Solving Schwarzian derivative differential equation in Hollowood & Kumar paper I Was re-deriving Hollowood & Kumar paper (here is arXiv link of it) which is about Anti-De Sitter Black-Holes with JT Gravity, anyway I got a problem with solving a schwarzian derivative differential equation in 3.19 of this paper which gives 3.22 solution without much explanation. I would be grateful if you could help me. i will attach images that needed in case you don't have access to paper.
in 3.19 image you see equation i'm trying to solve,in 2.20 gives us k definition,in 3.22 the answer we get and in 3.16 all information we need.




 A: The Schwarzian is defined as
$$
\{f(t), t\} = \left(\frac{f''(t)}{f'(t)}\right)' - \frac{1}{2}\left(\frac{f''(t)}{f'(t)}\right)^2 = \frac{f'''(t)}{f'(t)} - \frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2,
$$
as you can see in eq. (3.9). The authors propose as a particular solution (you can check it out) of eq. (3.19)
$$
\hat{f}(t) = \alpha \frac{K_\nu(\nu z)}{I_\nu(\nu z)}, \qquad  \nu=\frac{2\pi}{\beta k}, \qquad z=\sqrt{\frac{E_\text{shock}}{E_\beta}}e^{-k(t-t_0)/2},
$$
where $\alpha$ is a normalization constant and $I_n(x)$ and $K_n(x)$ are the modified Bessel functions of the first and second kind, respectively. Then, you can try to seek for your general solution applying a Möbius transformation
$$
f(t) = \frac{A\hat{f}(t) + B}{C\hat{f}(t) + D}
$$
to your particular solution $\hat{f}(t)$ (this general example may help you to convince yourself). See eq. (3.21) and the appendix B of the article to solve the constants $A$, $B$, $C$ and $D$, being mandatory that $AD-BC\neq0$. If you want a more detailed description, I'm all ears!
