This manuscript provides an introduction to ordinary differential equations and
dynamical systems. We start with some simple examples of explicitly solvable
equations. Then we prove the fundamental results concerning the initial value
problem: existence, uniqueness, extensibility, dependence on initial conditions.
Furthermore we consider linear equations, the Floquet theorem, and the autonomous
linear flow.

Then we establish the Frobenius method for linear equations in the complex domain
and investigate Sturm-Liouville type boundary value problems including oscillation
theory.

Next we introduce the concept of a dynamical system and discuss stability including
the stable manifold and the Hartman-Grobman theorem for both continuous and discrete
systems.

We prove the Poincare-Bendixson theorem and investigate several examples of planar
systems from classical mechanics, ecology, and electrical engineering. Moreover,
attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are
discussed as well.

Finally, there is an introduction to chaos. Beginning with the basics for iterated
interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for
homoclinic orbits.

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