Match List-I with List-II

Choose the correct answer from the options given below:

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

The correct answer is Option (3) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)

Each equation is reduced to linear form and integrating factor identified.

(A) $x\frac{dy}{dx}-y=2x^2$

$\frac{dy}{dx}-\frac{1}{x}y=2x$

Integrating factor $=e^{\int -\frac{1}{x}dx}=e^{-\log x}=\frac{1}{x}$

(A) → (II)

(B) $\frac{dy}{dx}+\frac{y}{x}=2x$

Integrating factor $=e^{\int \frac{1}{x}dx}=x$

(B) → (III)

(C) $x\frac{dy}{dx}+2y=x^2\log x$

$\frac{dy}{dx}+\frac{2}{x}y=x\log x$

Integrating factor $=e^{\int \frac{2}{x}dx}=e^{2\log x}=x^2$

(C) → (IV)

(D) $\frac{dx}{dy}-x=y$

$\frac{dx}{dy}-x=y$ is linear in $x$ with variable $y$

Integrating factor $=e^{\int (-1)dy}=e^{-y}$

(D) → (I)

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