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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.

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    Wait, $e^x$ is not a solution of that equation. The $4$-th derivative of $f$ is $f$ itself. Are you sure of the formula? – 2017-01-24
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    You probably mean $f''''(x) = f(x)$. – 2017-01-24
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    Woops, yes I do. I concentrated too much on the formatting rather than the content. – 2017-01-24
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    The first one is just $\frac{1}{2} (\cos (x)+\cosh (x))$ – 2017-01-24
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    Claude, yes they can all be expressed with similar formulae. So, I am not suggesting that there is anything new or fundamental here. It was just curiosity as to whether these functions were ever considered or had names. – 2017-01-24

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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 $f'''(0)$.

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    Thanks. I guess that is it. Although the power series look neat and attractive, given that sin, cos, sinh, and cosh are so well known, there is no need to pay much attention to these. – 2017-01-24
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    Part of my interest was that a naive look at your formulae might suggest that the functions have a wobble due to the sin or cos component but the power series quite clearly show that they don't. In fact, they are all very smoothly increasing for x > 0 as all the derivatives are strictly positive. – 2017-01-24