I'm facing this problem:
"Suppose you have this system of ODE's:
$\begin{pmatrix} \dot y (t)\\ \dot x (t) \end{pmatrix} = \begin{pmatrix} a & b\\ c & d \end{pmatrix} \begin{pmatrix} y (t)\\ x(t) \end{pmatrix}$
Find the solution which have a period of 2, i.e, $x(t+2) = x(t)$ and $y(t+2) = y(t)$, $\forall t \in \mathbb{R}$".
This is what I'm trying: Well, if that's periodic, the eigenvalues of the system's matrix cannot have a nonzero real part, i.e, the should be pure imaginary. Let's say $\lambda = \pm bi$. As cosines and sines and linearly independent functions, this should apply:
$\cos(bt) = \cos(b(t+2)) = \cos(bt +2b)\\ \sin(bt) = \sin(b(t+2)) = \sin(bt +2b)$
So I tried to solve for $b$. But the only $b$ I find that solves it all is $b=0$, which makes no sense for me. My approach was: as it should be valid for all $t$, it's valid for $t=0$. So, evaluate it at $t=0$ and trying solving for $b$. Am I doing something wrong? Thank you!