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where P (x ) – polynomial of degree with indefinite
s
coefficients.
We will notice that a number  can be a zero, in this case
right part of LNDE is a polynomial. All conclusions remain
suitable.
ІІ. f ( x)  e  x Q ( x cos) x   P ( x sin)  x  .
s 1 s 2

Where Q (x ), P (x ) - polynomials of degrees s ,s
1 s 2 s 1 2
accordingly with the set coefficients max( s , s )  s .
1 2
1) If a number    i is not the root of characteristic
Equation, we search the decision part *y in a kind:
y*  e  x u ( x cos)  x  ( x sin)  x  ,
s s
where u (x ), (x ) – polynomials of degree with indefinite
s s
coefficients;
2) If a number    i is a root of characteristic equation,
we search the decision part *y in a kind:
y*  xe  x u ( x cos)  x  ( x sin)  x  ,
s s
where u (x ), (x ) – polynomials of degree with indefinite
s s
coefficients.
Thus all conclusions remain suitable and in the cases when
  0 or in right part absent element, that contains cos x  or
sin x  .
We pass to solving of examples.

Example 6.4 To find common decision of equation:
2
y   2y   2y  x .

 We have LNDE with constant coefficients with right part
of the special kind. According to the LHDE:

   y 2  y  2 y 0 .

We make characteristic equation:

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