@article{Withers2022, 
author = {Christopher Withers and Saralees Nadarajah},
title = {Some linear differential equations generated by matrices},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {6},
pages = {9588-9602},
keywords = {characteristic matrix, Floquet's theorem, planetary perturbations},
url = {https://www.sciopen.com/article/10.3934/math.2022533},
doi = {10.3934/math.2022533},
abstract = {Given matrices    N  ∈      C          s      ×      s       and        S    0    ,  …  ,      S    q    ∈      C          s      ×      s      , we solve the linear differential equation                                 ∑                      n            =            0                    q                          T          n                (        t        )                (        d                  /                d        t                  )          n                f        (        t        )        =        g        (        t        )        ,            where    t  ∈  R,        T    n    (  t  )  =      e          t      N            S    n        e          −      t      N      , and    f  (  t  )  :  R  →      C    s  , using the roots of    d  (  ν  )  =  det    D  (  ν  ), where                       D        (        ν        )        =                  ∑                      n            =            0                    q                          S          n                                              (                          ν                              I                r                            +              N                        )                    n                .            For example,                       N        =                  (                                                    0                                            −                1                                                                    1                                            0                                              )                    implies                                 e                      t            N                          =                  (                                                    cos                ⁡                t                                            −                sin                ⁡                t                                                                    sin                ⁡                t                                            cos                ⁡                t                                              )                ,            so that        T    n    (  t  ) are periodic, giving an explicit solution to a form of Floquet's theorem.}
}