Solutions to $f^{(4)}(x) =f(x )$ This is my first question, I hope that it is not too simple and vague.  
There are several obvious well known solutions to this differential equation:
$$f^{(4)}(x) = f(x)$$
e.g. $e^x$, $e^{-x}$, $\sin(x)$, $\cos(x)$, $\sinh(x)$, and $\cosh(x)$ but, looking at the power series, suggests that the following would be an attractive basis to the solutions.  
$$1 + \frac{x^4}{4!} + \frac{x^8}{8!} + \frac{x^{12}}{12!} + . . .$$
$$x + \frac{x^5}{5!} + \frac{x^{9}}{9!} + \frac{x^{13}}{13!} + . . .$$
$$\frac{x^2}{2!} + \frac{x^6}{6!} + \frac{x^{10}}{10!} + \frac{x^{14}}{14!} + . . .$$
$$\frac{x^3}{3!} + \frac{x^7}{7!} + \frac{x^{11}}{11!} + \frac{x^{15}}{15!} + . . .$$
Of course, these could all be expressed in terms of $e^x$, $e^{-x}$, etc.  
Do these functions have names?  Are they ever studied?  It could be my searching skills at fault but I cannot find any reference to them.  

They do not have special names, because they can very easily be expressed in terms of $\cos(x),\;\sin(x),\;\cosh(x),\;\sinh(x)$. They are
$$
\frac{1}{2}(\cosh(x)+\cos(x)) \\
\frac{1}{2}(\sinh(x)+\sin(x)) \\
\frac{1}{2}(\cosh(x)-\cos(x)) \\
\frac{1}{2}(\sinh(x)-\sin(x)) \\
$$
Studying them would not result in much more than what we already know about $\cos(x),\;\sin(x),\;\cosh(x)$ and $\sinh(x)$.
One special thing about them is that you can easily derive the particular solution of the given differential equation, if you know $f(0)$, $f'(0)$, $f''(0)$ and $f'''(0)$.
