Is this system of differential equations linear? x ˙ = x + y + 5 y ˙ = x − yIs this system linear or nonlinear differential equations?

Question

Is this system of differential equations linear?            x      ˙        =  x  +  y  +  5            y      ˙        =  x  −  yIs this system linear or nonlinear differential equations?

Nancy Phillips · Accepted Answer

Explanation:The system is linear, yes. It can be written as      [                                                      x              ˙                                                                                      y              ˙                                            ]    =      [                            1                          1                                      1                          −          1                      ]        [                            x                                      y                      ]    +      [                            5                                      0                      ]

Medenovgj · Accepted Answer

Step 1You could write your system of equations as      (                                                      x              ˙                                                                                      y              ˙                                            )    =      (                            1                          1                                      1                          −          1                      )        (                            x                                      y                      )    +      (                            5                                      0                      )  and solve it this way:1. Solve your homogeneous system. In this case, it would be                    x        →              h    =      (                                        c            1                                e            t                    +                      c            2                                e                          −              t                                                            −          2                      c            2                                e                          −              t                                            )    .2. Solve for your particular solution.2.1. Rewrite your system as             x      →        (  t      )    ′    =  A            x      →        (  t  )  +            g      →       where             g      →        =      (                            5                                      0                      )  . Then guess a particular solution. In this case,                     x        →              p    =            a      →        .2.2. Take the derivative,                    x        →              p    =  0  ,and plug things in.  0  =  A            a      →        +            g      →        A            a      →        =  −            g      →        .2.3. Solve the system.      (                            1                          1                                      1                          −          1                      )        (                                        a            1                                                            a            2                                )    =      (                            −          5                                      0                      )        {                                        a            1                    +                      a            2                    =          −          5                                                  a            1                    −                      a            2                    =          0                              a    1    =  −  2.5   and       a    2    =  −  2.5            a      →        =      (                            −          2.5                                      −          2.5                      )    .Step 2So, the solution is            x      →        (  t  )  =      (                                        c            1                                e            t                    +                      c            2                                e                          −              t                                −          2.5                                      −          2                      c            2                                e                          −              t                                −          2.5                      )    .

Is this system of differential equations linear? Suppose I have a system of differential equations like below: dotx = x + y + 5, doty = x - y

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