Question

solve the homogeneous equation:

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

I know the answer is y = -x/(ln|x| + C) and y = 0

Answer 1

Hence dy/dx = y/x + (y/x)^2 ------- (*)

Set u = y/x. Then y = ux => y' = u + xu'

Combining with (*)
xu' + u = u + u^2
=> xu' = u^2
=> du/u^2 = dx/x
=> -1/u = ln|x| + C
=> u = -1/( ln|x| + C )

Ok, you see what to do now