Solve `(x^(2) + y^(2))(dy)/(dx) = 2xy`

Explanation

`(dy)/(dx) = (2xy)/(x^(2) + y^(2))`
Set `y = vx`, so that `(dy)/(dx) = v + x(dv)/(dx)`
`rArr x(dv)/(dx) = (2v)/(1+v^(2)) - v = (2v - v - v^(3))/(1+ v^(2)) = (v - v^(3))/(1 + v^(2))`
`rArr (dx)/(x) = (1 + v^(2))/(v(1-v^(2))) dv`
`rArr (dx)/(x) = ((1)/(v) + (1)/(1-v) - (1)/(1+v))dv` (Using partial fractions)
Integrating
`ln|x| = ln|v| - ln |1-v| - ln|1+v| + ln|k|`,
`rArr |x| = |(v)/((1-v)(1+v))||k|, k gt 0`
`rArr |x| = |((y)/(x))/(1-(y^(2))/(x^(2)))k|`
`rArr |x| = |(xy)/(x^(2)-y^(2))k|`
`rArr = ky = +- (x^(2) - y^(2))` where k is the constant of integration.