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is the decision of linear homogeneous differential equation (see
a theorem 4.1). But the system of terms (4.11) means that
Y ( )x  0 , Y  ( )x  0, that is Y ( )x fulfils the same
0 0
conditions, as well as ( )Y x  .
0
Because of unique of decision of LHDE, we come to
contradiction C y ( )x  C y ( )x  0: at C , C .
1 1 2 2 1 2
Consequently, our supposition, that point x  ( , )a b , in
0
whichW ( )x  0 wrong, that is a Wronski-Determinante W ( )x
0
does not grow into a zero in any point on ( , )a b that and it was
needed to lead to.

4.3.2. Structure of Common Decision of LHDE

We pass now to the question about the structure of common
decision LHDE.
Definition 4.7 Any aggregate n linearly independent on
,
( , )a b of partial decisions of y y , , y linear homogeneous
1 2 n
differential equation of the n-th order (4.2) is named the
fundamental system of decisions of this equations.
It is possible to show that for every LHDE - n with
continuous coefficients the fundamental system of decisions
exists. Moreover, if Equation has one fundamental system of
decisions even, it has their endless great number. We will pay
attention, that amount linearly independent decisions of LHDE
- n, from which the fundamental system of decisions of this
Equation is coincides with the order of this Equation.
Theorem 4. 4 (about the structure of common decision of
LHDE - n).
,
If y y , , y – fundamental system of decisions of
1 2 n
LHDE - n (4.2), function
y  C y ( )x  C y ( )x   C y ( )x (4.12)
1 1 2 2 n n

where C , C , , C – arbitrary constant, is the common
1 2 n
decision of this equation.

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