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Constant of integration are C and C . We determine them
1 2

from initial conditions y (0) 1 , y (0)  2 . We have got:
C   1 ln 2 , C  3 ln 2 .
1 2
Answer:
y    e x (3 ln 2  x ) (1 e  x )ln(1 e x ) 2 ln 2 .  
x
Note 6.1 The method of variation of arbitrary constant for
LNDE- n is like used. If y , y ,  , y are formed
1 2 n
fundamental system of decisions of LHDE of the n-th order.

That is y  C 1 ( )x y  C 2 ( )x y   C y - its general
1
n
2
n
decision. We search the partial decision of LNDE of the n-th
order as a sum

*
y  C ( )x y  C ( )x y   C y (6.10)
1 1 2 2 n n
in what function C k ( ) ,x k  1,2, ,n we find from the system
of equations
 C y   C y    C y   0
 1 1 2 2 n n
 C y    C y     C y    0
1 1
n
n
2
2

  . (6.11)
 C y  (n 2) 1 C y  (n 2) 2   C y  (n 2) n  0

 1 2 n
 C y  (n 1) 1 C y  (n 1) 2   C y  (n 1) n  f ( )x


1 2 n
According to Wronski-Determinante the last system of
linear differential equations is consistent.

6.2 Linear Heterogeneous Differential Equations of the
n-th Order With Constant Coefficients and Special Type of
the Right Part

We will consider the LNDE of the n-th order with the
constant coefficients of kind

y (n )  a y (n ) 1  ... a y  a y  f (x ) (6.12)
1 n 1 n
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