A novel Bäcklund invariance of a nonlinear differential equation

Here the nonlinear ordinary differential equation yy″ =S(x) is investigated. The interest of the proposed study is twofold: indeed, the high nonlinearity exhibited by the considered equation does not allow the application of any linearization method; on the other hand, it turns out, under suitable conditions, to be equivalent to a nonlinear integral equation arising in extended kinetic theory. The equivalence between the two nonlinear problems is exploited; in particular, conditions which need to be prescribed to establish such an equivalence are considered. Bäcklund transformations are applied to study the problem of interest. Specifically, it is proved that the nonlinear differential equation enjoys an invariance property when the “source term” S(x) is represented by a solution of a suitable functional equation. The latter is discussed and some solutions are explicitly written; thus, the corresponding Bäcklund charts are depicted to show the obtained new invariances.