Page 26 - 4549

We will increase the given equation on the found
integrating cofactor:
 1  x
 x dx  dy  0.
  2
 y  y

We will make sure, that the condition of complete
differential is now executed. Indeed

 1   x 
  x      2  


 y   y  1
   .
y  x  y 2

We find a function  yxu ,  after a formula (1.28),
considering x  ; 0 y  1 :
0 0

x  1  y 0  x 2 x  x x 2 x
u     x   dx   2 dy         .

0  y  1 y  2 y  0 2 y

Writing down a general integral in a kind (2.8), let as
multiply expression by y2 for the receipt of more perfect form:
x 2 y  2x  Сy .

Here, as well as in a previous example, some corrected
record of arbitrary constant tracing is suggested after it
independently. 

2.3 Homogeneous Equations

At first we will enter notion of homogeneous function.
Definition 2.2 A function f  yx,  is named the
homogeneous function of the k measuring ( k degree) in
relation to variables x and y , if at any   0 the correlation
takes place

24