I need to solve the following system of differential equations:
¨x=8x+4y¨y=−4x
Here's what I've done so far: I have reduced this system to a first order system, by saying x1:=x, x2:=˙x, x3:=y, x4:=˙y. This yields the system ˙X=A⋅X with
A=(010080400001−4000) X=(x1x2x3x4)
Then I've determined the eigenvalues λ1=2, λ2=−2, with the corresponding eigenvectors v1=(12−1−2)T and v2=(1−2−12)T.
Now what I'm struggling with is: how do I determine my set of fundamental solutions? I know that the terms c1e2t and c2e−2t are part of it for sure, but since I have two double eigenvalues, I also should have a solution somewhat like te2t resp. te−2t. But I just don't see how they alle come together.
