Schwarzian derivative, Painlevé XXV–Ermakov equation, and Bäcklund transformations

The role of Schwarzian derivative in the study nonlinear ordinary differential equations is revisited. Solutions and invariances admitted by Painlev\'e XXV-Ermakov equation, Ermakov equation third order linear a normal form are shown to be based on solutions equation. Starting from Riccati second element chian as simplest examples linearizable equations, introducing suitable change variables, it how represents key tool construction solutions. Two families B\"acklund transformations which link under investigation obtained. Some with relevant applications given discussed.