Systems of ODE: Converting complex solution to real

[Shaybay92]

Homework Statement

So, I have found a general solution to a system of linear first order ODE's and this is what I got:

X = c1v1e^(-1+2i)t + c2v2e^(-1-2i)t

where v1 = [-1+2i, 5], v2=[-1-2i,5]. The question is, how do I now change this solution into its real equivalent? i.e. I don't want any complex numbers in my solution.

I have a textbook which explains but it still doesn't make sense to me how they manage to go from a complex to real solution. Could someone explain step by step? Thanks.

 

[hunt_mat]

I think they're saying that X=x+yi, just resolve in real and imaginary parts...

 

[Shaybay92]

I don't think so, because it has to still span the solution space, and merely dropping the imaginary parts will not ensure this.