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10.4 Exponential Matrix Solutions  317




                                 EXAMPLE 10.14
                                        Let

                                                                             2  −5
                                                                        A =         .
                                                                             1   4
                                        MAPLE returns the exponential matrix
                                                                                        5
                                                                          1           − sin(2t)
                                                               3t cos(2t) − sin(2t)
                                                          At
                                                                                        2
                                                         e = e        1  sin(2t)  cos(2t) − sin(2t).
                                                                          2
                                                                                           1
                                                                      2                    2

                                        This is a fundamental matrix for the system X = AX. We could also solve this system by
                                        diagonalizing A, which has eigenvalues 3 ± 2i.
                                 EXAMPLE 10.15
                                        Let
                                                                          ⎛         ⎞
                                                                           2  1   0
                                                                      A = 0   3 −2 ⎠  .
                                                                          ⎝
                                                                           0  1   1
                                        Then
                                                                ⎛                                ⎞
                                                                 1sin(t) − cos(t) + 1  2(cos(t) − 1)
                                                          At
                                                         e = e 2t  ⎝ 0  sin(t) + cos(t)  −2sin(t)  ⎠  .
                                                                 0        sin(t)     cos(t) + sin(t)
                                        This is a fundamental matrix for X = AX.

                                           The fundamental matrix  (t) = e  At  is sometimes called a transition matrix for the system

                                        X = AX. This is a fundamental matrix satisfying  (0) = I n .
                                        Variation of Parameters and the Laplace Transform
                                        We will briefly mention a connection between the Laplace transform, the exponential matrix and
                                        the variation of parameters method for finding a particular solution   p (t) of X = AX + G,in

                                        which A is an n × n real, constant matrix.
                                           The variation of parameters method is to write   p (t) =  (t)U(t), where


                                                                              −1
                                                                    U(t) =     (t)G(t)dt
                                        and  (t) is a fundamental matrix for X = AX.

                                           Write U(t) as a definite integral with s as the variable of integration:
                                                                            t
                                                                              −1
                                                                    U(t) =    (s)G(s)ds.
                                                                           0
                                        Then
                                                                               t
                                                                                 −1
                                                                   p (t) =  (t)    (s)G(s)ds
                                                                             0
                                                                           t
                                                                                −1
                                                                      =     (t)  (s)G(s)ds.
                                                                         0
                                                                                                              At
                                        In this,  (t) can be any fundamental matrix for the system. If we choose  (t) = e , then
                                         −1
                                          (t) = e −At  and
                                                                   −1
                                                                          At −As
                                                              (t)  (s) = e e   = e A(t−s)  =  (t − s).

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                                   October 14, 2010  20:32  THM/NEIL   Page-317        27410_10_ch10_p295-342
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