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.