Mathematica only gives the real part of the solution. The complete solution can be complex. In general, the solution of linear differential equations with constant coefficients and distinct eigenvalues, is the linear combination of exponential functions with the roots in characteristic equation times independent variable t in the exponent. Thus when you have complex roots, you have a complex solution!
When Mathematica gives a sum of sines and cosines as the solution, that's just the real part of the solution - not the complete complex solution.