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k
~ ~
function y( x)   y ( x) is the partial decision of equation
i
 i 1
k
y ( n)  a ( x) y n (  )1  ...  a ( x) y   a ( x) y   f ( x) .
1 n 1 n i
 i 1

Example 6.7 To find common decision of equation
x
y   y  xe  e 2  x .

 We have LNDE second order with constant coefficients,
right part of which appears by the sum of two functions of the
special kind.
Proper LHDE  y  y  0.
We make characteristic equation  2  1  0 roots of which
are   i  .
2 , 1
Common decision of homogeneous equation:
~
y  C cos x  C sin x .
1 2
On principle of superposition of decisions we search the
partial decision of LNDE in a kind

x
x

y * y *  y *  ( Ax  B) e  Ce ,
1 2

because of y * we have: f 1 (x )  xe x , s , 1   1, a number 
1
is not the root of characteristic equation; for y * we have:
2
f (x )  xe x , s , 1   1, a number  is not the root of
1
characteristic equation.
We find y , * y * and will put *, yy * in the set
differential equation

x
1 y  (* Ax  B) e  Ce  x
x
0 y  (*  Ax  B  A) e  Ce  x
x
1 y  *  ( Ax  B  A  A) e  Ce  x
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