1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} + 2xy = 0$$. 2. **Identify the type:** This is a first-order linear differential equation of the form $$\frac{dy}{dx} + P(x)y = 0$$ where $$P(x) = 2x$$. 3. **Use the integrating factor method:** The integrating factor $$\mu(x)$$ is given by $$\mu(x) = e^{\int P(x) dx} = e^{\int 2x dx} = e^{x^2}$$. 4. **Multiply both sides by the integrating factor:** $$e^{x^2} \frac{dy}{dx} + 2x e^{x^2} y = 0$$ 5. **Recognize the left side as a derivative:** $$\frac{d}{dx} \left( y e^{x^2} \right) = 0$$ 6. **Integrate both sides:** $$\int \frac{d}{dx} \left( y e^{x^2} \right) dx = \int 0 dx$$ $$y e^{x^2} = C$$ where $$C$$ is the constant of integration. 7. **Solve for $$y$$:** $$y = C e^{-x^2}$$ **Final answer:** $$y = C e^{-x^2}$$