dy}{dx}\right)^2 + \displaystyle\frac{dy}{dx} + 1 = 0\) | (II) | order = 3, degree not defined |
| (C) | \(2x^2\displaystyle\frac{d^2y}{dx^2} - 3\left(\displaystyle\frac{dy}{dx}\right)^2 + y = 0\) | (III) | order = 2, degree = 3 |
| (D) | \(\displaystyle\frac{d^3y}{dx^3} + 2\left(\displaystyle\frac{dy}{dx}\right)^2 + \displaystyle\frac{dy}{dx} = 0\) | (IV) | order = 2, degree = 1 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
The order is the order of the highest derivative present. The degree is the power of that highest derivative, and it is defined only when the equation is a polynomial in the derivatives.
(A) \( \frac{d^3y}{dx^3} + y^2 + e^{dy/dx} = 0 \)
This matches (II) order = 3, degree not defined. The highest derivative is \( \frac{d^3y}{dx^3} \), but the term \(e^{dy/dx}\) is not a polynomial in \( \frac{dy}{dx} \), so the degree is not defined.
(B) \( \left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} + 1 = 0 \)
This matches (III) order = 2, degree = 3. The highest derivative \( \frac{d^2y}{dx^2} \) appears to the power 3.
(C) \( 2x^2\frac{d^2y}{dx^2} - 3\left(\frac{dy}{dx}\right)^2 + y = 0 \)
This matches (IV) order = 2, degree = 1. The highest derivative \( \frac{d^2y}{dx^2} \) appears to the power 1; the square on \( \frac{dy}{dx} \) does not count, since that is not the highest derivative.
(D) \( \frac{d^3y}{dx^3} + 2\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0 \)
This matches (I) order = 3, degree = 1. The highest derivative \( \frac{d^3y}{dx^3} \) appears to the power 1.
So the matching is (A)-(II), (B)-(III), (C)-(IV), (D)-(I).