Some Remarks on a Class of Ordinary Differential Equations: the Riccati Property

Here ordinary differential equations of third and higher order are considered; in particular, a class of equations which can be solved by quadratures is exploited. Indeed, crucial to obtain our result is the property of the Riccati equation, according to which, given one particular solution, then its general solution can be determined explicitly. Thus, what we term the “Riccati” Property is introduced to point out that the members of such a class are differential equations which are of a generalized form of Riccati equation. Trivial examples of differential equations which enjoy the Riccati Property are all linear second order ordinary differential equations. Here some further examples of ordinary differential equations which enjoy the same Property are considered. In particular, on the basis of group invariance requirements, a method to construct ordinary differential equations which enjoy the Riccati Property is given. Remarkably, it follows that ordinary differential equations enjoying the Riccati Property are related to nonlinear evolution equations which admit a hereditary recursion operator. Finally, further connections with nonlinear evolution equations are mentioned.