Match List-I with List-II

Choose the correct answer from the options given below:

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

The correct answer is Option (2) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)

$\text{(A) } \frac{d^{4}y}{dx^{4}}+2\log\!\left(\frac{d^{3}y}{dx^{3}}\right)=0$

$\text{Contains }\log\!\left(\frac{d^{3}y}{dx^{3}}\right)\Rightarrow\text{not a polynomial in derivatives}\Rightarrow\text{degree not defined}$

$\Rightarrow\ \text{(A)}\to\text{(III) not defined}$

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

$\text{Highest-order derivative is } \frac{d^{2}y}{dx^{2}}\Rightarrow \text{order}=2$

$\Rightarrow\ \text{(B)}\to\text{(IV) }2$

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

$\text{Polynomial in derivatives; highest-order term } \frac{d^{4}y}{dx^{4}}\text{ has power }1\Rightarrow\text{degree}=1$

$\Rightarrow\ \text{(C)}\to\text{(I) }1$

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

$\text{Highest-order derivative } \frac{d^{4}y}{dx^{4}}\Rightarrow \text{order}=4$

$\Rightarrow\ \text{(D)}\to\text{(II) }4$

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