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(C) For statement $A$:The given differential equation is $\frac{dy}{dx} + y = x^2$.Comparing this with the standard linear form $\frac{dy}{dx} + P(x)y

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$A$ is true, $R$ is true, $R \Rightarrow A$

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(C) For statement $A$:The given differential equation is $\frac{dy}{dx} + y = x^2$.Comparing this with the standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$, we get $P(x) = 1$.The integrating factor $(IF)$ is given by $e^{\int P(x) dx} = e^{\int 1 dx} = e^x$.Thus, statement $A$ is true.For statement $R$:The standard form of a linear differential equation is $\frac{dy}{dx} + P(x)y = Q(x)$.The integrating factor is defined as $e^{\int P(x) dx}$.Thus, statement $R$ is true.Since statement $A$ is derived directly from the formula given in statement $R$, $R \Rightarrow A$ is true.Therefore, both statements are true and $R$ is the correct explanation for $A$.

Observe the following statements:
$A$. Integrating factor of $\frac{dy}{dx} + y = x^2$ is $e^x$.
$R$. Integrating factor of $\frac{dy}{dx} + P(x)y = Q(x)$ is $e^{\int P(x) dx}$.
Then, the true statement among the following is:

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Observe the following statements:$A$. Integrating factor of $\frac{dy}{dx} + y = x^2$ is $e^x$.$R$. Integrating factor of $\frac{dy}{dx} + P(x)y = Q(x)$ is $e^{\int P(x) dx}$.Then, the true statement among the following is: