Professional Advice on Solving these Equations Would be Appreciated

[Perspicacious]

There is a smooth real-valued function f(x) defined on the positive reals such that, for all x>0 and for all y>0, the following identities are always true:

f(y) + 1/[(x^2)f(x)] = S/(xy)

f(x) – f(S) = y/(xS)

1/[(y^2)f(y)] - 1/[(S^2)f(S)] = x/(yS)

S is merely a function of x and y and is defined explicitly by the first equation.

It's easy to see that f(x)=1/x is one function that satisfies this system. Are there any other solutions?

 

[mathwonk]

it looks to me as if S depends on f, in that first equation. what am i not seeing?

 

[Perspicacious]

it looks to me as if S depends on f

You are indeed correct. And I said that S is a function of x and y. For example, if f(x)=1/x, then S=x+y and all these equations are satisfied. I am asking if there are any other solutions.