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dy P  yx, 
transformation   or got untied directly by
dx Q  yx, 
y
replacement u  , about which more detailed there will be
x
the question below.
The method of solving of homogeneous equation (2.15)
consists in the report of him to equation with the separated
variables.
y
For this purpose we will do replacement u  , that is
x
dy du
y  ux,  x  u .
dx dx
Indeed, after such replacement we will get equation
du
x  u    u variable in which it is easily to separate:
dx
du du dx
x     uu  ;  , whereupon integration is
dx    uu  x
carried out:
du
.   ln x  C (2.17)
   uu

We will notice that   uu  0 , because in other case we
y

would have equation y  , in which variables become
x
separated from directly.

Example 2.4 To untie differential equation
x 2   y y 2  xy  x 2  . 0

 We will write down equation in a kind
2
2
2
dy y  xy  x dy  y  y
  or       1 . We see that it is
dx x 2 dx  x  x
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