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Example 3.1 To find the common decision of equation
8
y IY  .
( x ) 3 5

 Pursuant to a formula (1.6) and rules of integration, we
have
8dx 2 ~

   y  y IY dx      .c
1
(x  )3 5 (x  )3 4

We find farther
2 ~ 2 ~ ~

  y  y    dx    (  c 1 )dx   xc 1  c 2 .
(x  )3 4 ( 3 x  )3 3

Let us integrate last equality one more and we will get the
common decision of the given differential equation:

~ 2
2 ~ ~ 1 c x ~ ~
 y  y  dx  (  xc 1  c 2 )dx    1  c 2 x  c 3 ,


( 3 x  )3 3 (x  )3 2 2
~ 2
1 c x ~ ~

y  y dx    (  1  c 2 x  c ) dx 
3
x (  )3 2 2
~ 3 ~ 2
1 c x c x ~ ~
  1  2  c 3 x  c 4 .
( 3 x  ) 3 6 2

Arbitrary constant it is possible remark for more compact
record. Then the sought common decision after will collect a
1
2
3
kind: y   c 1 x  с 2 x  c 3 x  c 4 . 
( 3 x  ) 3

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