Motivation and real life examples: an introduction about differential equations and examples.
Preliminaries: basic concepts from linear algebra and some important preliminaries from analysis like uniform convergence, Arzela-Ascoli theorem, fixed point theorems etc.
First and second order linear equations: examples, a systematic procedure to solve first order and development of the concept integrating factor, Second order homogeneous and non-homogeneous equations.
General existence and uniqueness theory: Picard's iteration, Peano's existence theory, Existence via Arzela Ascoli theorem, non-uniqueness, continuous dependence.
Linear systems: understanding linear system via linear algebra, stability of linear systems, explicit phase portrait in 2D linear with constant coefficients.
Qualitative analysis: examples of nonlinear systems, Stability analysis, Lyapunov stability, phase portrait of 2D systems, Poincare-Bendixson theory.
Introduction to two-point boundary value problems: linear equations, Green's function, nonlinear equations, existence and uniqueness. (from nptel.ac.in)