Answer : The characteristic polynomial is λ 3 (λ − 2)(λ − 3)2 . Thus, λ = 0 is a root of multiplicity 3, so it contributes the basic solutions e 0x , xe0x , x2 e 0x , i.e., 1, x, x2 . We have λ = 2 as a root of multiplicity 1, so it contributes the basic solution e 2x . Finally, we have λ = 3 as a root of multiplicity 2, so it contributes the basic solutions e 3x , xe3x . The general solution of the equation is a linear combination, with arbitrary coefficients, of the basic solutions, so the general solution is y = C1 + C2x + C3x 2 + C4e 2x + C5e 3x + C6xe3x .