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Its roots are real and multiple     1.
1 2
decisions parts will be written down:
x
x
y  e 1  x  e ; y  хe 1  x  хe .
1 2
Common decision is such therefore:
x
x
x
y  С e  С xe  С (  С x) e .
1 2 1 2

Note 5.1 If all roots characteristic equation (5.2) for LHDE
(5.1) levels between itself and are evened  the fundamental
0
system of decisions can be taken as:


e
e 0  x , хe 0  x ,..., x n 1  0 x ,

Then General decision will be written down


e
y  C e 0  x  C хe 0  x  ... C x n 1  0 x (5.10)
1 2 n

5.1.4 The roots of Characteristic Equation Are Complex
Conjugate

We will consider equation of the 2-nd order at first (5.6). If
discriminator of characteristic equation (5.7) is negative number
will complex conjugate by its solutions:      і ,     і 
1 2
а а 2
where    1 ,  а  1 .
2
2 4
Partial decisions of equations (5.6) will be written as:

y  e (  і x ) y ;  e (  і x ) (5.11)
1 2

These complex functions with respect of real argument
satisfy the equation (5.6). We will consider some complex
function of real argument:
y  u (x ) iv (x ) (5.12)
We will show that it satisfies the equation (5.6), and let this
equation will be accordingly actual that imaginary parts –
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