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 C  )(x y  C  ( yx )  ,0
 1 1 2 2 (6.8)
 C 1  )(x 1  y  C 2  (x )  y 2  (xf ).

As a determinant of this system   W ( )x  (Wronskian
0
of linearly independent decisions of LHDE) system of equations
(6.6), (6.7) has the upshots.
 
Solving of the system (6.8) we will find С (х ),С (х )
1 2
and than C ( ),x C ( )x .
1 2
 ~   ~
С (х )  С 1  (х )dx  С (х ),С (х )  С 2  (х )dx  С (х ) (6.9)
1 1 2 2

As we search the partial decision of LNDE only during this
integration arbitrary constant it is possible to scorn.
Putting the found values C ( )x and C ( )x in (6.5) we shall
1 2
get obsessed partial decision.

Example 6.1 To untie a task Cauchy
1
2
y  y  , y (0) 1 , y (0)  .

1 e x

 a) At first we will search the common decision of the proper
homogeneous equation y  y  0.
Characteristic equation:
 2    , 0
 (  ) 1  , 0
 1  , 0  2  1

x
partial decision: y  1, y  e they are linearly independent.
1 2
Consequently, the common decision of homogeneous equation
has a kind :
x

y  C  C e .
1 2

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