Finding general solution for a nonhomogeneous system of equations { x 1 ′ = x 2 + 2 e t x 2 ′ = x 1 + t 2 I want to find the general solution for it. I started by finding the general solution for the homogeneous equations: ( x 1 x 2 ) = C 1 ( e t + e − t e t − e − t ) + C 2 ( e t − e − t e t + e − t ) Now I need to find a "specific" solution for the nonhomogeneous equations but I have problems applying the method in which I make constants C 1 and C 2 a variable.

Question

Finding general solution for a nonhomogeneous system of equations      {                                        x            1            ′                    =                      x            2                    +          2                      e            t                                                            x            2            ′                    =                      x            1                    +                      t            2                                  I want to find the general solution for it. I started by finding the general solution for the homogeneous equations:      (                                        x            1                                                            x            2                                )    =      C    1        (                                        e            t                    +                      e                          −              t                                                                        e            t                    −                      e                          −              t                                            )    +      C    2        (                                        e            t                    −                      e                          −              t                                                                        e            t                    +                      e                          −              t                                            )  Now I need to find a &quot;specific&quot; solution for the nonhomogeneous equations but I have problems applying the method in which I make constants       C    1   and       C    2   a variable.

Keegan Barry · Accepted Answer

System is equivalent with      x    2    =      x    1    ′    −  2      e    t    ,      x    2    ′    =      x    1    +      t    2  or      x    2    =      x    1    ′    −  2      e    t    ,      x    1    ″    =      x    1    +      t    2    +  2      e    t    .Firstly, by using a method of variations of constants, we will solve the second equation:      x    1    ″    −      x    1    =      t    2    +  2      e    t    ,(after that, it will be easy to determine       x    2   from the first equation). Solution of homogenuous equation is       x    1    =      C    1        e    t    +      C    2        e          −      t      , so we have to solve the following system:      C    1    ′        e    t    +      C    2    ′        e          −      t        =  0  ,      C    1    ′        e    t    −      C    2    ′        e          −      t        =      t    2    +  2      e    t    ,where       C    1  ,       C    2   are functions of   t. It is easy to get that      C    1    ′    =  1  +            t      2        2        e          −      t        ,      C    2    (  t  )  =      C    2    −            e              2        t              2    −            e      t        2    (      t    2    −  2  t  +  2  )  .Finally,      x    1    =  (      C    1    −      1    2    )      e    t    +      C    2        e          −      t        +  t      e    t    −      t    2    −  2  ,      x    2    =  (      C    1    −      1    2    )      e    t    −      C    2        e          −      t        +  (  t  −  1  )      e    t    −  2  t  .

woowheedr · Accepted Answer

If       x    1    ,      x    2   are your solutions to the homogeneous equation (it doesn&#039;t actually matter what they are), look for solutions to the inhomogeneous equation in the form       y    i    =      v    i        x    i    .. If you try that, you will have simple equations for       v    1    ,      v    2    .

Finding general solution for a nonhomogeneous system of equations { <mtable columnal

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