Differential equations systemI have found the following example in one of my courses but I don't have any similar exercises resolved so I would like to know how to solve this:The differential equations system is the following: x 1 ′ = 3 x 1 − 2 x 2 + e t x 2 ′ = 2 x 1 − x 2 + 2 e 2 t a) Write the system in the matriceal form x ′ = A x + b ( t )b) Determinate the solutin of the system.

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Differential equations systemI have found the following example in one of my courses but I don&#039;t have any similar exercises resolved so I would like to know how to solve this:The differential equations system is the following:      x    1    ′    =  3      x    1    −  2      x    2    +      e    t        x    2    ′    =  2      x    1    −      x    2    +  2      e          2      t      a) Write the system in the matriceal form       x    ′    =  A  x  +  b  (  t  )b) Determinate the solutin of the system.

smh3402en · Accepted Answer

Step 1Writing the system in matrix form is nearly identical to writing any linear (non-differential-equation) system in matrix form:      {                            a          x          +          b          y          =          c                                      d          x          +          e          y          =          f                            ⟺                                (                                                    a                                            b                                                                    d                                            e                                              )                ⏟                            A                                            (                                                    x                                                                    y                                              )                ⏟                            x              =                              (                                                    c                                                                    f                                              )                ⏟                            b            The main components are the coefficient matrix A, what I&#039;ll call the solution vector x, and what I&#039;ll call (for lack of a better name) the right hand side vector b.In the case of a system of linear ODEs, the same       A        x    =      b   framework can be used. For example, you can rewrite the following general system as:      {                            a                      x            1                    +          b                      x            2                    =                                                    x                1                                      ′                                                d                      x            1                    +          e                      x            2                    =                                                    x                2                                      ′                                      ⟺        (                            a                          b                                      d                          e                      )        (                                        x            1                                                            x            2                                )    =      (                                                                      x                1                                      ′                                                                                          x                2                                      ′                                )    =            (                                                  x              1                                                                          x              2                                          )        ′  Step 2This extra term(s) is what your question refers to as b(t).For your particular system, the matrix form would simply be      (                            3                          −          2                                      2                          −          1                      )        (                                        x            1                                                            x            2                                )    +      (                            1                                      0                      )        e    t    +      (                            0                                      2                      )        e          2      t        =            (                                                  x              1                                                                          x              2                                          )        ′

Differential equations system. The differential equations system is the following: x_1' = 3x_1 - 2x_2 + e^t

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