Observe the following statements A. Integrating factor of $\frac{d y}{d x}+y=x^2$ is $e^x$ R. Integrating…
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
$A$ is true, $R$ is false
$A$ is false, $R$ is true
$A$ is true, $R$ is true, $R \Rightarrow A$
Both are false
Solution
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 $\mathrm{R} \Rightarrow \mathrm{A}$