What are other solutions to this differential equation, "similar" to $\sin x$ and $e^x$? 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.
 A: They are all  solutions of some 
$$ f'' + A f' + B f = 0    $$
with constants $A,B.$ The constants can be real numbers for $e^x, \sin x,$ but the full story allows them complex as needed.
Don't get sidetracked by the first order delay differential equations $f'(x) = f(x+k).$ The reason that appears  is that there are identities for $\sin (x+k)$  and $\cos(x+k).$
As a simple example,
$$  f'' + 2 f' + 2 f = 0  $$ has solutions
$$  e^{-x} \sin x , \; \;   e^{-x} \cos x  $$
as (real-valued) solutions. So your $f(x)$ could be 
$$ f(x) = C   e^{-x} \cos x + D   e^{-x} \sin x   $$ with real constants $C,D.$ Note that damping effect of the $e^{-x},$ which says that any such $f$ oscillates but goes fairly rapidly to $0.$ This is the type of phenomenon worked with in an automobile spring/shock absorber system. 
The similar but unusual
$$  f'' + 2 f' +  f = 0  $$ has solutions
$$  e^{-x}  , \; \; x  e^{-x}   $$
as (real-valued) solutions. So your $f(x)$ could be 
$$ f(x) = C   e^{-x}  + D x  e^{-x}    $$ with real constants $C,D.$ 
A: There's a good reason why you're finding yourself unable to find solutions in the latter cases. Those are examples of delay differential equations and they have a theory that's substantially more complicated than that of ODEs. 
A: 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
$$f(x+k)=\sum_n a_n\ \text{e}^{M_n x/k}\ \text{e}^{M_n}=\sum_n a_n\ \text{e}^{M_n x/k}\ M_n/k=f'(x).$$
