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 a x  b y  c 
 y   f  1 1 1  (2.19)

 a 2 x  b 2 y  c 2 

a b
where  1 1  0 (2.20)
a 2 b 2

f – arbitrary function continuous in a the region and even one
of numbers c , c different from a zero, as at c  c  0
1 2 1 2
equation (2.19) would be already homogeneous.
The process of report to homogeneous equation begins from
the decision of the system of equations

a   b   c 1  ,0
1
1
 (2.21)
a 
 2  b   c 2  .0
2

A condition (2.20) guarantees existence of unique decision
of the system (2.21). After it we do replacement

  ux  , dx  du ,
 (2.22)
 y  v  , dy  dv .

Got homogeneous equation

dv  a u  vb 
  f  1 1  (2.23)

du  a 2 u  b 2 v 

we integrate by the method considered higher, namely by
v
replacement t  then go back to old variables x and y .
u

Example 2.7 To untie differential equation
3 x 7 y  3  y  3 y 7 x 7  . 0
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