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y
2
Finally u  u 1  Cx . Now we will put u  and
x
y y 2
write down a general integral:  1  Cx or
x x 2
2
2
2
y  y  x  Cx . 

Note 2.1 If equation of kind  , yxP dx  Q  , yx dy  0

it is simultaneously and homogeneous and in complete
differentials, a general integral can be found as

. Px  Qy  C . (2.18)

Example 2.6 To untie differential equation
x 2  y 2 dx  2xy  y 2 dy  . 0

2
2
 Here are  , yxP   x  y 2 ; Q  , yx   2xy  y . We will
check that the condition of complete differential is executed.
Indeed
2
 x  y 2    xy2  y 2 
  2 y .
y  x 

At the same time functions  yxP ,  and  yxQ ,  are
homogeneous of the 2-th measuring, that is the given equation is
homogeneous. After a formula (1.33) we write down a general
integral:
2
x  y 2  2xyx   y 2 y  . С 

2.3.1 Equations That Are Taken to Homogeneous

Before homogeneous equation the equations of kind are
erected

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