If for the differential equation y′=yx+ϕ(xy), the general solution is y=xlog|Cx|, then ϕ(x/y) is given by

[d] Putting v=y/x so that xdvdx+v=dvdx We have xdvdx+v=v+ϕ(1/v) ⇒dvϕ(1/v)=dxx;⇒log|Cx|=∫dvϕ(1/v) But y=xlog|Cx| is the general solution, So xy=1v=log|Cx|=∫dvϕ(1/v) ⇒ϕ(1/v)=−1/v2 (Differentiating w.r.t.v both sides) ⇒ϕ(x/y)=−y2/x2