The distant origins of this book are found in my lecture notes that I wrote and rewrote for several years. The traces are the constant smudges that I made from time to time when I rediscovered these topics, you may not like the way of presentation, but I put in the best possible effort, it is not something new either since there are many references at the end of the book; I'll be happy if it at least works for a study group, it's impossible to avoid slipping mistakes. Some frequently asked questions that we ask ourselves, What exactly is a differential equation? Where and how did they originate, what is their use? What is done with them, how are the results of such manipulations analyzed? These questions indicate guidelines to follow, study aspects: theoretical, methodological and various fields of applications. There are historical facts that inspire; for example, obtaining the equation of the tangent line to a given curve was one of the latent problems until the end of the 17th century, on the eve of the birth of differential calculus. From then on the job consisted of solving the inverse problem, finding different methods in view of the fact that very few equations can be solved by specific rules. After a little more than two centuries, it is still not possible to establish general rules for all types of differential equations The physical world is constantly changing, it is also up to differential equations to study this change as a natural phenomenon, the laws that govern it, search for a mathematical model that governs its laws and interpret it. Differential equations in general, constitute the mathematical essence for modeling and comprise a large number of events in: physics, economics, engineering, chemistry, biology, physiology and economics, medicine, mathematics, movement of celestial bodies such as planets, moons and artificial satellites, astronomy, among others. Therefore, differential equations play a crucial role for their study. The content is thinking above all to facilitate understanding by the student, providing the basic technique to face a problem. Which is why numerous examples are included and most proofs are carefully detailed. Although abstraction is important, it makes no sense to study differential equations without reference to practical problems. I want to express my appreciation: to the Faculty of the Department of Mathematics; the director of the general office of interdisciplinary research; to the Rector of the San Luis Gonzaga National University, for crystallizing my sabbatical year and allowing a happy ending for the culmination of this book; to the physicist Mg. Carlos Tenorio, for the paintings alluding to the theme and to my dear Ica; special thanks to the Editorial Fund IDICAP PACÍFICO for the publication in digital version; there are many people who read the manuscript, their comments were valuable; thank you all.

This book is included in DOAB.