Match List-I with List-II

List-I	List-II
Differential Equation	Sum of order and degree
(A) $\frac{d^2y}{dx^2} + \frac{dy}{dx} + 3y = \sin x$	(I) 2
(B) $\frac{dy}{dx} = \sin(x + y)$	(II) 3
(C) $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2y}{dx^2}$	(III) 4
(D) $x^2(\frac{d^2y}{dx^2})^3 + y(\frac{dy}{dx})^4 + y^5 = 0$	(IV) 5

Choose the correct answer from the options given below:

Order is the highest derivative in the equation. Degree is the power of the highest order derivative after removing all roots and fractions involving derivatives.

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(A) $\frac{d^2y}{dx^2} + \frac{dy}{dx} + 3y = \sin x$

Highest derivative is $\frac{d^2y}{dx^2}$

Order $= 2$

Power of $\frac{d^2y}{dx^2}$ is $1$

Degree $= 1$

Sum $= 2 + 1 = 3$

Matches with (II)

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(B) $\frac{dy}{dx} = \sin(x + y)$

Highest derivative is $\frac{dy}{dx}$

Order $= 1$

Power of $\frac{dy}{dx}$ is $1$

Degree $= 1$

Sum $= 1 + 1 = 2$

Matches with (I)

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(C) $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2y}{dx^2}$

Highest derivative is $\frac{d^2y}{dx^2}$

Order $= 2$

The equation contains a square root, so it must be removed:

$1 + (\frac{dy}{dx})^2 = (\frac{d^2y}{dx^2})^2$

Power of $\frac{d^2y}{dx^2}$ is $2$

Degree $= 2$

Sum $= 2 + 2 = 4$

Matches with (III)

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(D) $x^2(\frac{d^2y}{dx^2})^3 + y(\frac{dy}{dx})^4 + y^5 = 0$

Highest derivative is $\frac{d^2y}{dx^2}$

Order $= 2$

Power of $\frac{d^2y}{dx^2}$ is $3$

Degree $= 3$

Sum $= 2 + 3 = 5$

Matches with (IV)

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Final Matching:

(A) → (II)

(B) → (I)

(C) → (III)

(D) → (IV)