Some linear differential equations generated by matrices

Given matrices $ N\in C^{s\times s} $ and $ S_0, \ldots, S_q\in C^{s\times s} $, we solve the linear differential equation \begin{document}$ \begin{align*} \sum\limits_{n = 0}^q T_n(t)\ (d/dt)^n f(t) = g(t), \end{align*} $\end{document} where $ t\in R $, $ T_n(t) = e^{tN}S_ne^{-tN} $, and $ f(t):R\rightarrow C^s $, using the roots of $ d(\nu) = \det\ D(\nu) $, where \begin{document}$ \begin{align*} D(\nu) = \sum\limits_{n = 0}^q S_n\ \left(\nu I_r+N\right)^n. \end{align*} $\end{document} For example, \begin{document}$ \begin{align*} N = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{align*} $\end{document} implies \begin{document}$ \begin{align*} e^{tN} = \begin{pmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{pmatrix}, \end{align*} $\end{document} so that $ T_n(t) $ are periodic, giving an explicit solution to a form of Floquet's theorem.