Solve the system { x 1 ′ ( t ) = 3 x 1 ( t ) − 2 x 2 ( t ) + e 2 t , x 1 ( 0 ) = a x 2 ′ ( t ) = 4 x 1 ( t ) − 3 x 2 ( t ) , x 2 ( 0 ) = b by using the method of diagonalization. Substitution x = T z

Question

Solve the system      {                                        x                      1            ′                    (          t          )          =          3                      x                      1                    (          t          )          −          2                      x                      2                    (          t          )          +                      e                          2              t                                ,                      x                      1                    (          0          )          =          a                                                          x                      2            ′                    (          t          )          =          4                      x                      1                    (          t          )          −          3                      x                      2                    (          t          )          ,                      x                      2                    (          0          )          =          b                                by using the method of diagonalization. Substitution   x  =  T  z

rynosluv101swv2s · Accepted Answer

We have:      x    ′    (  t  )  =      [                                        x            1            ′                    (          t          )                                                  x            2            ′                    (          t          )                      ]    =  A  x  (  t  )  +  f  (  t  )  =      [                            3                          −          2                                      4                          −          3                      ]    x  (  t  )  +      [                                        e                          2              t                                                            0                      ]    ,  x  (  0  )  =      [                            a                                      b                      ]  Diagonalize (not always possible) the matrix   A and arrive at:  A  =  T  D      T          −      1        =      [                            1                          1                                      2                          1                      ]        [                            −          1                          0                                      0                          1                      ]        [                            −          1                          1                                      2                          −          1                      ]  We want to use   x  =  T  z    ⟹    z  =      T          −      1        x    ⟹        z    ′    =      T          −      1            x    ′  , which will allow us to decouple the equations. We have:-       x    ′    =  A  x  +  f  (  t  )-       T          −      1            x    ′    =      T          −      1        (  T  D      T          −      1        )  x  +      T          −      1        f  (  t  )-       z    ′    =  D  z  +      f    ^    (  t  )  ,         where           f    ^    (  t  )  =      T          −      1        f  (  t  )  ,     z  (  0  )  =      T          −      1        x  (  0  )These equations are now decoupled and more easily solved. We have:      z    1    ′    (  t  )  =  −      z    1    (  t  )  −      e          2      t        ,      z    1    (  0  )  =  −  a  +  b        z    2    ′    (  t  )  =      z    2    (  t  )  +  2      e          2      t        ,      z    2    (  0  )  =  2  a  −  bYou can now find   z  (  t  ) and then write:  x  (  t  )  =  T  z  (  t  )The solution will be:  x  (  t  )  =      [                                        x            1                    (          t          )                                                  x            2                    (          t          )                      ]    =      [                                        e                          −              t                                            (            a                          (              2                              e                                  2                  t                                            −              1              )                        +            b                          (              −                              e                                  2                  t                                            )                        +            b            )                                                            e                          −              t                                            (            2            a                          (                              e                                  2                  t                                            −              1              )                        −            b                          (                              e                                  2                  t                                            −              2              )                        )                                ]

Solve the system { <mtable rowspacing="4pt" columnspacing="1em"> <mtr>

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