ERMAKOV-PINNEY AND EMDEN-FOWLER EQUATIONS: NEW SOLUTIONS FROM NOVEL BACKLUND TRANSFORMATIONS

摘要

We study the class of nonlinear ordinary differential equations y '' y = F(z, y(2)), where F is a smooth function. Various ordinary differential equations with a well-known importance for applications belong to this class of nonlinear ordinary differential equations. Indeed, the Emden-Fowler equation, the Ermakov-Pinney equation, and the generalized Ermakov equations are among them. We construct Backlund transformations and auto-Backlund transformations: starting from a trivial solution, these last transformations induce the construction of a ladder of new solutions admitted by the given differential equations. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficult to apply.