Set of ODE: how can I solve it?I want to solve this system, but I have never solved a system of ODE, can you help me? { d A d t = − a A d B d t = a A − b B d C d t = b B I have solved the first equation: A ( t ) = A 0 e − a t

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Set of ODE: how can I solve it?I want to solve this system, but I have never solved a system of ODE, can you help me?      {                                                      d              A                                      d              t                                =          −          a          A                                                                d              B                                      d              t                                =          a          A          −          b          B                                                                d              C                                      d              t                                =          b          B                        I have solved the first equation:  A  (  t  )  =      A    0        e          −      a      t

Danika Rojas · Accepted Answer

After putting solution of first equation in 2nd, we get,            d      B              d      t        +  b  B  =  a      A    0        e          −      a      t      This is a linear differential equation in   B which can be solved by multiplying integrating factor   (      e          b      t        ) on both sides.EDIT:      e          b      t                  d      B              d      t        +  b      e          b      t        B  =  a      A    0        e          (      b      −      a      )      t      Now L.H.S. is equal to             d      (      B              e                  b          t                    )              d      t      Hence,                    (k is some constant)                    B                  e                      b            t                          =        a                  A          0                ∫                  e                      (            b            −            a            )            t                          d        t        =        a                  A          0                                      e                          (              b              −              a              )              t                                            (            b            −            a            )                          +        k            which gives   B  =  a      A    0              e              −        a        t                    (      b      −      a      )        +  k      e          −      b      t      Now, if we add three ODE&#039;s, we have,                              (                              c            0                     is some constant)                                                  d            (            A            +            B            +            C            )                                d            t                          =        0                ⟹                A        +        B        +        C        =                  c          0                    Now put   A,  B to get   C.

antennense · Accepted Answer

If you solve the first two equations as Avatar suggested you don&#039;t really need to substitute into   (  3  ) to find   C since   (  1  )  +  (  2  )  +  (  3  ) gives      d          d      t        (  A  +  B  +  C  )  =            d      A              d      t        +            d      B              d      t        +            d      C              d      t        =  0  ⟺  A  +  B  +  C  =  const  ⟺  C  =  const  −  A  −  B  .

Set of ODE: how can I solve it? I want to solve this system, but I have never solved a system of OD

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