Assertion: Integrating factor of `(dy )/(dx) + y = x^2 ` is ` e^x`
Reason: Integrating factor of ` ( dy )/(dx) + P( x) y= Q (x) ` is ` e^(int p(x) dx)`

Options

A. If both assertion and reason are CORRECT and the reason is CORRECT explanation of the assertion.
B. If both assertion and reason are CORRECT and the reason in INCORRECT explanation of the assertion.
C. If assertion is CORRECT and the reason in INCORRECT
D. If assertion in INCORRECT and the reason is CORRECT.

Answer

To solve the problem, we need to analyze the assertion and the reason given in the question.

### Step 1: Identify the differential equation
The given differential equation is:
\[ \frac{dy}{dx} + y = x^2 \]
This is a first-order linear ordinary differential equation of the form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \( P(x) = 1 \) and \( Q(x) = x^2 \).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by the formula:
\[ \mu(x) = e^{\int P(x) \, dx} \]
Substituting \( P(x) = 1 \):
\[ \mu(x) = e^{\int 1 \, dx} = e^{x} \]

### Step 3: Verify the assertion
The assertion states that the integrating factor of the equation is \( e^x \). From our calculation, we found that:
\[ \mu(x) = e^x \]
Thus, the assertion is correct.

### Step 4: Analyze the reason
The reason states that the integrating factor of the general form \( \frac{dy}{dx} + P(x)y = Q(x) \) is \( e^{\int P(x) \, dx} \). This is indeed the standard formula for finding the integrating factor for a first-order linear differential equation. Since we derived the integrating factor using this formula, the reason is also correct.

### Conclusion
Both the assertion and the reason are correct, and the reason correctly explains the assertion.

### Final Answer
Both assertion and reason are correct, and the reason is the correct explanation of the assertion.

---