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There are such properties of Wronskian.

,
Theorem 4.1 If functions y y , , y linearly dependent
1 2 n
on( , )a b their Wronski-Determinante of equals a zero on this
interval.
At n=2 we have y  ky from here
2 1

y 1 y 2 y 1 k y 1 y 1 y 1
W ( )x    k  0 ,
y ' y ' y ' ky ' y ' y '
1 2 1 1 1 1

that and it was needed to lead to.
,
Theorem 4.2 Let functions y y , , y be the upshots
1 2 n
of LHDE – n (4.2) on ( , )a b . If for some
0
x  ( , )a b W ( )x  0 , W ( )x  for all x  ( , )a b .
0 0
As well as higher, we will consider a case n=2. Lets
y  y – decisions LHDE (4.3).
1 2
Then
 
y  a y  a y  0
1
1
1
1
2

 
y  a y  a y  0
2 1 2 2 2
We will increase the first equation on y , and second - on
2
y . We will take away from got the second equality first, will
1
get :
   
y y  y y  a ( yy  y y )  0 .
1 2 2 1 1 1 2 2 1

If to take (4.7) into account, the last equality will acquire a
kind :
W (  x )  Wa (x )  0 . (4.8)
1

We will find the decision of this equation subject to the
condition
W ( )x 0  W  0 . (4.9)
0
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