This question is based a bit on an "Ansatz" but I hope my intuition is right. I was practising functional equations for a mathematics competition and I encountered the following equation:
$f(xy) = yf(x) + x + f(f(y) − f(x))$
After substituting for $x=1$, I get:
$f(n) = nf(1) + f(f(n)-f(1))$
which looks a lot like Cauchy's equation. I was wondering whether you could solve this using dynamic programming, after all
$f(n)-f(1) = \textbf{(n-1)}f(1)+ f(f(n)-f(1))$
In the likely case that my hunch is wrong, it would be great if you could show me how to solve this equation (or give me hints) and tell me when dynamic programming is appropriate.