Direct Integration of Ordinary Differential Equations

The process of solving an ordinary differential equation can be reduced to one or, in the case of high-order equations or systems of such equations, to several integration operations. This work is devoted to the presentation of such methods - the methods of direct integration, developed by the authors. A Nobel laureate, the outstanding physicist L. D. Landau (1908-1968) once wrote, "e;For the study of theoretical physics, first of all, knowledge of mathematics is necessary. What is needed is not all kinds of theorems about existence, which mathematicians are so generous with, but mathematical technique, that is, the ability to solve specific mathematical problems. First, you need to learn how to correctly differentiate (and, if possible, quickly), integrate and solve ordinary differential equations in quadrature; as well as the study of vector analysis and tensor algebra (that is, the ability to operate with tensor indices). The main role in this study should be played not by the textbook, but by the problem book - which one is not very important, the main thing is that it has enough tasks. I am categorically convinced that all theorems about existence, excessively strict proofs, etc., should be completely banished from mathematics studied by physicists"e;. Agreeing with these words said to specialists in theoretical physics, we believe that they are equally true for specialists in all fields of engineering, economics and other sciences. Therefore, special attention should be paid to methods that allow the direct integration of ordinary differential equations. Unfortunately, there are practically no books in which it would be possible to study differential equations in the way that Landau pointed out. We believe that this work will fill this gap.