Question

What could be an approach to solving such equations?

$$f'(x)=C \prod_{k=0}^x f(k)$$

$$\frac{g'(x)}{g(x)}=C+ \sum_{k=0}^{x-1} g(k)$$

Here the product and the sum are understood as indefinite sum and product (i.e. generalized to real x), although a solution which holds only for integer x would also be appreciated.

Are there any methods available?

More than in a concrete solution I am interested to learn general methods of solving such equations.

Answer 1

I recognize the differential function equation you are discussing because it is produced by power towers of height x.

Let $f(x)=a^x$, then $f^n(x)$ produces a power tower.

Typically functional equations are solved by having some preexisting idea as to the form of the solution. I do recommend "Iterative Functional Equations" by Kuczma if you haven't read it.

Answer 2

Instead of "indefinite sum and product" make a change of the independent variable to convert to a conventional delay-differential equation. For example, in the first one, let $F(x) = \prod_{k=0}^x f(x)$ so that $f(x) = F(x)/F(x-1)$, then write everything in terms of $F$.
Of course there is no reason to think there is any "formula" for the solutions...