  The class of nonlinear ordinary differential equations $y^{\prime\prime}y =
F(z,y^2)$, where F is a smooth function, is studied. Various nonlinear ordinary
differential equations, whose applicative importance is well known, belong to
such a class of nonlinear ordinary differential equations. Indeed, the
Emden-Fowler equation, the Ermakov-Pinney equation and the generalized Ermakov
equations are among them. B\"acklund transformations and auto B\"acklund
transformations are constructed: these last transformations induce the
construction of a ladder of new solutions adimitted by the given differential
equations starting from a trivial solutions. Notably, the highly nonlinear
structure of this class of nonlinear ordinary differential equations implies
that numerical methods are very difficulty to apply.
