Some linear differential equations generated by matrices

Given matrices $N\in C^{s\times s}$ and $S_0,\dots ,S_q\in C^{s\times s}$, we solve the linear differential equation

$$\begin{array}{r}\sum _{n=0}^q{T}_n(t)(d/dt{)}^{n}f(t)=g(t),\end{array}$$

where $t\in R$, $T_n(t)=e^{tN}{S}_n{e}^{-tN}$, and $f(t):R\to C^s$, using the roots of $d(\nu )=detD(\nu )$, where

$$\begin{array}{r}D(\nu )=\sum _{n=0}^q{S}_n{\left(\nu I_r+N\right)}^n.\end{array}$$

For example,

$$\begin{array}{r}N=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\end{array}$$

implies

$$\begin{array}{r}{e}^{tN}=\left(\begin{array}{cc}\cos t& -\sin t\\ \sin t& \cos t\end{array}\right),\end{array}$$

so that $T_n(t)$ are periodic, giving an explicit solution to a form of Floquet's theorem.