This course offers an introduction to the study of ordinary differential equations. Topics include: Explicit solutions of some linear and non-linear equations (using tools that span: separation of variables, integrating factors, Greens functions, and Laplace transform methods); Series solutions of ODEs with analytic coefficients and with regular singular points; Fundamental existence and uniqueness theorems (Peano, Picard-Lindelof, and Osgood); Introduction to dynamical systems (Poincare-Bendixon theorem); Stability of equilibrium points and periodic orbits.