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(a  ) r a a
1 2 3
  b (b  ) r b  0
1 2 3 (7.11)
c c (c  ) r
1 2 3

Equation (7.11) is algebraic equation relatively r . Name
him characteristic equation of the system of linear differential
equations with constant coefficients.
There is this equation of the third degree in our case.
We will consider a case, when equation (7.11) has three
actual different roots r , ,r r . By turns we put each of these
1 2 3
roots r in the system (7.10) and determine unknowns
i
 ,  , .
i i i
It is possible to lead to that the common decision of the
system of differential equations (7.8) in this case has a kind:
2 r t
1 rt
x  C  e  C  e  C  e 3 r t ;
1 1 2 2 3 3
1 rt
2 r t
y  C  e  C  e  C  e 3 r t ;
1 1 2 2 3 3 (7.12)
1 rt
2 r t
z  C  e  C  e  C  e 3 r t .
1 1 2 2 3 3
In a vector form we will write down the common decision
so:
   
( )Y t  C Y ( )t  C Y ( )t  C Y ( )t (7.13)
1 1
2 2
3 3

where

 x ( )t    1    2    3 
   
Y ( )t   y ( ) , ( )t  Y t     e 1 rt ,Y ( )t     e 2 r t , ( )Y t     e 3 r t .
  1  1  2  2  3  3 
       
 z ( )t    1    2    3 
(7.14)
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