### On a functional-differential equation related to Golomb's self-described sequence

The functional-differential equation ${f}^{\text{'}}\left(t\right)=1/f\left(f\left(t\right)\right)$ is closely related to Golomb’s self-described sequence $F$, $$\underset{1,}{\underbrace{1,}}\phantom{\rule{4pt}{0ex}}\underset{2,}{\underbrace{2,2,}}\phantom{\rule{4pt}{0ex}}\underset{2,}{\underbrace{3,3,}}\phantom{\rule{4pt}{0ex}}\underset{3,}{\underbrace{4,4,4}}\phantom{\rule{4pt}{0ex}}\underset{3,}{\underbrace{5,5,5,}}\phantom{\rule{4pt}{0ex}}\underset{4,}{\underbrace{6,6,6,6,}}\cdots .$$ We describe the increasing solutions of this equation. We show that such a solution must have a nonnegative fixed point, and that for every number $p\ge 0$ there is exactly one increasing solution with $p$ as a fixed point. We also show that in general an initial condition doesn’t determine a unique solution: indeed the graphs of two distinct increasing solutions cross each other infinitely...