1. **Stating the problem:** We want to solve the differential equation $$\frac{dy}{dx} - \frac{y}{x} = 2x$$ where $y$ is a function of $x$. 2. **Identify the standard form:** The equation is linear and can be written as $$\frac{dy}{dx} + P(x)y = Q(x)$$ where here, $$P(x) = -\frac{1}{x}$$ and $$Q(x) = 2x$$. 3. **Find the integrating factor (IF):** The integrating factor is given by $$u = e^{\int P(x) dx} = e^{\int -\frac{1}{x} dx} = e^{-\ln x} = x^{-1} = \frac{1}{x}$$. 4. **Multiply the entire differential equation by the integrating factor:** $$\frac{1}{x} \frac{dy}{dx} - \frac{1}{x^2} y = 2x \cdot \frac{1}{x}$$ which simplifies to $$\frac{d}{dx} \left( \frac{y}{x} \right) = 2$$ This step uses the product rule in reverse, recognizing the left side as the derivative of $\frac{y}{x}$. 5. **Integrate both sides with respect to $x$:** $$\int \frac{d}{dx} \left( \frac{y}{x} \right) dx = \int 2 dx$$ which gives $$\frac{y}{x} = 2x + C$$ where $C$ is the constant of integration. 6. **Solve for $y$:** $$y = x(2x + C) = 2x^2 + Cx$$ **Final answer:** $$y = 2x^2 + Cx$$ This means the general solution to the differential equation is a quadratic function plus a linear term multiplied by an arbitrary constant $C$. If you want, I can explain any step in more detail!