So two solutions are of the form a+ bi and a- bi and the other two are ak+ bki and ak- bki for some constant k?

If two solutions to the characteristic equation are a+ bi and a- bi then two independent solutions are

and

. The other solutions to the characteristic, ak+ bki and ak- bki, give

and

.

(The fact that the second pair of solutions to the characteristic equation are proportional to the first pair is not really of importance. If one pair of solutions is

,

and the other is l

and

then the four independent solutions to the differential equation are

,

,

, and

.