The book "Geometric methods in the theory of ordinary differential equations" by Vladimir Igorevich Arnold is an outstanding work that opens the world of mathematical analysis and geometry to readers. Arnold, known for his in-depth research in mathematics, in this paper offers a unique approach to the study of ordinary differential equations (ODS), using geometric methods as the main tool of analysis. From the first pages of the book, the reader is immersed in an exciting journey through the complex but fascinating concepts that underlie ODU. Arnold masterfully combines theory with practical examples, making the material available to both students and professionals seeking to deepen their knowledge in the field. The book is intended for a wide range of readers: from senior students of mathematical and physical faculties to graduate students and researchers working in the field of mathematical modeling and theory of dynamical systems. One of the key topics raised in the book is the relationship between the geometric properties of solutions of differential equations and their analytical characteristics. Arnold demonstrates how visualization and geometric methods can greatly simplify the understanding of complex mathematical concepts. This makes the book especially valuable for those who prefer visual approaches in teaching and research. Readers interested in topics such as system dynamics, sustainability, and quality ADT theory will find many useful ideas and tools in this book. Arnold not only explains the theoretical foundations, but also shows how to put them into practice, making the book an indispensable tool for students and researchers. Arnold's style is clear and logical, which makes it easy to assimilate even the most complex ideas. He uses many illustrations and graphs to illustrate his thoughts, which makes the material more visual and understandable. This quality of the author's style has already become his calling card, and his other works, such as "Methods of Mathematical Physics" and "Geometry and Dynamics", also show his ability to explain complex concepts in a simple and accessible language. The book "Geometric Methods in the Theory of Ordinary Differential Equations" deserves attention not only for its scientific value, but also for its ability to inspire readers to new discoveries. It emphasizes the importance of an interdisciplinary approach, connecting mathematics with geometry, which opens up new horizons for research. If you are looking for literature that will help you deepen your knowledge in the field of differential equations and teach you to see mathematics from a new perspective, this book will become your faithful companion. It will not only enrich your scientific arsenal, but also give you new ideas for research and practical applications. Do not miss the opportunity to get acquainted with the work of one of the greatest mathematicians of our time - Vladimir Igorevich Arnold