Match List-I with List-II

List-I	List-II
(A) Degree of this differential equation $\frac{d^4y}{dx^4} + 2\log_e\left(\frac{d^3y}{dx^3}\right) = 0$	(I) 1
(B) Order of this differential equation $e^{\left(\frac{dy}{dx}\right)^3} + 3y\left(\frac{d^2y}{dx^2}\right)^3 = 0$	(II) 4
(C) Degree of $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^2 = 0$	(III) not defined
(D) Order of the differential equation $2\frac{d^4y}{dx^4} + \left(\frac{d^2y}{dx^2}\right)^5 = 0$	(IV) 2

Choose the correct answer from the options given below:

Let's understand Order and Degree first:

Order: The highest derivative in the equation

Degree: The power of the highest derivative when the equation is a polynomial in derivatives. If derivatives appear inside log, exponential, or other transcendental functions, degree is not defined.

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(A) Degree of:

$\frac{d^4y}{dx^4} + 2\log_e\left(\frac{d^3y}{dx^3}\right) = 0$

The derivative $\frac{d^3y}{dx^3}$ appears inside a logarithm.

When derivatives are inside log, sin, cos, or exponential functions, the degree is not defined.

(A) matches with (III) not defined

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(B) Order of:

$e^{\left(\frac{dy}{dx}\right)^3} + 3y\left(\frac{d^2y}{dx^2}\right)^3 = 0$

Identify the highest derivative:

$\frac{dy}{dx}$ has order $1$

$\frac{d^2y}{dx^2}$ has order $2$

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

Order $= 2$

(B) matches with (IV) 2

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(C) Degree of:

$\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^2 = 0$

This is a polynomial in derivatives (no log or exponential of derivatives).

The highest order derivative is $\frac{d^4y}{dx^4}$

The power of $\frac{d^4y}{dx^4}$ is $1$

Degree $= 1$

(C) matches with (I) 1

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(D) Order of:

$2\frac{d^4y}{dx^4} + \left(\frac{d^2y}{dx^2}\right)^5 = 0$

Identify the highest derivative:

$\frac{d^4y}{dx^4}$ has order $4$

$\frac{d^2y}{dx^2}$ has order $2$

The highest derivative is $\frac{d^4y}{dx^4}$

Order $= 4$

(D) matches with (II) 4

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Final Matching:

(A) $\rightarrow$ (III)

(B) $\rightarrow$ (IV)

(C) $\rightarrow$ (I)

(D) $\rightarrow$ (II)

Correct Answer: A-III, B-IV, C-I, D-II