The following system of differential equations describes a charged particle in a viscous medium enveloped in an EM field, $$\partial^2_t x(t) = a\cos(\omega t +\phi)y +bx +c\partial_t y -d\partial_t x$$ $$\partial^2_t y(t) = a\cos(\omega t +\phi)x +by -c\partial_t x -d\partial_t y$$
All of the coefficients are real and positive. It's well known there are two circular eigenmotions. Given this, it seems useful to make the change of variables $u(t)=x(t)+i y(t)$. Doing so produces, $$\partial^2_t u(t) = ai\cos(\omega t +\phi)u^* +bu -(d+ic)\partial_t u$$
In the case of $a=d=0$, a solution of $u(t)=\exp(-iw_0t)$ exists with $w_0$ holding two values based on $b$ and $c$. It can be assumed $\partial_t x(0)=\partial_t y(0)=0$
Any suggestions on how to proceed further? If it makes the problem significantly easier, the case of $d=0$ is also of great interest.