How to solve this system of ODE's? [ x ˙ 1 x ˙ 2 ] = [ cos ⁡ t − sin ⁡ t sin ⁡ t cos ⁡ t ] [ x 1 x 2 ]

Question

How to solve this system of ODE&#039;s?      [                                                                x              ˙                        1                                                                                    x              ˙                        2                                ]    =      [                            cos          ⁡          t                          −          sin          ⁡          t                                      sin          ⁡          t                          cos          ⁡          t                      ]        [                                        x            1                                                            x            2                                ]

Angelo Murray · Accepted Answer

This is a linear system of the form      x    ′    =  A  (  t  )  x  ,where   x  ∈        R    2  ,   A  (  t  )  ∈  C  (  R  ;        R          2      ×      2        ), and most important                    (1)                    A        (        s        )        A        (        t        )        =        A        (        t        )        A        (        s        )        .            Satisfaction of   (  1  ) implies that the solution of      x    ′    =  A  (  t  )  x  ,        x  (  0  )  =      ξ    0    ,In our case  A  (  t  )  =      (                            cos          ⁡          t                          −          sin          ⁡          t                                      sin          ⁡          t                          cos          ⁡          t                      )    ,then      ∫    0    t    A  (  s  )    d  s  =      (                            sin          ⁡          t                          cos          ⁡          t          −          1                                      1          −          cos          ⁡          t                          sin          ⁡          t                      )    .Next we use use the fact that  exp  ⁡      (                            a                          −          b                                      b                          a                      )    =      (                                                  e            a                    cos          ⁡          b                          −                                e            a                    sin          ⁡          b                                                            e            a                    sin          ⁡          b                                                e            a                    cos          ⁡          b                      )    ,and finally we obtain that  x  (  t  )  =      (                                                  e                          sin              ⁡              t                                cos          ⁡          (          1          −          cos          ⁡          t          )                          −                                e                          sin              ⁡              t                                sin          ⁡          (          1          −          cos          ⁡          t          )                                                            e                          sin              ⁡              t                                sin          ⁡          (          1          −          cos          ⁡          t          )                                                e                          sin              ⁡              t                                cos          ⁡          (          1          −          cos          ⁡          t          )                      )          ξ    0    .

How to solve this system of ODE's? [ <mtable rowspacing="4pt" columnspacing="1em">

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