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where f – continuous function and terms are executed
a  ; 0 b  0 . Indeed, then we do replacement,
t   a


ax  by  c  t , t  a  y b , y  and equation (1.20)
b

acquires a kind t  a  f   t , that is get equation with the
separated variables. 

Example 1.4 To find the common decision of equation

y  sin x  . y
dy dt
 We will do substitution x  y  t , then 1  and
dx dx
the given equation will be written down in a kind
dt
 sin t 1.
dx
We will consider two cases.

1) Lets sin t 1  . 0 Then we separate variables
dt dt

dx   0 and integrate dx    .C
sin t 1 sin t 1

We will find more difficult integral separately by universal
trigonometric substitution:

t 2zdz
z  tg , sin t ,
dt 2 1 z 2 2dz
    
sin t 1 2dz 2  2z 
dt  , t  2arctg z 1 z  2  1
1 z 2 1   z 
dz dz 2 2
 2   2     C 2    C 2 .
z 2  2 z 1  z  1 2 z  1 t
tg  1
2

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