INTEGRATION OF A THIRD-ORDER ODE VIA ANALYTICAL AND GEOMETRICAL METHODS

Abstract

Analytical and geometrical methods are applied to integrate an ordinary differential equation of third order. The main objective is to compare both approaches and show the possibilities that each one of them offers in the integration process of the considered equation, specially when not only Lie point symmetries but also generalized symmetries are involved. The analytical method of order reduction by using a generalized symmetry provides the general solution of the equation but in terms of a primitive that cannot be explicitly evaluated. On the other hand, the application of geometrical tools previously reported in the recent literature leads to two functionally independent first integrals of the equation without any kind of integration. In order to complete the integration of the given third-order equation, a third independent first integral arises by quadrature as the primitive of a closed differential one-form. From these first integrals, the expression of the general solution of the equation can be expressed in parametric form and in terms of elementary functions.