Novel kinds of generalized Clairaut-type equation and their singular solutions

Abstract

The aim of this paper is twofold. First, we investigate the solutions of classical Clairaut-type equations armed with special function methods for the ODE, PDE and functional counterparts, generalizing some of the results found in the literature. On the other hand, we introduce a new generalization of Clairaut-type equations and investigate the properties of their solutions, with specific focus on the singular solutions. Though the extension of Clairaut’s equations we provide
can be considered as their linear or minimal extension, interestingly enough, we find that the singular solutions are non-linearly extended, and include extra functional terms that endow these singular solutions with very different properties when compared to those of classical Clairaut’s equations. We show that this feature of Generalized Clairaut’s equations is exclusive of the singular solutions,
since the general solutions do not have such attribute.