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Observe the following statements
A. Integrating factor of $\frac{d y}{d x}+y=x^2$ is $e^x$
R. Integrating factor of $\frac{d y}{d x}+P(x) y=Q(x)$ is $e^{\int P(x) d x}$
Then, the true statement among the following is
Options:
A. Integrating factor of $\frac{d y}{d x}+y=x^2$ is $e^x$
R. Integrating factor of $\frac{d y}{d x}+P(x) y=Q(x)$ is $e^{\int P(x) d x}$
Then, the true statement among the following is
Solution:
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Verified Answer
The correct answer is:
$A$ is true, $R$ is true, $R \Rightarrow A$
Statement A
Integrating factor of $\frac{d y}{d x}+y=x^2=e^{\int 1 \cdot d x}=e^x$
Statement R
$$
\begin{aligned}
\frac{d y}{d x}+P(x) y & =Q(x) \\
\operatorname{IF} & =e^{\int P(x) d x}
\end{aligned}
$$
$\therefore$ Both statements $A$ and $B$ are true and statement $\mathrm{R} \Rightarrow \mathrm{A}$.
Integrating factor of $\frac{d y}{d x}+y=x^2=e^{\int 1 \cdot d x}=e^x$
Statement R
$$
\begin{aligned}
\frac{d y}{d x}+P(x) y & =Q(x) \\
\operatorname{IF} & =e^{\int P(x) d x}
\end{aligned}
$$
$\therefore$ Both statements $A$ and $B$ are true and statement $\mathrm{R} \Rightarrow \mathrm{A}$.
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