(* defined a function y[x] in terms of other functions like this *)
y[x_]:=(((C f1[x]+f2[x]))/(C f3[x] +f4[x]));

(* lets just check we have solve for C correctly ... yes *)
Solve[y==y[x],C]

(* then the first derivative of y[x] with C replaced by the above expression is *)
rHS=Simplify[D[(((C f1[x]+f2[x]))/(C f3[x] +f4[x])),x]/.{C->((-f2[x]+y[x] f4[x])/(f1[x]-y[x] f3[x]))}]

(* lets get terms prop. to y[x] *)
yTerm=Coefficient[rHS,y[x]]

(* lets get the terms prop. to ((y[x])^2)*)
y2Term=Coefficient[rHS,((y[x])^2)]

(* lets get the constant term *)
constTerm=Simplify[rHS-yTerm y[x] - y2Term ((y[x])^2)]

Thus we have shown that y'[x]=constTerm[x]+yTerm[x] y[x]+y2Term[x] ((y[x])^2) or a Ricatti equation as we were to show.