Qualitative behavior of dynamical vector fields

Differential Equations come in two classes, deterministic and stochastic. The first part of this document analyzes some of the stable properties of the set of all trajectories in the real plane converging on a critical point defined by two distinct negative eigenvalues---a so-called node.;Secondly, also in the deterministic class, we offer a method new for finding closed-form primitives for a great variety of differential forms, through a reduction process faciltated by a Lyapunov-type Energy function. Many of these forms lie in classes which heretofore have not been shown to be solvable in closed form.;In the stochastic section, the third part of this work outlines the appropriate procedures for calculating differentials and solutions for fields perturbed by random processes.;For the final chapter, we present the development of a theory of Laplace Transforms for stochastic calculations. The resulting Table of Transforms has been initiated, and shall eventually be enlarged. Applications are offered to demonstrate the utility of this approach.