Match List-I with List-II

List-I	List-II
(Differential equation)	(Order and Degree)
(A) $\frac{d^3y}{dx^3} + y^2 + e^{dy/dx} = 0$	(I) order = 3, degree = 1
(B) $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} + 1 = 0$	(II) order = 3, degree not defined
(C) $2x^2\frac{d^2y}{dx^2} - 3\left(\frac{dy}{dx}\right)^2 + y = 0$	(III) order = 2, degree = 3
(D) $\frac{d^3y}{dx^3} + 2\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0$	(IV) order = 2, degree = 1

Choose the correct answer from the options given below:

Order is determined by the highest derivative in the equation. Degree is the power of the highest derivative when the equation is a polynomial in derivatives. If the equation contains transcendental functions (exponential, trigonometric, logarithmic) of derivatives, the degree is not defined.

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For equation (A): $\frac{d^3y}{dx^3} + y^2 + e^{dy/dx} = 0$

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

The equation contains $e^{dy/dx}$, which is an exponential function of a derivative. The equation is not a polynomial in derivatives, so degree is not defined.

Match: (A) → (II)

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

The highest derivative is $\frac{d^2y}{dx^2}$, so order $= 2$.

The equation is a polynomial in derivatives. The highest derivative $\frac{d^2y}{dx^2}$ appears with power $3$, so degree $= 3$.

Match: (B) → (III)

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

The highest derivative is $\frac{d^2y}{dx^2}$, so order $= 2$.

The equation is a polynomial in derivatives. The highest derivative $\frac{d^2y}{dx^2}$ appears with power $1$ (the coefficient $2x^2$ does not affect the degree), so degree $= 1$.

Match: (C) → (IV)

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

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

The equation is a polynomial in derivatives. The highest derivative $\frac{d^3y}{dx^3}$ appears with power $1$, so degree $= 1$.

Match: (D) → (I)

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Final matching:

(A) → (II), (B) → (III), (C) → (IV), (D) → (I)