Match List-I with List-II

List-I	List-II
Differential Equations	Order and degree
(A) $ydx + x\log(y/x)dy - 2xdy = 0$	(I) Order : 2, degree:1
(B) $\left(\frac{d^3y}{dx^3}\right)^2 + 3\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^4 = y^2$	(II) Order :1, degree:1
(C) $\frac{dy}{dx} + \log\left(\frac{dy}{dx}\right) + x = y$	(III) Order : 3, degree:2
(D) $\left(\frac{ds}{dt}\right)^4 + 2s\frac{d^2s}{dt^2} = 0$	(IV) Order : 1, degree: Not defined

Choose the correct answer from the options given below:

Order of a differential equation is the highest derivative present.

Degree of a differential equation is the power of the highest derivative, provided the equation is polynomial in derivatives.

If the derivative appears inside transcendental functions (logarithm, exponential, trigonometric), the degree is not defined.

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For equation (A): $ydx + x\log(y/x)dy - 2xdy = 0$

Converting to standard form by dividing by $dx$:

$y + x\log(y/x)\frac{dy}{dx} - 2x\frac{dy}{dx} = 0$

The highest derivative is $\frac{dy}{dx}$, so Order $= 1$

Power of $\frac{dy}{dx}$ is $1$, so Degree $= 1$

(A) matches with (II)

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For equation (B): $\left(\frac{d^3y}{dx^3}\right)^2 + 3\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^4 = y^2$

The derivatives present are $\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$, and $\frac{d^3y}{dx^3}$

The highest derivative is $\frac{d^3y}{dx^3}$, so Order $= 3$

Power of $\frac{d^3y}{dx^3}$ is $2$, so Degree $= 2$

(B) matches with (III)

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For equation (C): $\frac{dy}{dx} + \log\left(\frac{dy}{dx}\right) + x = y$

The highest derivative is $\frac{dy}{dx}$, so Order $= 1$

The derivative appears inside a logarithm $\log\left(\frac{dy}{dx}\right)$

The equation is not polynomial in the derivative, so Degree is not defined

(C) matches with (IV)

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For equation (D): $\left(\frac{ds}{dt}\right)^4 + 2s\frac{d^2s}{dt^2} = 0$

The derivatives present are $\frac{ds}{dt}$ and $\frac{d^2s}{dt^2}$

The highest derivative is $\frac{d^2s}{dt^2}$, so Order $= 2$

Power of $\frac{d^2s}{dt^2}$ is $1$, so Degree $= 1$

(D) matches with (I)

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Final matching:

(A) - (II) Order: 1, degree: 1

(B) - (III) Order: 3, degree: 2

(C) - (IV) Order: 1, degree: Not defined

(D) - (I) Order: 2, degree: 1