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Consequently, the sought decision after has a kind:
y 1 x (  e ) 1 x  2 cos x 2 sin x . 

6.3 Linear Heterogeneous Differential Equations of the
Second Order With Constant Coefficients

If n  2 we have the LNDR of the second order with

 
constant coefficients y  y p  qy  f (x ) , where p, are
q
numbers, (xf )  0 is a function.
Clearly, that it is a case part of the LNDE of the n-th order
considered higher, and which it is yet simpler to analyse. Like to
previous two special types of function (xf ) are examined.

s
I f ( x)  Q ( x) e  x  ( A x  A x s 1  ...  A e )  x ,
s 0 1 s

where Q (x ) – polynomial of degree with the set coefficients.
s
1) If  is not the root of characteristic equation of LHDE,
we search partial decision *y as:

s
y*  P ( x) e  x  ( B x  B x s 1  ...  B e )  x ,
s 0 1 s

where P (x ) – polynomial of degree with indefinite
s
coefficients;

2) If  – single (simple) root of characteristic equation, we
search the decision part *y in a kind:
s
y*  xP ( x) e  x  x( B x  B x s 1  ...  B e )  x ,
s 0 1 s
where P (x ) – polynomial of degree with indefinite
s
coefficients;

3) If  – double root of characteristic equation, we search
the decision part *y in a kind:
s
2
y*  x 2 P ( x) e  x  x ( B x  B x s 1  ... B e )  x ,
s 0 1 s
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