1. **State the problem:** Solve the differential equation $$y'' - 3y' + 2y = 6$$ using the method of undetermined coefficients. 2. **General approach:** For a linear differential equation with constant coefficients and a nonhomogeneous term, the solution is $$y = y_c + y_p$$ where $$y_c$$ is the complementary (homogeneous) solution and $$y_p$$ is a particular solution. 3. **Find the complementary solution $$y_c$$:** Solve the characteristic equation: $$r^2 - 3r + 2 = 0$$ Factor: $$(r - 1)(r - 2) = 0$$ So, $$r = 1$$ or $$r = 2$$. 4. The complementary solution is: $$y_c = C_1 e^{x} + C_2 e^{2x}$$ 5. **Find a particular solution $$y_p$$:** The right side is a constant 6, so try a constant solution: $$y_p = A$$ 6. Substitute $$y_p$$ into the differential equation: $$0 - 0 + 2A = 6$$ So, $$2A = 6 \implies A = 3$$ 7. **Write the general solution:** $$y = y_c + y_p = C_1 e^{x} + C_2 e^{2x} + 3$$ 8. This is the complete solution to the differential equation. **Final answer:** $$\boxed{y = C_1 e^{x} + C_2 e^{2x} + 3}$$