So I am experimenting with a linear systems solver to find new exciting applications for it. While it is possible to play around to solve some of the more basic functional equations, I am trying to be able to do more advanced things. Would it make sense to approach non-linear functional equations like for example $$f(x+1)f(x) = f'(x)$$ with some sequence of linear solutions? I am thinking iterations involving things like $$f(x_{n+1}) = \frac{f'(x_n)}{f(x_n+1)} \hspace{1cm} f'(x_{n+1}) = \frac{f(x_{n})}{f(x_{n}+1)} \hspace{1cm} f(x_{n+1}+1) = \frac{f'(x_{n})}{f(x_{n})}$$ (or minimizing weighted sums of their norm when equating to 0), like this:
$$\left\|f(x_{n+1}) - \frac{f'(x_n)}{f(x_n+1)}\right\|+ \left\|f'(x_{n+1}) - \frac{f(x_{n})}{f(x_{n}+1)}\right\| + \left\|f(x_{n+1}+1) - \frac{f'(x_{n})}{f(x_{n})}\right\|$$