Differential equations for which the cross-ratio of four solutions is weakly monotone
A Riccati equation is a scalar differential equation \(x'=f(t,x)\) for which the derivative of \(f\) with respect to \(x\) is linear in \(x\). It is well-known that the cross ratio of any four distinct solutions of a Riccati equation is constant. In this paper, the authors prove that the derivative of \(f\) with respect to \(x\) is weakly convex (resp. concave) if and only if the cross ratio of any four solutions \(x(t)<y(t)<u(t)<v(t)\) is weakly increasing (resp. decreasing).
cross ratio of any four distinct solutions
scientific article; zbMATH DE number 930183