TITLE: What are other solutions to this differential equation, "similar" to $\sin x$ and $e^x$?
QUESTION [7 upvotes]: I've been studying electronics, where they make great use of the relationship between the sine and exponential functions ($e^{i \omega t} = \cos{\omega t} + i \sin \omega t)$. This relationship is confusing to me, so I started digging into it, and thinking about how they have similar definitions, in terms of differential equations.
$f(x) = e^x$ is the solution to this differential equation:
$$
f'(x) = f(x)
$$
and $f(x) = \sin x$ is a solution to this similar equation:
$$
f'(x) = f(x + \pi/2)
$$
I wanted to see solutions to the following, for other values of constant $k$.
$$
f'(x) = f(x + k)
$$
but my differential equation solving skills are non-existent.  So my main question is: What are these functions, and what do their graphs look like?  A secondary question is: Do you know how to write the bit of code necessary to solve that third DE (for some value of $k$) using sage or wolfram alpha?  I have sage but don't know what to write.

REPLY [4 votes]: A year ago I asked a similar question here. So let me generalize GEdgars nice answer. Choose a $k$ and consider
$$f(x):=\sum_n a_n\ \text{e}^{M_n x/k},$$
with some magic numbers $M_n$ fulfilling the relation $\text{e}^{M_n}=M_n/k$. These are related to the Lambert W function as explained in the question I linked you to above. Then
$$f(x+k)=\sum_n a_n\ \text{e}^{M_n x/k}\ \text{e}^{M_n}=\sum_n a_n\ \text{e}^{M_n x/k}\ M_n/k=f'(x).$$