Many problems of higher analysis presuppose a knowledge of ordinary differential equations; for example, problems of potential theory, of the calculus of variations, of theoretical physics and of partial differential equations (see Chapter 37.). Beyond this, a wide field of applications is opened up by ordinary differential equations; for example, the calculation of pendulum oscillations, satellite trajectories, load carrying wings, dams, earthquake tremors, heat propagation, speeds of chemical reactions and of radioactive decay, as well as calculations in electrotechnology and ship building. Only differential equations for real variables and real-valued functions will be examined here and, renouncing full mathematical rigour, methods of solution will be given that occur frequently in practice. A first glance will also be given at typical problems in this field, at its vast and often difficult theory.