Schwarzian derivative in higher-order Riccati equations

The Sturm–Liouville equation represents the linearised form of the first-order Riccati equation. This provides an evidence for the connection between Schwarzian derivative and this first-order nonlinear differential equation. Similar connection is not obvious for higher-order equations in the Riccati chain because the corresponding linear equations are of order greater than two. With special attention to the second- and third-order Riccati equations we demonstrate that Schwarzian derivative has a natural space in higher Riccati equations. There exist higher-order analogues of the Schwarzian derivative. We demonstrate that equations in the Riccati hierarchy are embedded in these higher-order derivatives. (author)