Match List-I with List-II

Choose the correct answer from the options given below:

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

The correct answer is Option (1) → (A)-(IV), (B)-(I), (C)-(II), (D)-(III)

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

Square both sides after isolating radical

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

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

Highest order derivative is $\frac{d^2y}{dx^2}$ and its power is $2$

$(A)\rightarrow(IV)$

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

$\sqrt{\frac{d^3y}{dx^3}}=\sqrt[12]{\frac{d^2y}{dx^2}}$

Raise both sides to power $12$

$\left(\frac{d^3y}{dx^3}\right)^6=\frac{d^2y}{dx^2}$

Highest order derivative is $\frac{d^3y}{dx^3}$ with power $6$

$(B)\rightarrow(I)$

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

Contains exponential of derivative, cannot be made polynomial in derivatives

Degree is not defined

$(C)\rightarrow(II)$

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

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

Cube both sides

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

Highest order derivative is $\frac{d^2y}{dx^2}$ with power $3$

$(D)\rightarrow(III)$

Final Matching: (A)-(IV), (B)-(I), (C)-(II), (D)-(III).