Discovering Repetitive Functions

Question

I am curious to know if there is a way to have discovered the exponential function e, trigonometric functions and whatever other function out there which is equal to itself multiplied by some constant (or perhaps variable), k, before any of them were even discovered. In other words, is there a general way to find all the possible functions which has the property that it equals itself at least after some N number of derivatives if not for the first derivative. For example, e^x becomes e^x or e^3x becomes 3e^3x so d(f(x))/dx = kf(x)

Answer 1

All such functions are linear combinations of exponential functions. $f(x)=e^{{\root n\of k}x}$ satisfies $f^{(n)}(x)=kf(x)$. Let $\zeta_n=e^{2\pi i/n}$ be a primitive complex $n$th root of unity, then $f_j(x)=e^{\zeta_n^j{\root n\of k}x}$ is a solution for $j=0,1,\dots,n-1$. The general solution is $\sum_{j=0}^{n-1}a_jf_j(x)$. This includes such functions as $\cos x=(1/2)e^{ix}+(1/2)e^{-ix}$ which is the case $n=2$, $k=-1$.