As the title suggests, this book aims to introduce the theory of differential equations. In theory, finding closed expressions for integrals plays a minor role, as it is often easier to deduce the properties of solutions directly from the differential equation itself rather than from explicit expressions. However, the investigation of the nature of solutions is the task of the theory. Accordingly, I provide in the introduction the existence and uniqueness theorem for the simplest case of an ordinary differential equation dy/dx = j(x, y), under the assumption of a Lipschitz condition for j(x, y). This way, the reader approaches the systematic presentation, which begins with section 1, with a certain level of knowledge about differential equations. This section addresses existence problems and questions regarding the entirety of all solutions for all ordinary differential equations.