Let X ( t ) , A , B , C be matrices and A , B , C are constant matrices. Does the following linear differential equation have a closed form solution? d X ( t ) d t = A + B X C .

Question

Let   X  (  t  )  ,  A  ,  B  ,  C be matrices and   A  ,  B  ,  C are constant matrices. Does the following linear differential equation have a closed form solution?            d      X      (      t      )              d      t        =  A  +  B  X  C  .

Zayden Wiley · Accepted Answer

First, note that            d                      d            t            [          ∑              k        ≥        0                            t                  k                            k        !                    B              k              M          C              k              ]    =  B      (          ∑              k        ≥        1                            t                  k          −          1                                      (          k          −          1          )                !                    B              k        −        1              M          C              k        −        1              )    C  =  B      (          ∑              k        ≥        0                            t                  k                            k        !                    B              k              M          C              k              )    C  .Now, suppose   B and   C are nonsingular. Motivated by the above, take  X  (  t  )  =      ∑          k      ≥      0                  t              k                    k      !            B          k            (    X    (    0    )    +          B              −        1              A          C              −        1              )        C          k        −      B          −      1        A      C          −      1      as a solution. You can verify that the above is a solution directly:      X          ′        (  t  )  =  B      [          ∑              k        ≥        0                            t                  k                            k        !                    B              k                    (      X      (      0      )      +              B                  −          1                    A              C                  −          1                    )              C              k              ]    C  =  A  +  B  X  (  t  )  C  .

Let X ( t ) , A , B , C be matrices and A , B , C

Answered question

Answer & Explanation

New Questions in Algebra I