TITLE: Solving the differential equation $f'(x)=af(x+b)$
QUESTION [9 upvotes]: How does one find all the differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ which satisfy the equation
$$
f'(x)=af(x+b),\quad \text{for}\quad  a,b \in \mathbb{R}?
$$ 
I see that functions of the form $\alpha e^{\beta x}$ and $\alpha \sin(\beta x + \gamma)$ will be solutions, but I'm not sure how to exhaustively find all of the solutions.

REPLY [4 votes]: Try working with the Fourier transform.   The Fourier transform of a translated function like $f(x+b)$ is the function's transform multiplied by a unit-magnitude phase factor, something like $\exp(-i b k)$  while the derivative $f'$ is described in the Fourier world as multiplication by k.   I imagine a bit of fooling around with that will turn up useful solutions.