Saying that is a solution of a certain homogenreous second order linear equation with constant coefficents implies that a+ Bi is a solution to the characteristic equation.

IFthe differential equation (and therefore characteristic equation) hasrealcoefficients, then a- Bi must also be a solution to the characteristic equation and so is an independent solution to the differential equation. However, if any of the coefficients is non-real complex, that is not necessarily true.