# Solving tricky functional equation

1. Jul 15, 2014

### Frogeyedpeas

Consider the following linear functional operator:

$$Q_w[f(x)] = \lim_{h\rightarrow w} \lbrace \frac{f(x + h) - f(x)}{h} \rbrace$$

How does one solve the equation

$$a_0(x)Q_0[f(x)] = a_1(x)Q_1[f(x)]$$

Spelt out that is:

$$a_0(x)*f'(x) = a_1(x)(f(x+1) - f(x))$$

For the case of constant functions $a_0(x) = a_0$ and $a_1(x) = a_1$ the solution is simply found by assuming

$$f(x) =e^{Lx}$$

thereby implying:

$$a_0L e^{Lx} = a_1(e^{Lx}(e^{L} - 1))$$

which can be solved as

$$\frac{L}{1 - e^{L}} = =\frac{a_1}{a_0}$$

And L can be extracted through the use of Lambert-W function.

But what about more general functions?

2. Jul 16, 2014

### da_nang

This is a delay differential equation, so I'd start looking there. (Not my expertise unfortunately)