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Lecture 5 Solving of Homogeneous Linear Differential
Equations (LHDE) with Constant Coefficients

How we already marked in a previous lecture, does not
have the unique method of solving of linear differential
equations. The exception is made by differential equations with
constant coefficients.

5.1. Linear homogeneous differential Equations with
Constant Coefficients

5.1.1. Characteristic Equation

We will consider to the LHDE of n-th order with constant
coefficients
y (n )  a y (n ) 1   ...  a  y  a y  0, (5.1)
1 n 1  n

where а і - material numbers і=1,2,...,n.
At first we will define the class of functions ( )x which
can be the upshots of such equations. As substitution y   ( )x
in (5.1) converts this equality into an identity, the elements of
left part (5.1) at such substitution must be destroyed. Therefore
functions, which have similar to itself derivative, can be the
upshots of equation (5.1). Are such functions to the index:
x 
y  e (   const ). Indeed
x 
x 
x 
x 
х
х
( e )   e  ; ( e )     2 e ;.....; ( e ) ( mx)   m e ,

that is they differ by constant cofactors – degrees  .
In this communication we will search the upshot LHDE
(5.1) with constant coefficients in a kind
x 
y  e (5.2)

where  a size which it is needed to find became . Putting (5.2)
in (5.1), we will obtain a condition for determination  :

( n  a  n  1  ... a   a )e x   0 .
1 n 1  n
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