Differential equations are a powerful and pervasive mathematical tool in the sciences and are fundamental in pure mathematics as well. Almost every system whose components interact continuously over time can be modeled by a differential equation, and differential equation models and analyses of these systems are common in the literature in many fields including physics, ecology, biology, astronomy, and economics. For example, the following can all be modeled as a system of differential equations: planets, stars, electric circuits, predator and prey populations, epidemics, and economics. We will start by studying the classical theory of ordinary differential equations then will develop dynamical systems approaches to understanding more complex non-linear systems. The goal throughout the course will be to better understand the behavior of the system being studied.