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As e x   0, the last equality will be executed, if

 n  a  n 1   ...  a   a  0. (5.3)
1 n  1 n

For determination  got, thus, algebraic equation of the n-
th degree. It is called characteristic equation LHDE (5.1) with
constant coefficients.
Equation (5.3) turns out with (5.1) formal replacement of
)
(i
derivative y degrees ,  i i 1 , 0 ,... n (y ) 0 (  ) y .
As known, algebraic equation of the n-th degree has n roots
actual or complex. If every root so much one time, what his
multiple.
Such cases are possible:
1) the roots of characteristic equation are actual and
different;
2) among the roots of characteristic equation there are the
actual multiple;
3) among the roots of characteristic equation there are the
simple complex;
4) among the roots of characteristic equation there are the
complex multiple;
We will consider each of these cases.

5.1.2 The Roots of Characteristic Equation Real and
Simple

We will consider a case, when all roots of characteristic
equation are actual and different (speak yet - „simple"), we get
the fundamental system of decisions LHDE (5.1) :
x
e , e  2 x ,..., e n  x (5.4)
 1
Decision (5.4) content is evened, all of they are linearly
independent, consequently, the common decision LHDE is
found (5.1) :
y  С e  1 x  С e  2 x  .....  С e n  x (5.5)
1 2 n

Example 5.1 To find the common decision of equation

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