1. The problem is to solve the differential equation $y'' + 3y' + 2y = 6$ using the method of undetermined coefficients. 2. The general approach is to find the complementary solution $y_c$ by solving the homogeneous equation $y'' + 3y' + 2y = 0$ and then find a particular solution $y_p$ for the nonhomogeneous part. 3. Solve the characteristic equation for the homogeneous part: $$r^2 + 3r + 2 = 0$$ Factor: $$(r + 1)(r + 2) = 0$$ So, $$r = -1, -2$$ 4. The complementary solution is: $$y_c = C_1 e^{-x} + C_2 e^{-2x}$$ 5. For the particular solution, since the right side is a constant $6$, 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. The general solution is: $$y = y_c + y_p = C_1 e^{-x} + C_2 e^{-2x} + 3$$ This completes the solution for problem 1.