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.