Using substitution to solve first order differential equations

i have the equation

(x+2y) * y' = y

do i use a linear substitution or homogeneous substitution?

(x + 2y) y ' = y

divide by x

(1 + 2y/x) y ' = y/x

let y/x = v
y = vx
y ' = v + xv '

(1 + 2v)(v + xv ' ) = v

=> v + xv ' = v /(1 + 2v)

=> xv ' = v /(1+2v) - v

=> xv ' = ( v - v - 2v^2)/(1 + 2v)

=> xv ' = - 2v^2 /(1 + 2v)

separating variables

[(1 + 2v)/v^2 ] dv = -2 dx/x

=> [ (1/v^2) + (2/v) ] dv = -2 dx/x

integrating

=> -(1/v ) + 2 ln v = -2 ln x + c

=> 2 ln v + 2 ln x = c + (1/v)

=> v^2 x^2 = Ce^(1/v)

substitute v = y/x

=> y^2 = C e^(x/y)

=> y = K√e^(x/y)

(x + 2y) * y' = y

Dividing both sides by x:

y' + (2y/x)y' = y/x

Using the substitution v = y/x

xv' + v = y'

xv' + v + 2v*(xv' + v) = v

xv'(2v + 1) + v + 2v² = v

xv'(2v + 1) = -2v²

(2v + 1)/(-2v²) dv = dx/x

Integrating both sides:

-ln|v| + 1/(2v) = ln|x|

-ln|y/x| + x/(2y) = ln|x|

ln(y) = x/(2y)