1. **Problem statement:** Solve the second order differential equation $$y'' - 2y' + y = e^x$$. 2. **Find the complementary solution $g(x)$:** The complementary solution solves the homogeneous equation: $$y'' - 2y' + y = 0$$ 3. **Characteristic equation:** $$r^2 - 2r + 1 = 0$$ 4. **Solve the characteristic equation:** $$r^2 - 2r + 1 = (r - 1)^2 = 0$$ This has a repeated root $r = 1$. 5. **Complementary solution form for repeated root:** $$g(x) = (C_1 + C_2 x)e^{rx} = (C_1 + C_2 x)e^x$$ 6. **Explain why $y_p(x) = Ae^{2x}$ does not work:** The right side is $e^x$, but the trial solution $Ae^{2x}$ corresponds to $e^{2x}$, which is not the forcing function. Also, $e^{2x}$ is not a solution to the homogeneous equation, so it is not related to duplication here. The trial solution must be related to the forcing function $e^x$. 7. **Correct trial particular solution form after duplication rule:** Since $e^x$ is part of the complementary solution (due to repeated root $r=1$), the trial solution must be multiplied by $x^2$ to avoid duplication: $$y_p(x) = Ax^2 e^x$$ This is the correct form for the particular solution. **Final answers:** $$g(x) = (C_1 + C_2 x)e^x$$ $$y_p(x) = Ax^2 e^x$$