Given

      x² (dy/dx) = y² + 2xy

=> dy/dx = y² /x² + 2xy/x²

=> dy/dx = y² /x² + 2y/x ..............1

Now Let y = vx

=> dy/dx = v + x*(dvdx)

From equation 1, we get

v + x*(dv/dx) = v² + 2v

=> x*(dv/dx) = v² + 2v - v

=> x*(dv/dx) = v² + v

=> dv/(v² + v) = dx/x

=> dv/v(v + 1) = dx/x .............2

Now, 

1/v(v + 1) = A/v + B/(v+1)  {using partial fraction}

=> 1/v(v + 1) = {A(v+1) + Bv}/v(v+1)

=> 1 = A(v+1) + Bv

=> 1 = Av + A + Bv

=> 1 = (A+B)v + A

=> A = 1 and A + B = 0

=> A = 1 and B = -1

Now,

 1/v(v + 1) = 1/v - 1/(v+1)

From equation 2, we get

{1/v - 1/(v+1)}dv = dx/x

=> dv/v - dv/(v+1) = dx/x 

Integrate on both side, we get

      ∫dv/v - ∫dv/(v+1) = ∫dx/x

=>  logv - log(v+1) = logx + logc  {logc is a constant}

=> log(v/v+1) = logcx

=> cx = v/(v+1)

=> cx = (y/x)/(y/x + 1)

=> cx = (y/x)/{(y + x)/x}

=> cx = y/(y + x)

=> y = cx(y + x)