Match List-I with List-II

List-I	List-II
Differential equation	Degree
(A) $\frac{d^2y}{dx^2} + \sqrt{\frac{dy}{dx}} - y = 0$	(I) 6
(B) $\sqrt{\frac{d^3y}{dx^3}} - \sqrt[12]{\frac{d^2y}{dx^2}} = 0$	(II) Not defined
(C) $\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} + e^{\frac{dx}{dx}} = x^2$	(III) 3
(D) $\sqrt[3]{\frac{dy}{dx}} - \frac{d^2y}{dx^2} = e^x$	(IV) 2

Choose the correct answer from the options given below:

Analysis of (A):

$\frac{d^2y}{dx^2} + \sqrt{\frac{dy}{dx}} - y = 0$

• Isolate the radical term: $\sqrt{\frac{dy}{dx}} = y - \frac{d^2y}{dx^2}$
• Square both sides to eliminate the radical:

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

• The highest order derivative is $\frac{d^2y}{dx^2}$ (Order = 2).
• The power of this highest order derivative is $2$.
• Degree = 2 $\rightarrow$ (IV)

Analysis of (B):

$\sqrt{\frac{d^3y}{dx^3}} - \sqrt[12]{\frac{d^2y}{dx^2}} = 0 \implies \left(\frac{d^3y}{dx^3}\right)^{\frac{1}{2}} = \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{12}}$

• To eliminate fractional powers, raise both sides to the power of the LCM of denominators $(2, 12)$, which is $12$:

$\left[\left(\frac{d^3y}{dx^3}\right)^{\frac{1}{2}}\right]^{12} = \left[\left(\frac{d^2y}{dx^2}\right)^{\frac{1}{12}}\right]^{12} \implies \left(\frac{d^3y}{dx^3}\right)^6 = \frac{d^2y}{dx^2}$

• The highest order derivative is $\frac{d^3y}{dx^3}$ (Order = 3).
• The power of this highest order derivative is $6$.
• Degree = 6 $\rightarrow$ (I)

Analysis of (C):

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

• Since the derivative $\frac{dy}{dx}$ is an exponent inside a transcendental function ($e^{\frac{dy}{dx}}$), this equation cannot be expressed as a polynomial in terms of its derivatives.
• Degree = Not defined $\rightarrow$ (II)

Analysis of (D):

$\sqrt[3]{\frac{dy}{dx}} - \frac{d^2y}{dx^2} = e^x$

• Isolate the radical term: $\left(\frac{dy}{dx}\right)^{\frac{1}{3}} = \frac{d^2y}{dx^2} + e^x$
• Cube both sides to clear the radical power:

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

• The highest order derivative is $\frac{d^2y}{dx^2}$ (Order = 2).
• The power of this highest order derivative is $3$.
• Degree = 3 $\rightarrow$ (III)