The world around us is constantly evolving. From the orbit of a planet to the spread of a virus, Ordinary Differential Equations (ODEs) are the fundamental mathematical tool for describing and understanding these changes, bridging rigorous Mathematical Analysis with physical laws and enabling the prediction of complex phenomena.

This volume offers a comprehensive and modern introduction to the theory and applications of ODEs. It systematically covers the "classical" topics, such as first-order linear ODEs, proper formulation of Initial Value Problems, qualitative theory for first-order linear ODEs, linear ODEs of second and higher order, the method of power series, Boundary Value Problems, Sturm Theory and Sturm-Liouville Problems, systems of first-order linear ODEs, Laplace transform, qualitative theory of vector differential equations. Additionally, it includes elements of Painlevé equations theory and integrability, phase space analysis, normalization and perturbation methods, as well as the history of ODEs.

The textbook can be used for a basic or advanced undergraduate course on ODEs, as well as for a graduate course on ODEs or/and Dynamical Systems, in Departments of Mathematics, Physics, Computer Science, Engineering Sciences, and Economics. It combines mathematical rigor with "practical utility," including numerous solved examples and a wide range of applications from classical and modern Physics, Biology, and other natural sciences.