In my Ordinary Differential Equations class, we've learned the following method for finding a general solution for $x'=Ax$ (where $A$ is a $2x2$ matrix), if it gives two complex eigenvalues, $\mu+i\upsilon$ and $\mu-i\upsilon$ respectively.
Picking either of these eigenvectors, we should be able to reach a general real solution. Let's say we pick the eigenvalue $\mu+i\upsilon$ and then find two eigenvectors, $a+ib$ and $a+ib$ (where $a$ and $b$ are $2x1$ vectors).
My textbook then says that we can then find a general real solution by picking either one of these eigenvalues, and plugging it into the following:
$x = c_1e^{\mu t}(a*cos(\upsilon t)-b*sin(\upsilon t))+c_2e^{\mu t}(a*sin(\upsilon t)+b*cos(\upsilon t))$
(There's a proof that we can fold the imaginary part into the $c_2$ solution, since theoretically either $c$ could be complex, but we want only the real part.)
Here's the issue: In the past, I have gotten seemingly different general real solutions from picking either of the eigenvectors--but a solution is supposed to be made up of the fundamental set of solutions, and is also supposed to be unique!
So, the big question: How do we prove that whether we plug $a+ib$ or $a-ib$, we reach the same general real solution?
Or is the answer that we don't? And in fact we reach two different general real solutions with two different fundamental sets, because choosing either of the two eigenvectors gives a different fundamental set?
And if the above hypothesis is correct, then what is the meaning of a general real solution? Shouldn't they be solving the same equation?