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4.4 Linear Heterogeneous Differential Equations

4.4.1. Structure of Common Decision of Linear
Heterogeneous Differential Equation

Next to LNDE – n

y (n )  a (х )y (n ) 1  ... a (х )y   a (х )y  f (x )
1 n 1 n
y (n )  a (х )y (n ) 1  ... a (х )y   a (х )y  f (x ) (4.1)
1 n 1 n

we will examine the proper him (with the same left part!) LHDE
- n
y (n )  a 1 (х )y ( n ) 1  ... a n  1 (х )  ay  n (х ) y 0 (4.2)
The following theorem takes place:
Theorem 4.6 (about the structure of common decision of
LNDE - n).
The common decision LNDE (4.1) can be represented as a
sum of some his decision part y * ( )x and common decision
of homogeneous equation y  proper to him (4.2)
0
y  y * ( )x   ( )x . (4.18)
y

 (4.18) is a decision of the LNDE according to the
property of decisions LNDE.
We will show that the decision (4.18) is general. For this
purpose it is enough to show that a function (4.18) fulfils any
initial conditions. We will write down these terms for a case
n  2 :
 y ( )x 0  y * ( )x   y ( )x 0  y 0 ,

0
 *
  y ( )x 0  y ( )x  y ( )x 0  y 1 ,
0

(where x lies in the region of continuity of coefficients
0
p 1 ( ) ,x p 2 ( ) ;x y 0 , y – arbitrary numbers).
1

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