solve: y^3 dx+2(x^3-xy^2) dy=0

Question

Y^3 dx+2(x^3-xy^2) dy=0

Answer 1

→ dy/dx=(y/x)^3 /2(y/x)^2 -2 → (1)

Answer 2

Transform: y/x

Answer 3

F(x)_=F: y'

Answer 4

→ F+xF' → (2)

Answer 5

(1),(2)→ x dF/dx=F^3/2F^2-2 -F

Answer 6

→ x dF/dx=F(2-F^2)/2(F^2 -1) ,

Answer 7

That is,

Answer 8

{2F^2 -2/ F(2-F^2) dF

Answer 9

={dx/x → (3)

Answer 10

The integrand on the left can be written as:

Answer 11

2F^2 -2/F(2-F^2)=A/F+B/√2+F+c/√2-F

Answer 12

Where the A,B,C are given from the system,

Answer 13

-A-B+C=2

Answer 14

B+C=0

Answer 15

& 2A=-2

Answer 16

Solving, we get

Answer 17

A=-1, B=-1/2 , C=1/2

Answer 18

{2F^2 -2/F(2-F^2) dx=-{ dF/F-1/2•d(F+√2)/F+√2-1/2• { d(F-√2)/F-√2

Answer 19

=-lnF-1/2 ln(F+√2)-1/2 ln(F-√2)

Answer 20

=- lnF-1/2 ln(F^2-2)

Answer 21

= -lnF•√F^2 -2

Answer 22

And back in (3) we have

Answer 23

- ln F√F^2 -2

Answer 24

=lnx+c base 0

Answer 25

→ ln F•√F^2-2

Answer 26

=ln(1/xe^co)

Answer 27

→F^2•(F^2 -2)

Answer 28

= 1/x^2 e^2co

Answer 29

Inverse transform:

Answer 30

(Y/x)4 -2(y/x)^2

Answer 31

=y/x^2 e^-2co

Answer 32

= 1/x^2 ^-2co

Answer 33

→ y^4-2x^2 y^2-cx^2=0

Answer 34

We get an incomplete quartic equation.

Answer 35

This can be reduced to a quadratic

Answer 36

t^2-2x^2•t-cx^2=0

Answer 37

With D=4x^2•(x^2+c)

Answer 38

t=x^2 ± x|√x^2 + c|

Answer 39

t = y^2

Answer 40

± y=ab•sq•rt{x^2±x•ab•sq•rt^(x^2+c)}