1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} + \frac{y}{x} = x^2$$ for $y$ as a function of $x$. 2. **Identify the type of equation:** This is a first-order linear differential equation of the form $$\frac{dy}{dx} + P(x)y = Q(x)$$ where $$P(x) = \frac{1}{x}$$ and $$Q(x) = x^2$$. 3. **Find the integrating factor (IF):** The integrating factor is given by $$\mu(x) = e^{\int P(x) dx} = e^{\int \frac{1}{x} dx} = e^{\ln|x|} = |x|.$$ Since $x$ is typically positive in this context, we take $$\mu(x) = x.$$ 4. **Multiply the entire differential equation by the integrating factor:** $$x \frac{dy}{dx} + y = x^3.$$ 5. **Recognize the left side as a derivative:** $$\frac{d}{dx}(xy) = x^3.$$ 6. **Integrate both sides with respect to $x$:** $$\int \frac{d}{dx}(xy) dx = \int x^3 dx$$ $$xy = \frac{x^4}{4} + C,$$ where $C$ is the constant of integration. 7. **Solve for $y$:** $$y = \frac{x^4}{4x} + \frac{C}{x} = \frac{x^3}{4} + \frac{C}{x}.$$ **Final answer:** $$y = \frac{x^3}{4} + \frac{C}{x}.$$