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{align*} \sum\limits_{n = 0}^q T_n(t)\ (d/dt)^n f(t) = g(t), \end{align*} $
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{align*} D(\nu) = \sum\limits_{n = 0}^q S_n\ \left(\nu I_r+N\right)^n. \end{align*} $
For example,
$ \begin{align*} N = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{align*} $
implies
$ \begin{align*} e^{tN} = \begin{pmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{pmatrix}, \end{align*} $
so that $ T_n(t) $ are periodic, giving an explicit solution to a form of Floquet's theorem.
Citation: Christopher Withers, Saralees Nadarajah. Some linear differential equations generated by matrices[J]. AIMS Mathematics, 2022, 7(6): 9588-9602. doi: 10.3934/math.2022533
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{align*} \sum\limits_{n = 0}^q T_n(t)\ (d/dt)^n f(t) = g(t), \end{align*} $
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{align*} D(\nu) = \sum\limits_{n = 0}^q S_n\ \left(\nu I_r+N\right)^n. \end{align*} $
For example,
$ \begin{align*} N = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{align*} $
implies
$ \begin{align*} e^{tN} = \begin{pmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{pmatrix}, \end{align*} $
so that $ T_n(t) $ are periodic, giving an explicit solution to a form of Floquet's theorem.
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V. P. Derevenskii, First-order linear ordinary differential equations over special matrices, Differential Equations, 56 (2020), 696–709. https://doi.org/10.1134/S0012266120060038 doi: 10.1134/S0012266120060038
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M. Elishevich, Boundary-value problem for a system of linear inhomogeneous differential equations of the first order with rectangular matrices, Journal of Mathematical Sciences, 228 (2018), 226–244. https://doi.org/10.1007/s10958-017-3617-8 doi: 10.1007/s10958-017-3617-8
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Christopher Withers, Saralees Nadarajah. Some linear differential equations generated by matrices[J]. AIMS Mathematics, 2022, 7(6): 9588-9602. doi: 10.3934/math.2022533
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