Preliminaries; Basics from linear algebra and real analysis like concepts of dependence, independence, basis, Rank-Nullity theorem, determinants and eigenvalues, remarks on Jordan decomposition theorem - convergence, uniform convergence, fixed point theorems, Lipschitz continuity, 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, Wronskian, methods of solving.
General Existence and Uniqueness theory; Picard's iteration, Peano's exisentce 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.
Periodic Solutions; Stability, Floquet theory, particular case o second order equations-Hill's equation.
Sturm-Liouville theory; Oscillation theorems.
Qualitative Analysis; Examples of nonlinear systems, Stability analysis, Liapunov stability, phase portrait of 2D systems, Poincare Bendixon theory, Leinard's theorem.
Introduction to two-point Boundary value problems; Linear equations, Green's function, nonlinear equations, existence and uniqueness.