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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?