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
If to take into account ( )y x the structure, we will rewrite
0
the last system of equations so:

 y * ( )x  C y ( )x  C y 2 ( )x 0  y 0 ,

0
2
1 1
0
 *
  y ( )x  C y ( )x  C y 2 ( )x 0  y 1 ;
0
1 1
2
0

or

 C y ( )x  C y 2 ( )x 0  y  y * ( ) ,x 0

1 1
2
0
0
 *
  C y ( )x  C y 2 ( )x 0  y  y ( ) .x 0
2
0
1
1 1

The determinant of this system, is a determinant of linear
independent decisions LNDE - 2, therefore does not equal a
zero. And, consequentlyC and C are determined simply. A
1 2
theorem is proved. 
This theorem gives us one of methods of integration
LNDE. He can be named the method of selection of partial
decision.
He consists in the following :
а) we find some decision part LNDE ;
b) we find the common decision of proper LHDE.
Their sum will be the sought common decision after LNDE.

Example 4.2 To find the common decision of equation
y
y   3x (x  0) .
x

 Type of coefficients and right part of the given equation
„prompts" that the partial decision of this equation needs to be
m
searched in the class of functions x . By direct substitution we
*
3
make sure, that one of decisions of the set equation is y  x .
We will untie the proper homogeneous equation
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