Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(III), (B)-(I), (C)-(II), (D)-(IV)

The correct answer is Option (2) → (A)-(III), (B)-(I), (C)-(II), (D)-(IV)

Explanation:

Matching List-I with List-II:

(A) The degree of the differential equation $ \left( \frac{d^2 y}{dx^2} \right)^2 + \left( \frac{dy}{dx} \right)^2 = x \sin\left( \frac{dy}{dx} \right) $
→ Highest order derivative is $ \frac{d^2 y}{dx^2} $ raised to power 2, and the equation is polynomial in derivatives.
→ Degree = 2 but RHS involves $ \sin\left( \frac{dy}{dx} \right) $, so it is not polynomial in derivatives.
→ Degree is Not defined ⇒ (III)

(B) $ \frac{d^2 y}{dx^2} + \left( \frac{dy}{dx} \right)^{1/4} + x^{1/5} = 0 $
→ $\frac{dy}{dx} =( \frac{d^2 y}{dx^2}+ x^{1/5})^{4}$
→ Degree is 4 since $ \frac{d^2 y}{dx^2} $ degree is 4 → (I)

(C) $ \frac{d^2 y}{dx^2} \left( \frac{dy}{dx} \right)^3 + 6y^5 = 0 $
→ Highest derivative: $ \frac{d^2 y}{dx^2} $ raised to power 1 and equation is polynomial
→ Degree = 1 ⇒ (II)

(D) $ 1 + \left( \frac{dy}{dx} \right)^4 = 7 \left( \frac{d^2 y}{dx^2} \right)^3 $
→ Highest derivative: $ \frac{d^2 y}{dx^2} $ raised to power 3
→ Degree = 3 ⇒ (IV)