1. The problem is to solve the differential equation $$\frac{dy}{dx} = x + y$$. 2. This is a first-order linear differential equation. The standard form is $$\frac{dy}{dx} - y = x$$. 3. To solve, we find the integrating factor (IF): $$\mu(x) = e^{-\int 1 dx} = e^{-x}$$. 4. Multiply both sides of the equation by the integrating factor: $$e^{-x} \frac{dy}{dx} - e^{-x} y = x e^{-x}$$ 5. The left side is the derivative of $$y e^{-x}$$: $$\frac{d}{dx} (y e^{-x}) = x e^{-x}$$ 6. Integrate both sides with respect to $$x$$: $$y e^{-x} = \int x e^{-x} dx + C$$ 7. Use integration by parts for $$\int x e^{-x} dx$$: Let $$u = x$$, $$dv = e^{-x} dx$$, then $$du = dx$$, $$v = -e^{-x}$$. 8. Applying integration by parts: $$\int x e^{-x} dx = -x e^{-x} + \int e^{-x} dx = -x e^{-x} - e^{-x} + C'$$ 9. Substitute back: $$y e^{-x} = -x e^{-x} - e^{-x} + C$$ 10. Multiply both sides by $$e^{x}$$ to solve for $$y$$: $$y = -x - 1 + C e^{x}$$ Final answer: $$y = C e^{x} - x - 1$$