Match List - I with List - II. Match the integrating factors :

Choose the correct answer from the options given below :

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

B. $x \frac{d y}{d x}+y=3 x^2 \Rightarrow \frac{d y}{d x}+\frac{1}{x}y=\frac{3 x^2}{x}$    (dividing eq by leading coefficient of $\frac{dy}{dx}$)

So integrating factor

$=e^{\int \frac{1}{x} d x}=e^{\log x}=x$       →      III

C. $x \frac{d y}{d x}-y=3 x^2 \Rightarrow \frac{d y}{d x}-\frac{1}{x} y=\frac{3 x^2}{x}$      (dividing eq by leading coefficient of $\frac{dy}{dx}$)

So integrating factor

$=e^{\int p ~d x}=e^{\int \frac{-1}{x} d x}=e^{-\ln |x|}=e^{\ln (\frac{1}{x})} = \frac{1}{x}$             →            I

D. $\frac{d y}{d x} {-y}=x$

so integrating factor

$=e^{\int p d x}=e^{\int-1 d x}=e^{-x}$        →         II