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P  x Q  y
1 dx  2 dy  0 (1.16)
P  x Q  y
2 1

Q  y P  x
or 2 dy   1 dx (1.17)
Q  y P  x
1 2

which has a general integral

P  x Q  y
 1 dx   2 dy  C (1.18)
P  x Q  y
2 1

Q  y P  x
or  2 dy    1 dx  C . (1.19)
Q  y P  x
1 2

We will notice that division on     yQxP can be caused to
2 1
the loss of decisions of differential equation, that
P   0x , Q   0y . These cases are the upshots of equations
2 1
it is necessary to check additionally.
Now we will consider concrete examples.
Example 1.2 To find the common decision of equation
e y 1 x 2 dy  2x 1 e y dx  . 0
 For the separation of variables we will divide both parts
of equation on expression  1 x 2  1 e y  and get
e y dy 2xdx
  0 then let us integrate
1 e y 1 x 2
e y dy 2 xdx

 y  2  C , as a result will write down a general
1  e 1  x
integral ln 1 e y  ln  1 x 2  ln C , from where have
1 e  y C 1 x 2 . It is more comfortable type of general
integral, from which it is possible to get the common decision
y  ln C   1 x 2  .1

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