Match List-I with List-II

Choose the correct answer from the options given below:

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

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

$\text{For } \frac{dy}{dx}+P(x)\,y=Q(x),\ \text{I.F.}=e^{\int P(x)\,dx}$

$(A)\ \frac{dy}{dx}+2xy=1 \;\Rightarrow\; P(x)=2x \;\Rightarrow\; \text{I.F.}=e^{\int 2x\,dx}=e^{x^{2}}\ \Rightarrow\ (IV)$

$(B)\ x\frac{dy}{dx}+2xy=1 \;\Rightarrow\; \frac{dy}{dx}+2y=\frac{1}{x} \;\Rightarrow\; P(x)=2 \;\Rightarrow\; \text{I.F.}=e^{\int 2\,dx}=e^{2x}\ \Rightarrow\ (II)$

$(C)\ x\frac{dy}{dx}+y=1 \;\Rightarrow\; \frac{dy}{dx}+\frac{1}{x}y=\frac{1}{x} \;\Rightarrow\; P(x)=\frac{1}{x} \;\Rightarrow\; \text{I.F.}=e^{\int \frac{1}{x}\,dx}=x\ \Rightarrow\ (I)$

$(D)\ x\frac{dy}{dx}+2y=2 \;\Rightarrow\; \frac{dy}{dx}+\frac{2}{x}y=\frac{2}{x} \;\Rightarrow\; P(x)=\frac{2}{x} \;\Rightarrow\; \text{I.F.}=e^{\int \frac{2}{x}\,dx}=x^{2}\ \Rightarrow\ (III)$

Matching: $(A)\!\to\!(IV),\ (B)\!\to\!(II),\ (C)\!\to\!(I),\ (D)\!\to\!(III)$