Periodic solutions of a planar ODE Linearizing the equation of the two body problem at a circular solution I came accross the following planar second order system of differential equations
$$
\begin{cases}
2\frac{1}{\omega^2} \ddot{u}=3u\cos2\omega t+3v\sin2\omega t+u\\
2\frac{1}{\omega^2}\ddot{v}=3u\sin2\omega t-3v\cos2\omega t+v
\end{cases}
$$
It is easy to see that $(u,v)=(-\sin\omega t,\cos\omega t)$ is a $\frac{2\pi}{\omega}$-periodic solution. I'm wondering if there are other independent ones.
 Do you have any suggestion to find them?

Try $$u(t) = \sin(2 \omega t),\ v(t) = -\cos(2 \omega t) - 3$$
and
$$ u(t) = \cos(2 \omega t)-3,\ v(t) = \sin(2\omega t)$$
A fourth fundamental solution is non-periodic:
$$ u(t) = 3 t \omega \sin(\omega t) - 2 \cos(\omega t),\ v(t) = 3 t \omega \cos(\omega t) - 2 \sin(\omega t)$$
