let y = vx

\( dy = v dx +x dv\)

\( (x -2vx) dx + (2x +vx)(v dx + x dv) = 0\)

\( (1-2v)dx + (2+v) (v dx + x dv ) = 0\)

\( (1 - 2v) dx + v(2+v) dx +x (2+v) dv = 0 \)

\( (v^2 +2v +1 - 2v )dx + x (2 +v) dv = 0\)

\( ( v^2 + 1 ) dx + x(2 +v) dv = 0\)

\( \frac{dx}{x} + \frac{(2 + v)dv}{v^2 +1 } = 0\)

\( \frac{dx}{x} +\frac{2 dv}{v^2 +1 } + \frac{v dv }{v^2 +1}=0 \)

\( ln x + 2 arctan v + \frac{1}{2} ln (v^2 +1) = 0\)

\( 2 lnx + 4 arctan \frac{y}{x} + ln (\frac{x^2 + y^2}{x^2 }) = c\)

\(2 lnx + 4 arctan \frac{y}{x} + ln (x^2 + y^2) - 2 ln x = c \)

\(4 arctan \frac{y}{x} + ln (x^2 +y^2) = c \)