For clarity, we use the notations $f^{[n]}=f\circ \dots\circ f$ for nesting and $f^{(n)}=\frac d{dx}\left(\frac{d}{dx}\dots\right)f$ for differentiation.
After reading these two posts (here and here) on MO, I couldn't help but wonder what would happen if we were to iterate $f^{-1}$ on that DE. I first started by trying to merge arguments from both pages to see if it could work, to no avail: to my understanding, one has to simply further or adapt the techniques presented on the first page, as the presence of the inverse makes the problem fundamentally different (and more difficult). I haven't managed to go further than the trivial analysis:
Pick an ansatz of the form $ax^b$ and substitute. Equate the powers and the coefficients:
$\begin{cases} b-1=\frac{1}{b^n} \iff b^{n+1}-b^{n}-1=0 \\ a=b^{\frac{-b^2}{b^2+b+1}} \end{cases}$
Fortunately, the polynomial in $b$ always has a positive root so there is always this solution. What other solutions are there and how to find them? I have willingly not specified the domain for $x$, pick whatever suits your answers, or even better, discuss the several possibilities.
One can push this one step forward and explore DEs of the form: $$F(x,f',...,f^{(n)},f^{-1},(f^{-1})^{[2]},...,(f^{-1})^{[n]})=0$$ as presented similarly by this article.
Any and all ideas are appreciated!
