1. **State the problem:** Solve the differential equation $$\frac{d^2y}{dx^2} + y = e^x$$. 2. **Identify the type of equation:** This is a non-homogeneous linear second-order differential equation with constant coefficients. 3. **Solve the homogeneous equation:** $$\frac{d^2y}{dx^2} + y = 0$$. The characteristic equation is $$r^2 + 1 = 0$$. Solving for $$r$$: $$r^2 = -1 \implies r = \pm i$$. So the homogeneous solution is: $$y_h = C_1 \cos x + C_2 \sin x$$. 4. **Find a particular solution $$y_p$$:** Since the right side is $$e^x$$, try a particular solution of the form: $$y_p = Ae^x$$. Compute derivatives: $$\frac{dy_p}{dx} = Ae^x$$, $$\frac{d^2y_p}{dx^2} = Ae^x$$. Substitute into the differential equation: $$Ae^x + Ae^x = e^x$$, which simplifies to: $$2Ae^x = e^x$$. Divide both sides by $$e^x$$ (nonzero): $$2A = 1 \implies A = \frac{1}{2}$$. So the particular solution is: $$y_p = \frac{1}{2} e^x$$. 5. **Write the general solution:** $$y = y_h + y_p = C_1 \cos x + C_2 \sin x + \frac{1}{2} e^x$$. This is the complete solution to the differential equation.