Set of ODE: how can I solve it?I want to solve this system, but I...

Addison Trujillo

Addison Trujillo

Answered

2022-07-01

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

Answer & Explanation

Danika Rojas

Danika Rojas

Expert

2022-07-02Added 9 answers

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'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

antennense

Expert

2022-07-03Added 7 answers

If you solve the first two equations as Avatar suggested you don'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 .

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