Solução da equacao diferencial homogenea (x² + y²)dx + (2x + y)ydy=0?

n sei

dy/dx = (x²+y²) / (2x+y)y

substitua y=ux, então:

dy/dx = (x²+y²) / (2x+y)y

d(ux)/dx = (x²+(ux)²) / (2x+(ux))(ux)

d(ux)/dx = x²(1+u²) / x²(2+u)(u)

du/dx x + u.dx/dx = (1+u²) / (2+u)(u)

du/dx x + u = (1+u²) / (2+u)(u)

du/dx x = [(1+u²) / (2+u)(u)] - u

du/dx x = [(1+u²) / (2+u)(u)] - (2u²+u³)/(2+u)u

du/dx x = [(1+u²)- (2u²+u³)]/(2+u)u

du/dx x = [1+u²- 2u²-u³)]/(2+u)u

du/dx x = (-u³-u²+1)/(2+u)u