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
 We  consider, consequently, that vector-founctsii
Y ( ) ,t Y ( ) ,t Y ( )t form the fundamental system of decisions
1 2 3
of the set system of differential equations.
Note 7.2 As a determinant of the system (7.11) equals a
zero, even one of his lines is linear combination of other. And
to that in the homogeneous system (7.10) is „superfluous"
equation. This fact can be used yet and so: one of variables
 i ,  i , it is possible to take arbitrary, for example, 1 (one
i
only requirement : a coefficient near this variable at substitution
in the system (7.10) r  must not be zero).
r
i

Example 7. 3 To find Cauchy task:

 dx
 dt  2x  2 ;y 1

 x (0)  2 , (0)y  .
dy 2 (7.15)
  x  3 ;y
  dt
 We search the common decision of the system at first.
In our example:
 a a 2   2 2 
A   1     .
 b b 2   1 3 
1
rt
Lets x   e rt , y   e  one of decisions. Then for
,
we get finding   system of equations:

 (a  ) r   a   0,  (2 r )  2  0,
2
1
  
b  (b  ) r   0,   (3 r )  0, (7.16)
 1 2 
r  the root of characteristic equation:

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