solve: dy/dx=3x^2 Y+Y^2 /2x^3+3xy

Question

dy/dx=3x^2 Y+Y^2 /2x^3+3xy

Answer 1

Let W_=W(x)

Answer 2

& y=x^2•w → (1)

Answer 3

Then y'

Answer 4

→ x^2•w'+2wx

Answer 5

The given is transformed into:

Answer 6

x^2 w'

Answer 7

=3x^4 w+x^4 w^2 /2x^3 +3x^3 w -2xw →

Answer 8

→ x^2 w'=

Answer 9

=-x(5w^2 +w) /3w +2

Answer 10

And (3w+2)/w(5w+1) dw

Answer 11

=-dx/x → (2)

Answer 12

Now 3w+2/w(5w+1)

Answer 13

_= A/W + B/5w+1

Answer 14

Which gives,

Answer 15

A= 2 &

Answer 16

B=-7

Answer 17

That is,

Answer 18

That is,3w+2/w(5w+1) dw

Answer 19

=2 dw/w - 7dw/5w+1

Answer 20

=2dw/w -7/5 d(5w+1)/5w+1 → (3)

Answer 21

From (2) and (3)

Answer 22

After performing the integration we get:

Answer 23

lnw^2 - 7/5 ln(5w+1)

Answer 24

=c power ~ - lnx

Answer 25

→ 5 lnw^2+5 lnx- ln(5w+1)^7= c power ~

Answer 26

ln(w^10 X^5)+ ln(5w+1)^-7

Answer 27

= c power ~

Answer 28

→ Ln[w^10 X^5/(5w+1)^7]= c power ~

Answer 29

W→ Y/X^2 :

Answer 30

W^10 X^5_= Y^10/X^20• X^5

Answer 31

=Y^10/X^15

Answer 32

(5y/x^2+1)^7•e^ c power~

Answer 33

→ y^10

Answer 34

= x^15 •(5y+x^2)^7/x^14 •e^c power ~

Answer 35

The solution then is given in implicit form as:

Answer 36

Y^10 = c•x•(5y+x^2)^7

Answer 37

Therefore,