Match List-I with List-II

List-I	List-II
Differential Equation	Order and Degree
(A) $\left(\frac{d^2y}{dx^2}\right)^2 = e^x\left(\frac{dy}{dx}\right)^4 + 1 = 0$	(I) order = 1 and degree = 2
(B) $\left(\frac{dy}{dx}\right)^2 + xy = 0$	(II) order = 2 and degree = 1
(C) $\left(1 + \frac{dy}{dx}\right)^{3/2} = 4\left(\frac{d^2y}{dx^2}\right)^2$	(III) order = 2 and degree = 2
(D) $\sqrt\frac{d^2y}{dx^2} + 1 = \frac{dy}{dx}$	(IV) order = 2 and degree = 4

Choose the correct answer from the options given below:

Order is the highest order derivative present in the equation. Degree is the power of that highest order derivative, after the equation is made free of radicals/fractions in derivatives.

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(A) $\left(\dfrac{d^2y}{dx^2}\right)^2 + e^x\left(\dfrac{dy}{dx}\right)^4 + 1 = 0$

Highest order derivative is $\dfrac{d^2y}{dx^2}$ → Order $= 2$

Power of $\dfrac{d^2y}{dx^2}$ is $2$ → Degree $= 2$

(A) → (III)

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(B) $\left(\dfrac{dy}{dx}\right)^2 + xy = 0$

Highest order derivative is $\dfrac{dy}{dx}$ → Order $= 1$

Power of $\dfrac{dy}{dx}$ is $2$ → Degree $= 2$

(B) → (I)

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(C) $\left(1 + \dfrac{dy}{dx}\right)^{3/2} = 4\left(\dfrac{d^2y}{dx^2}\right)^2$

There is a fractional power $\dfrac{3}{2}$, so squaring both sides:

$\left(1 + \dfrac{dy}{dx}\right)^{3} = 16\left(\dfrac{d^2y}{dx^2}\right)^{4}$

Highest order derivative is $\dfrac{d^2y}{dx^2}$ → Order $= 2$

Power of $\dfrac{d^2y}{dx^2}$ is $4$ → Degree $= 4$

(C) → (IV)

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

There is a square root on a derivative, so rearranging and squaring:

$\sqrt{\dfrac{d^2y}{dx^2}} = \dfrac{dy}{dx} - 1$

$\dfrac{d^2y}{dx^2} = \left(\dfrac{dy}{dx} - 1\right)^2$

Highest order derivative is $\dfrac{d^2y}{dx^2}$ → Order $= 2$

Power of $\dfrac{d^2y}{dx^2}$ is $1$ → Degree $= 1$

(D) → (II)

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List-I	List-II
(A)	(III) — Order = 2, Degree = 2
(B)	(I) — Order = 1, Degree = 2
(C)	(IV) — Order = 2, Degree = 4
(D)	(II) — Order = 2, Degree = 1