Page 20 - 4549

x
. u   P  yx, dx    y (2.5)

x 0

In this case a variable is fixed and the arbitrary became
integration has the appearance of arbitrary function from y .
We will choose   y thus, that the second was executed with
(2.4). For this purpose we integrate (2.5) on and will equate
to  yxQ , :
x
u   P
  dx     Qy   , yx .
y  y 
0 x

 P  Q
As from a condition (2.3)  , it is possible to write
y  x 
x  Q
down  dx     Qy   yx, , from where, taking into account
x 
x 0
scopes of integration Q  yx,  Q x ,  y     Qy   yx, ,
0
obsessed, consequently

    Qy  x ,  y (2.6)
0

y
    Qy  x , y dy (2.7)
0
y 0

Such integration is possible only on condition that equality
(2.6) does not contain a variable x obviously, in opposite case it
is needed to search an error: either equation was not in complete
differentials or calculation were viciously.
We will put a result (2.7) in (2.5):
x y
. u   P  yx, dx   Q x , y dy (2.8)
0
x 0 y 0
18