This is a translation of a book that has been used for many years in Sweden in courses on ordinary differential equations for advanced undergraduate and beginning graduate students. It gives a careful and thorough introduction to the main areas of the field and should also be useful for engineers and applied mathematicians. Topics covered are existence, uniqueness and approximation of solutions, linear systems with constant coefficients, power series solutions, eigenfunction expansions and qualitative methods for non-linear systems.

The book contains many illustrative examples and almost three hundred exercises.