What is the integrating factor $I(x)$ for the first-order linear differential equation $$\frac{dy}{dx} + \left(\frac{3}{x}\right) y = x^2, \quad x > 0$$?

(A) x^3

(B) 3\ln(x)

(C) e^{3x}

(D) x^{-3}

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Standard Form:** $\frac{dy}{dx} + P(x)y = Q(x)$ where $P(x) = 3/x$. 2. **Stage 2: Integrating Factor Formula:** $I(x) = \exp\left(\int P(x) dx\right) = \exp(3\ln x) = \exp(\ln x^3) = x^3$. 3. **Stage 3: Distractor Elimination:** Option A is $x^3$; Option B is the exponent argument; Option C assumes constant coefficient. 4. **Stage 4: Selection:** Option A is correct.

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