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Lecture 4 Linear Differential Equations of Higher
Orders (elements of general theory)

4.1 Basic Concepts. Classification of Linear Differential
Equations

Definition 4.1 Linear differential equation of the n-th
order – Equation of kind is named to the go (LDE-n)

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

where y – a function with respect of argument x ,
a (x )(k  3 , 2 , 1 ,...n ) and f ( )x - the set functions continuous
k
on ( , )a b .
Definition 4.2 Functions a (x )(k  3 , 2 , 1 ,...n ) are named the
k
0
coefficients of equation, and ( )f x - its right part. If f ( )x  ,
such equation, that is

y (n )  a (x )y ( n ) 1  ....  a (x )  ay  (x ) y 0 (4.2)
1 n  1 n

name linear homogeneous equation (LHDE-) or equation
without right part. Speak also, that LHDE- (4.2) answers
LNDR- (4.1).
Farther we will formulate and lead to some properties of
homogeneous equations. We will formulate theorems in a
general kind, and will lead to LHDE 2-nd order with the purpose
of simplification of records

y   a (x )y   a (x )y  f (x ) ; (4.3)
1 2

   y a (x )   ay (x ) y 0 (4.4)
1 2

according to LNDE-2 and LHDE-2.
Theorem 4.1 Linear combination of decisions LHDE also
is decision of LNDE.

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