Regarding the possible sizes of families of functions which differentiate in a circle or cycle.

Question

We all probably know of the functions $$\underset{\text{1 period or cycle}}{\underbrace{\cos(x) \to -\sin(x) \to -\cos(x)\to \sin(x)}} \to \cos(x) \to \cdots$$

Which are generated by the differential operator so that the sequence repeats after some steps. Do there exist such families of different sizes? Say for example period 3 or 5?

Answer 1

For period three, you are effectively asking for the solution of the differential equation: $y = y'''$.

This has solution $y(x) = c_1e^x+ c_2e^{-x/2} \sin( \frac{\sqrt{3}x}{2}) + c_3e^{-x/2} \cos(\frac{\sqrt{3}x}{2})$.

You similarly solve a differential equation for $y=y'''''$ in the period 5 case.

Answer 2

Hint:For $$\sin x, \cos x$$ we have $y^{(4k+r)}=y^r\\r=0,1,2,3$

foe example $y=\sin x \to y^{(47)}=?$ $$y=\sin x \\y^{(1)}=\cos x\\y^{(2)}=-\sin x\\y^{(3)}=-\cos x\\y^{(4)}=y=\sin x\\y^{(5)}=y^{(4k+1)}=y^{(1)}=\cos x\\\vdots\\47=44+3 =4k+3 \to \\y^{(47)}=y^{(3)}=y'''=-\cos x$$

Answer 3

If we choose a function $\phi(x)$ such that $\phi^n(x)=1$, for instance $x^n=1$ and $x$ is a complex root of 1, then we have $(e^{\phi(x)})^{(n)}=e^{\phi(x)}$ (one has a bit cheating) :)