Match List-I with List-II :

Choose the correct answer from the options given below:

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

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

(A) $x d y-\left(y+2 x^2\right) d x=0$

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

$I.F.=e^{\int -\frac{1}{x}dx}=e^{-\log x}=\frac{1}{x}$ (I)

(B) $\left(2 x^2-3 y\right) d x=x d y$

$\frac{dy}{dx}+\frac{3y}{x}=2x$

so $I.F.=e^{\int\frac{3}{x}dx}=e^{3\log x}=x^3$ (IV)

(C) $\left(2 y+3 x^2\right) dx+x d y=0$

$\frac{dy}{dx}+\frac{2y}{x}=-3x$

so $I.F.=e^{\int\frac{2}{x}dx}=e^{2\log x}=x^2$ (III)

(D) $2 x d y+\left(3 x^3+2 y\right) d x=0$

$\frac{dy}{dx}+\frac{y}{x}=\frac{-3x^2}{2}$

$I.F.=e^{\int\frac{1}{x}dx}=e^{\log x}=x$ (II)