Time dependent differential equation system with periodicity

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I'm doing research as a grad student and I'm stuck on solving a system of differential equations, with a periodic matrix:

$\dot{x}=\begin{bmatrix}a-\cos(t)& b\\b &a+\cos(t) \end{bmatrix}x$

Is there a way to find solutions for $x(t)$ ?

In my readings, I found Floquet theory might be useful, but I can't make enough sense of it to apply it to my problem.

Also, the following link might be useful, though I couldn't make it work here: Decoupling coupled differential equations with time dependent coefficients.