Sophus Lie had a tremendous impact in several areas of mathematics. His work centered on understanding continuous transformation groups and showing how these groups supply an organizing principle for different areas of mathematics. One of those areas is differential equations, and this book is his magnum opus on the subject.

One of Lie's major interests was to develop a theory for differential equations in analogy to Galois theory for polynomial equations. He showed how one could naturally associate a continuous group to a differential equation, so that the solvability of the group (in the sense of algebra) is related to the solvability of the differential equation (in the sense of “quadrature”, meaning integration and algebraic manipulations). The book also discusses Lie's remarkable classification of all three-dimensional groups and their possible actions on the plane. The exposition in the book is elementary and contains numerous examples.

This is a textbook on the integration of ordinary and partial differential equations in which the Lie theory for solving such equations is expounded. The text is in German.