| In this dissertation we applied geometric methods to study underdetermined second order scalar ordinary differential equations (called general Monge equations), nonlinear involutive systems of two scalar partial differential equations in two independent variables and one unknown and non-Monge-Ampere Goursat parabolic scalar PDE in the plane. These particular kinds of differential equations are related to general rank-3 Pfaffian systems in five variables. Cartan studied these objects in his 1910 paper. In this work Cartan provided normal forms only for some general rank-3 Pfaffian systems with 14-, 7-, and 6-dimensional symmetry algebras.;In this dissertation we provided normal forms of all general rank-3 Pfaffian systems in five variables with a freely acting transverse 3-dimensional symmetry algebra. We applied our normal forms to [i] sharpen Cartan's integration method of nonlinear involutive systems, [ii] classify all general Monge equations with a freely acting transverse 3-dimensional symmetry algebra, of which many new examples are presented, and [iii] provide a broad classification of non-Monge-Ampere Darboux integrable hyperbolic PDE in the plane.;We developed a computer software, called FiveVariables, that classifies general rank-3 Pfaffian systems. FiveVariables runs in the environment DifferentialGeometry of Maple, version 11 and later. |
