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3
4
*
0 y  Ax  Bx  Cx 2 ,
3
0 (y  4Ax  3Bx  2Cx ,
2
*
) 
-3 (y  12Ax  6Bx  2C ,
2
*
) 
0 (y   24Ax  6B ,
*
) 
1 (y * IV  24A ,
)

2
y *IV  3y *   36Ax  18Bx  6C  24A  9x 2 .

Equating coefficients at the identical degrees х in left and
right parts of the last identity, we have the system of linear
algebraic Equations for determination А, В, С:

2
х -36А=9,
х 1 -18В=0,
0
х -6С+24А=0,
1 2
2
*
So А=-1/4, B=0, C=-1. So, y  x ( x  ) 1 . Consequently,
4
1 2
*
2
y  x ( x  ) 1 is partial decision of given LNDE.
4

A function is the common decision of equation
*
4
y  ~ y  c  c x  c e 3x  c e  3x  1 x  x 2 . 
y 
1 2 3 4
4

Example 6.3 To find partial decision of equation
y   2y   y  4 (sin x  cos ) x , which satisfies initial

conditions (y ) 0  , 2 ) 0 (  y  , 2 ) 0 (   y   1.

 We have the LNDE third order with constant coefficients
with right part of the special kind. Proper to him LHDE:

    y 2  y   y  0.

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