Derivatives and Integrals of superexponential functions

Question

It is known the exponential functions are proportional to the their own derivatives and integrals, Are their any other kinds of functions that have this kind of property? How about tetrations, hexation and other hyper operations are these functions known to be proportional to their derivatives or their integrals? or is this fact generally false for hyper operations?

Answer 1

The only functions of a single real or complex variable that are directly proportional to their derivatives are the exponential functions $f(x)=ce^{kx}$ where $c$ and $k$ are constants. The same is true for functions proportional to their (indefinite) integrals.

I'll show that true for the derivatives: you can show this also for integrals by differentiating the integral equation. So let's assume that, for a given constant $k$,

$$y'=ky$$

which is the same as

$$y'-ky=0$$

We could solve that by separation of variables, but that has some problems in rigor. Instead, let's use the product rule to find the derivative of $e^{-kx}\cdot y$:

$$\begin{align} (e^{-kx}\cdot y)' & = e^{-kx}\cdot y' + (e^{-kx})'\cdot y \\ & = e^{-kx}\cdot y'-ke^{-kx}\cdot y \\ & = e^{-kx}\cdot (y'-ky) \\ & = e^{-kx} \cdot 0 \\ & = 0 \end{align}$$

Therefore the function $e^{-kx}\cdot y$ is a constant: let's call it $c$. Then we end up with

$$y = c\cdot e^{kx}$$

which is an exponential function. We have shown that those are the only possible functions that are directly proportional to their derivatives.

Now, if you meant inversely proportional, there are other possibilities.