Do the functions f(x), g(x), and h(x) exist so that f'(x)=g(x), g'(x)=h(x), and h'(x)=f(x), but none of the functions are multiples of each other?

Question

I know that there are functions where, if you take the derivative of that function a multiple of n times, the $n^{th}$ derivative of that function is equal to the original function (e.g. $e^x$ and zero for n = 1, $e^{-x}$ for n = 2, $\sin{x}$ and $\cos{x}$ for n = 4). I am curious as to whether or not there are functions where the smallest possible value of n is not equal to 1, 2, or 4. My gut feeling says that there should be, at the very least, functions where the smallest value of n is 3. However, I have not been able to find any. I am curious about the n = 3 case in particular, but a general solution for all n would be fantastic.

Answer 1

Take $f$ to be the sum of every third term of the Taylor series for exp, i.e., $$ f(x)=1+\frac{x^3}{3!}+\frac{x^6}{6!}+\frac{x^9}{9!}+\dots. $$ Define $g=f'$ and $h=f''$. Observe that $h'=f'''=f$.

Clearly the same idea also works to produce longer derivative-cycles.

Answer 2

Yes. There is standard method of solving the DE $f'''-f=0$ (by solving the cubic equation $\lambda^{3}-1=0$). One of the solutions is $e^{-x/2} \cos (\frac {\sqrt 3 x } 2)$ and this function has the property that no two of the functions $f, f' f''$ are constant multiples of each other.