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We will show that if y  y (x ) and y  y (x ) is
1 1 2 2
decisions of equation (4.3), a function
y (x )  C y  (x ) C  y (x ) also will be decision of this
1 1 2 2
equation at any values С and С . Indeed, as y and y -
1 2 1 2
decisions (4.3)
   
y  a y  a y  0 and y  a y  a y  0 .
1 1 1 2 1 2 1 2 2 2
Taking into account these to equality, we will get

(C  y C  y )   a (  C  y C  y )   a (  C  y C  y ) 
1 1 2 2 1 1 1 2 2 2 1 1 2 2
   
C  y C  y  a (  C  y C  y )  a (  C  y C  y ) 
1 1 2 2 1 1 1 2 2 2 1 1 2 2
   
C (y  ya  a y ) C (y  ya  a y )  C  0C  0  0
1 1 1 1 2 1 2 2 1 2 2 2 1 2

And it means that a function (xy )  C y  (x ) C  y (x )
1 1 2 2
is a decisions of equation (4.3), that and it was needed to lead to.
Consequently, if y , y ,..... y - decisions of equation (4.2),
1 2 n
a function, y( x ) C 1 y  1 ( x ) C 2 y  2 ( x .....)  С n y  n where
C , C ,..... С – arbitrary constant, also is a decision of equation
1 2 n
(4.2).

4.2. Linear Dependence and Linear Independence of
Systems of Functions

Definition 4.3 Linear function y y , , y is named
,
1 2 n
combination of functions

Y  C y   C y (4.5)
1 1 n n

it is formed from the set functions by linear operations, that is
operations of increase on a number and addition.
,
Definition 4.4 System of functions y y , , y certain on
1 2 n

 ,a b is named linearly dependent on this interval, if it is
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