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We will consider concrete examples, which illustrate the
considered method of selections of partial decision LNDE.

Example 6.2 To find the common decision of equation
 
y IV  3y  9x 2 .

 We have the LNDE of the 4-th order with constant
coefficients and special type of right part.
We make characteristic Equation, we find its solves of
~
fundamental system of decisions and common decision y of
proper LHDE:
 4  3 2  , 0  2 ( 2  ) 3  , 0     , 0    ; 3
1 2 4 , 3
y  e 0x  , 1 y  xe 0x  , x y  e 3x , y  e  3x ;
1 2 3 4
~ 3x  3x
y  c  c 2 x  c 3 e  c 4 e .
1

Right part of given LNDE is polynomial, that is case part of
function of the special kind. It is needed to check in this case,
whether there is a number the zero by the root of characteristic
equation. Indeed, characteristic equation has a double root the
zero:     , 0 therefore we search *y the partial decision
1 2
of LNDR y* in a kind:
* 2 2
y  x (Ax  Bx  C ),

as right part is polynomial of exactly the 2-nd degree.
IV
*
Putting y )( * ( ,   y ) in given LNDE, we will get an identity for
finding of indefinite coefficients. Here and farther 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:

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