Given xy'=y2/(x+y).
Let y=vx, y'=v+xv', where v is a function of x. Rewrite DE:
xv+x2v'=v2x2/(x+vx)=v2x/(1+v); take out common factor x:
v+xv'=v2/(1+v); multiply through by 1+v:
v+v2+xv'(1+v)=v2; v2 cancels out on both sides,
v+xv'(1+v)=0; divide through by v:
1+xv'(1/v+1)=0, xv'(1/v+1)=-1; separate variables:
(1/v+1)dv=-dx/x (because v' means dv/dx), integrating:
ln|v|+v=ln|a/x| (a is a constant),
v=ln|a/x|-ln|v|=ln|a/(vx)|; y=vx, so:
y/x=ln|a/y|,
y=xln|a/y|, or ey=ax/yx.