Match List-I with List-II:

List-I	List-II
Differential Equations	Degree/Order
(A) Degree of the differential equation $\frac{d^3y}{dx^3} + 2 \log x.y = 0$	(I) 3
(B) Order of the differential equation $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^4 + xy = 0$	(II) 2
(C) Degree of the differential equation $\left(\frac{d^4y}{dx^4}\right)^2 + \left(\frac{dy}{dx}\right)^3 + x^2y = 0$	(III) 1
(D) Order of the differential equation $\frac{d^3y}{dx^3} + y\left(\frac{dy}{dx}\right)^3 = 0$	(IV) 4

Before we solve, let's be clear on two terms:

$\textit{Order}$ — The highest order derivative in the equation. Just look for the biggest $n$ in $\dfrac{d^n y}{dx^n}$.

$\textit{Degree}$ — The exponent (power) of the highest order derivative, but only when the equation is polynomial in its derivatives (no logs, square roots, or other functions wrapped around derivatives).

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(A) Degree of $\dfrac{d^3y}{dx^3} + 2\log x \cdot y = 0$

The highest order derivative is $\dfrac{d^3y}{dx^3}$, and it is raised to the power $1$.

Note: $\log x$ is a function of $x$, not of a derivative — it's just a coefficient and doesn't affect the degree.

$\text{Degree} = 1$

$\Rightarrow$ (A) $\to$ (III)

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(B) Order of $\dfrac{d^4y}{dx^4} + \left(\dfrac{dy}{dx}\right)^4 + xy = 0$

The derivatives present are $\dfrac{d^4y}{dx^4}$ (4th order) and $\dfrac{dy}{dx}$ (1st order).

The highest is $4$. The power on $\dfrac{dy}{dx}$ is irrelevant — order counts how many times you differentiate, not the exponent.

$\text{Order} = 4$

$\Rightarrow$ (B) $\to$ (IV)

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(C) Degree of $\left(\dfrac{d^4y}{dx^4}\right)^2 + \left(\dfrac{dy}{dx}\right)^3 + x^2y = 0$

The highest order derivative is $\dfrac{d^4y}{dx^4}$, and it is raised to the power $2$.

The equation is polynomial in its derivatives, so the degree is simply the exponent of the highest order derivative.

$\text{Degree} = 2$

$\Rightarrow$ (C) $\to$ (II)

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(D) Order of $\dfrac{d^3y}{dx^3} + y\left(\dfrac{dy}{dx}\right)^3 = 0$

The derivatives present are $\dfrac{d^3y}{dx^3}$ (3rd order) and $\dfrac{dy}{dx}$ (1st order).

The highest is $3$.

$\text{Order} = 3$

$\Rightarrow$ (D) $\to$ (I)

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Final Matching:

$(A) \to (III) \implies 1$

$(B) \to (IV) \implies 4$

$(C) \to (II) \implies 2$

$(D) \to (I) \implies 3$