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We make characteristic equation:  3  2 2    0 and
  , 0     1.
1 2 3
~
x
Common decision of LHDE is y  C  C ( 2  C 3 x) e .
1
We search the partial decision of LNDR as
y*  Acos x  Bsin x, because right part of the equation is
f (x )  4 (sin x  cos ) x (special type of right part (6.17)):
f ( x ) e  x P ( x cos)  x  Q ( x sin)   x max(, s , s )  s and as
s 1 s 2 1 2
if  s , 0  , 0   1 thus a number   i  i is not the root of
characteristic equation.
Farther, finding * yy , * ,  y *  we will put in given LNDE
and, applying the method of indefinite coefficients (see previous
Example 2.), will find coefficients: A  B  2.
Thus, the partial decision of LNDE will be

y*  2 cos x 2 sin x.

Common decision of LNDE is
x
y 
y  ~ y*  C  C (  C x) e  2 cos x 2 sin x.
1 2 3
Then we will find:

x

y  (C  C 1 (  x ))e  ( 2  sin x  cos ) x ,
2 3
x
 
y  (C  C 2 (  x ))e  2 (cos x  sin ) x .
2 3

Taking advantage of initial conditions
) 0 ( y  , 2 ) 0 (  y  , 2 ) 0 (   y   1, we will obtain the system of
equations
 C 1  C 2  2  ,2

 C 2  C 3  2  ,2

 C 2  2C 3  2   ,1

from where C  , 1 C   , 1 C  1 .
1 2 3
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