change use different words Because coefficients are real, 3+i implies 3i is also a root. Characteristic polynomial: (r + 1/2)(r (3 + i))(r (3 i)) = (r+1/2)((r 3)^(2) + 1) = (r + 1/2)(r^(2) 6r + 10) =r^(3) (11/2)r^(2) + 7r + 5. So a corresponding homogeneous linear ODE is: y 211y + 7y + 5y = 0 This answer is unique up to a nonzero constant multiple of the entire equation. The scaling doesn't change the roots. The solution form would be: y=C1e^(x/2) + e^(3x)(C2cos(x) + C3sin(x))