I'm trying to find non-trivial functions $f \colon \mathbb R \to \mathbb R $ that $f'(x) = f(x+k)$ with $k \in \mathbb R$.

For $k \le 0$, I've found functions based on $f(x)= e^x$, such as $f(x) = e^{x \frac{W(k)}{k}}$, where $W(k)$ is the Lambert W function.

However, for $k>0$, I can only find a solution if $k=2\pi n+\frac{\pi}{2} $ with $n\in\mathbb N$. The solution is $f(x) = \sin x$. Are there solutions for other values of $k$?

I was hoping that for $k>0$, a function whose graphic is similar to $f(x)=-e^x$ could exist, but it seems it doesn't exist. Or does it?

The method I used is basically trial and error. What other method could I have used?

Thanks.