Page 68 - 4549

Equation (4.8) is differential equation with the separated
variables. We separate variables and, integrating find
dW
 dxa 1  0.
W
x
ln W  a dx  ln C .
 1
x 0
x
 a dx
 1
W  Ce x 0 .

Taking into account an initial condition, we get
x
 a dx
 1
W ( x) W ( x e ) x 0 (4.10)
0

This formula is named a formula Ostrogradsky - Liouville.
It is visible from her, that for, that x  ( , )a b W ( )x  and it
0
was needed to lead to.
Theorem 4.3 If functions y y , , y are linearly
,
1 2 n
independent by the upshots LHDE-n (4.2) on ( , )a b W ( )x  ,
0
for all x  ( , )a b .
We will prove a theorem „from opposite" (at n  ). Lets
2
y ( ) ,x y ( )x – linearly independent upshots of Equation (4.4)
1 2
and there is a point x  ( , )a b such, that W ( )x  0 . We will
0 0
make such system of equations :

 C y ( )x  C y ( )x  0 ,
 1 1 0 2 2 0 (4.11)
 C y ( )x  C y 2 ( )x 0  0 ,
2
0
1 1

in which are unknown - C and C . The system (4.11) is the
1 2
system of linear homogeneous algebraic equations, its
determinant   W ( )x 0  0. Therefore function
Y ( )x  C y ( )x  C y ( )x
1 1 2 2
66