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We will illustrate this fact on the resulted higher example.
dx dy
We will note for this purpose:  u ,  v . Then the system
dt dt
of equations (6.12) will be:

 dx  , u
 dt

 dy  , v
 dt
 du (7.2)
 m  X ( , , , , ) ,t x y u v
 dt
 dv
 m  ( , , , , ) .t x y u v
 dt

In this connection we will examine only the systems of the
first order, untied in relation to derivative ; such systems are
named normal, their kind:

 y  f 1 ( , ,x y y 2 , , y n ) ,
1
1

 y  f 2 ( , ,x y y 2 , , y n ) ,
1
2

 (7.3)

 y  f n ( ,x y y 2 , , y n ) .
,

n
1

Under the decision of the system of differential equations
(7.3) we will understand the system of functions

 y   1 ( ,x C 1 , ,C n ) ,
1

  (7.4)
 ,
 y   n ( ,x C 1 ,C n ) ,
n
103