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We search the partial decision LNDE in a kind
* y  x 2 e x (Ax  ) B , because right part of the set equation
x
x

f ( x ) xe has a kind f ( x)  Q ( x) e at s  , 1   1 thus a
s
number   1 is the root of characteristic equation of multiple
2
r  2 . The last predetermines the presence of cofactor x .
We will find * yy , * and put *, yy , * y * in LNDE.

3
1 * y  e x (Ax  Bx 2 ),
2
3

 2 * y  e x 3 ( Ax  2Bx ) e x (Ax  Bx 2 ),
2
3
1 * y   e x (Ax  x 2 3 ( A  ) B  2Bx ) e x 3 ( Ax  2x 3 ( A  ) B  2B ).
3
2
x
y  * 2 y  *  y * e ( x ( A  2 A  A ) x 3( A  B  3 A  6 A  2 B  B )
 x 2( B  6 A  2 B  4 B  2) B ) xe x

x
Shortening on e after transformations we have
6 Ax  2 B  x .
Equating coefficients at identical degrees x , we have
x 1 6 A 1 1
and A  ,B  0 .
x 0 2 B 0 6

1 3 x
Thus, the partial decision of LNDE is y*  x e .
6

Common decision of heterogeneous differential equation:
~ x 1 3 x
y  y  y*  C (  C x) e  x e .
1 2
6

6.4 Principle of superposition of decisions

~
If functions y (x ) are the upshots parts of equations
i

y (n )  a (x )y (n ) 1  ... a (x  a (x )y  f (x ),i  , 1 , k
)y 
1 n 1 n i
98