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 2  6  13  0
D 36  4  13  16
6  i 4 6  i 4
 1   3  2 i; 2   3  i 2
2 2

Its roots complex and simple:   , 3   2 .
Partial decisions will be written down:
3
x
3
x
y  e cos 2 x; y  e sin x 2
2
1
The common decision is such therefore:
y  C y  C y  e 3x (C cos 2x  C sin 2x ).
1 1 2 2 1 2

Note 5.1 If for LHDE with the constant coefficients of 2к
order characteristic equation has к root of multiple complex
number    і ,    і . System of functions are:

e
e  x cos x ;хe  x cos x ;......., х к 1  x cos x
(5.16)
e
e  x sin x ; хe  x sin x ;........, х к 1  x sin ; x 

The fundamental system of decisions is formed. General
decision will be written down as:

y  (( C  C 2 х ..... С n x k 1 ) cos x   ( C  C 2 х ..... С n x k 1 ) sin x  e )  x
1
1
(5.17)

5.1.5. The roots of Characteristic Equation are
Arbitrary

We will consider a general case, when among the roots of
characteristic equation (5.2) there are real simple, real multiple,
complex conjugate both simple and multiple. Summing up all
said higher, it is possible to formulate such rule:
To find general decision LHDE it is needed with the
constant coefficients of the n-th order (5.1):
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