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We will notice that the arbitrary became was written down
in a logarithmic kind as ln C , to it will be succeeded and farther
with the purpose of simplification of eventual result. We will
mark in addition, whatever no upshots were lost, as
1 x 2 1 e y  0 at x  R, y  R . Thus, sought common
decision after 

Example 1.3 To find the decision of Cauchy task
x 3   y y  , 7 y   51  .
 We will rewrite equation x 3 y  7  , y in a kind

7  y

y  , and then answers as (1.15). To separate variables in
x 3
this case it is needed to write down derivative in
dy

differentials y  :
dx
dy 7  y dy dx
 and farther   .
dx x 3 y  7 x 3

Equation with the separated variables, which we integrate
dy dx
    3 and will as a result get a general integral
y  7 x
1
ln y  7   C , was got . We will find an integral part now
x 2 2
that satisfies an initial condition   51 y :
1 1
ln 5  7   C , C  ln 2  . Consequently, the decision
2 1  2 2
1 1
of Cauchy task will be ln y 7   ln 2  .
2x 2 2

Before equation with the separated variables the equations
of kind are erected


y  f ax  by  , c (1.20)
14