Match List-I with List-II

List-I	List-II
(A) The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x\sin\left(\frac{dy}{dx}\right)$	(I) 4
(B) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0$	(II) 1
(C) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + 6y^5 = 0$	(III) Not defined
(D) The degree of differential equation $1 + \left(\frac{dy}{dx}\right)^4 = 7\left(\frac{d^2y}{dx^2}\right)^3$	(IV) 3

Choose the correct answer from the options given below:

The degree of a differential equation is the highest power of the highest order derivative, only when the equation is written as a polynomial in its derivatives. If derivatives appear inside transcendental functions (sin, cos, log, etc.), the degree is not defined.

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(A) $\left(\dfrac{d^2y}{dx^2}\right)^2 + \left(\dfrac{dy}{dx}\right)^2 = x\sin\!\left(\dfrac{dy}{dx}\right)$

The right-hand side has $\sin(dy/dx)$, a transcendental function wrapping a derivative.

This cannot be rewritten as a polynomial in derivatives.

$(A) \rightarrow \text{(III) Not Defined}$

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(B) $\dfrac{d^2y}{dx^2} + \left(\dfrac{dy}{dx}\right)^{1/4} + x^{1/5} = 0$

The term $(dy/dx)^{1/4}$ has a fractional power, so it is not yet polynomial in derivatives. Note that $x^{1/5}$ is just a function of $x$, not a derivative, so it doesn't affect the degree.

$\left(\dfrac{dy}{dx}\right)^{1/4} = -\dfrac{d^2y}{dx^2} - x^{1/5}$

Raising both sides to the 4th power:

$\dfrac{dy}{dx} = \left(\dfrac{d^2y}{dx^2} + x^{1/5}\right)^4$

Now it is polynomial in derivatives. The highest order derivative is $\dfrac{d^2y}{dx^2}$, and when the right side is expanded, its highest power is 4.

$(B) \rightarrow \text{(I) } 4$

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(C) $\dfrac{d^2y}{dx^2} + \left(\dfrac{dy}{dx}\right)^3 + 6y^5 = 0$

Already a polynomial in derivatives.

$\dfrac{d^2y}{dx^2}$ → 2nd order (highest) → power $= 1$

$\left(\dfrac{dy}{dx}\right)^3$ → 1st order (not highest) → power $= 3$

$6y^5$ → not a derivative

The degree is determined by the power of the highest order derivative, which is $\dfrac{d^2y}{dx^2}$ with power 1, not $(dy/dx)^3$.

$(C) \rightarrow \text{(II) } 1$

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(D) $1 + \left(\dfrac{dy}{dx}\right)^4 = 7\left(\dfrac{d^2y}{dx^2}\right)^3$

Already a polynomial in derivatives.

The highest order derivative is $\dfrac{d^2y}{dx^2}$, and its power is 3.

$(D) \rightarrow \text{(IV) } 3$

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$(A) \rightarrow \text{(III) Not Defined}$

$(B) \rightarrow \text{(I) } 4$

$(C) \rightarrow \text{(II) } 1$

$(D) \rightarrow \text{(IV) } 3$