Match List-I with List-II

List-I	List-II
Differential Equation	Degree
(A) $xy\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0$	(I) 3
(B) $\frac{d^2y}{dx^2} + \log\left(\frac{dy}{dx}\right) = 0$	(II) 1
(C) $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + \frac{dy}{dx} + 1 = 0$	(III) not defined
(D) $2x^2\left(\frac{d^2y}{dx^2}\right)^3 - 5\left(\frac{dy}{dx}\right)^3 + y = 0$	(IV) 2

Choose the correct answer from the options given below:

The degree of a differential equation is the power of the highest order derivative in the equation when it is a polynomial in derivatives. If the equation is not a polynomial in derivatives (contains log, exponential, or fractional powers of derivatives), the degree is not defined.

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For equation (A): $xy\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0$

The highest order derivative is $\frac{d^2y}{dx^2}$ (second order).

It appears as $xy\frac{d^2y}{dx^2}$, which has power $1$.

The equation is a polynomial in derivatives.

Degree $= 1$

(A) matches with (II)

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For equation (B): $\frac{d^2y}{dx^2} + \log\left(\frac{dy}{dx}\right) = 0$

The equation contains $\log\left(\frac{dy}{dx}\right)$, which is not a polynomial term.

The log function cannot be eliminated.

Degree is not defined.

(B) matches with (III)

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For equation (C): $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + \frac{dy}{dx} + 1 = 0$

The highest order derivative is $\frac{d^2y}{dx^2}$ (second order).

It appears as $\left(\frac{d^2y}{dx^2}\right)^2$, which has power $2$.

The equation is a polynomial in derivatives.

Degree $= 2$

(C) matches with (IV)

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For equation (D): $2x^2\left(\frac{d^2y}{dx^2}\right)^3 - 5\left(\frac{dy}{dx}\right)^3 + y = 0$

The highest order derivative is $\frac{d^2y}{dx^2}$ (second order).

It appears as $\left(\frac{d^2y}{dx^2}\right)^3$, which has power $3$.

The equation is a polynomial in derivatives.

Degree $= 3$

(D) matches with (I)

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Final matching:

(A) - (II): Degree $= 1$

(B) - (III): Degree not defined

(C) - (IV): Degree $= 2$

(D) - (I): Degree $= 3$