Explore a comprehensive course on ordinary differential equations and their applications. Delve into topics such as general and linear second-order equations, periodic orbits, Poincare-Bendixon theory, stability of equilibrium points, and phase plane analysis. Learn about Picard's existence and uniqueness theorem, Gronwall's Lemma, and well-posedness of initial value problems. Study series solutions, continuation of solutions, and various types of systems including 2x2 and general systems. Gain a deep understanding of first and second-order linear equations, diagonalizability, and the application of fixed point theorems in existence proofs.