Question

`y' = x/y - x/(1+y)` ? by using separation of variable

Answer 1

`3y^2 + 2y^3 = 3x^2 + A`

Explanation:

We have:

`y' = x/y - x/(1+y)`

Which we can write as:

`y' = (x(1+y) - xy) / (y(1+y))`

`\ \ \ = (x+xy - xy) / (y(1+y))`

`\ \ \ = x / (y(1+y))`

`:. y + y^2 \ dy/dx = x`

Which is now a seperal DE, so we can "separate the variables" to get:

`int \ y + y^2 \ dy = int \ x \ dx`

Which we can now readily integrate:

`1/2y^2 + 1/3y^3 = 1/2x^2 + C`

`:. 3y^2 + 2y^3 = 3x^2 + A`

Which is the Implicit General Solution