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Match each differential equation in List-I with its solution in List-II. List-I: (I) y - x(dy/dx) = y² + dy/dx (II) (2x - 10y³) dy/dx + y = 0 (III) sec² y dy + tan y dx = dx (IV) sin y dy/dx = cos y (1 - x cos y) List-II: (P) xy² = 2y⁵ + c (Q) sec y = x + 1 + c e^x (R) (x + 1)(1 - y) = c y (S) tan y = 1 + c e^(-x)

1. I -> P; II -> Q; III -> R; IV -> S
2. I -> R; II -> P; III -> S; IV -> Q
3. I -> P; II -> R; III -> S; IV -> Q
4. I -> R; II -> S; III -> P; IV -> Q

Correct answer: I -> R; II -> P; III -> S; IV -> Q

Solution

(I) gives (x+1)(1-y)=cy (R); (II) gives xy²=2y⁵+c (P); (III) gives tan y = 1 + c e^(-x) (S); (IV) gives sec y = x + 1 + c e^x (Q).

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