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 2  2  2  0, and   1  i .
2 , 1

~ x
Common decision of LHDE is y  e (C 1 cos x  C 2 sin ) x .
We search the partial decision LNDE in a kind
2
y * Ax  Bx  C .
2
Right part f (x )  x corresponds to special type
x

f ( x)  Q ( x) e of right part and  s , 2  0 are not the solves
s
of characteristic equation.
We shall find y , * y * and put y *, y , * y * in given
LNDE.
How agreed already, for comfort of calculations we will
write expressions for the function of *y and its derivative in
separate lines and to the left after a vertical hyphen to write
down the proper coefficients, with which they enter to left part
of given LNDE. Executing the increase of expressions on these
coefficients and subsequent addition, we will get:

2
2 y*  Ax  Bx  C

 2 y*  2 Ax  B
 
1 y*  2 A
2
y *   2y *   2y *  2Ax  x 2 ( B  4A )  2 ( C  2B  2A )  x 2

We equate coefficients at identical degrees of x :
x 2 2 A 1
x 1 2 B 4 A 0

x 0 2 C 2 B 2 A 0
1 1
From where A  , B  , 1 C  .
2 2

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