DESCRIPTION
The book is an introduction to ordinary differential equations, addressed to undergraduate students in exact sciences. Mathematical concepts are introduced in a carefully manner followed by a large number of applications. These applications include two extremely important features: the solution of the differential equations, showing the strength of the theory, and the detailed interpretation of the solutions obtained.
The text exposes the subject step by step. It contains elementary methods to obtain solutions, selected topics of the basic theory of the differential equations, and techniques used to describe the configuration spaces and the asymptotic behavior of the solutions.
Question: It is said that every book has a message. What was the motivation for writing the text “Applied Differential Equations”?
Answer: The great relevance of Mathematics lies in the fact that in addition of having a life of its own as science, with its theories and its problems, it has the unique characteristic of being able to penetrate, as an important and sometimes indispensable weapon in many other branches of the human knowledge. We should not forget that fact, when we do our job, as a teacher or a researcher. When teaching mathematics to students of other areas it is essential to motivate them, showing them the importance of what they are learning to the problems of their specialties. To the students of the mathematical area it is useful to show them a science rich in applications, to tell them that the roots of so many mathematical theories are in the facts of Nature. Through these roots came the force that propelled the remarkable growth of much of a large part of Mathematics in the past. Nobody ignores the work of Newton, Leibniz and others in the creation and development of Calculus, pari passu with Mechanics and other branches of Physics. More recently, were identified characteristics that are similar in the works of Poincaré and of Hilbert. Differential equations is one of the branches of Mathematics that, in our view, should not be studied forgetting these roots.
Question: What kind of students is the book intended for and how is it organized?
Answer: The text is accessible to undergraduate students in the area of exact sciences who have taken a one-year course in Calculus. Some sections may be omitted in a first course. For example, the Picard’s existence and uniqueness theorem is presented in a modern way, because we think that even for beginners, this attitude is instructive. However, the demonstration could eventually be considered too abstract for most students and therefore be omitted. In a first course the emphasis should be on the techniques of obtaining solutions. Therefore the course should be concentrated on chapters 2, 4 and 5, where these techniques are developed. The force of these techniques is on the applications study . These applications can be chosen among those presented in these same chapters. We suggest starting with chapter 2, commenting on the Theorem of Existence and Uniqueness in chapter 3, developing chapters 4 and 5, and finishing with autonomous systems in the plane, studying, for example, the techniques of obtaining solutions of the Linear Systems with Constant Coefficients, sections 6.2.1 e 7.1.2.
The book is organized as follows: the first sections of each chapter are intended to describe the various methods of obtaining solutions; the exercises section follows; and then the applications. The text presents mathematical concepts in a carefully way and is very rich in applications. It contains more applications than is normally studied in traditional courses, giving the reader and teacher the possibility of choosing according to the interests of the class. Some concepts are first introduced in an introductory way, and later, in subsequent chapters, they are presented again in a more thoroughness and complete way. For example, the concept of stability is introduced in section 2.3, and complemented later in chapters 6 and 7. In a more specialized course, whose students have already made the Calculus course, the course can and should be complemented, with more theoretical emphasis, studying chapters 3, 6 and 7, and the corresponding applications.
Question: The sections of the book on some branches of physics are longer than are usually found in a book on mathematics. Wouldn’t it be better to refer the reader to books of physics?
Answer: These sections were precisely the ones that gave us more work to write, because we are not experts in these areas. But we believe the effort was worth it given the goals we have in mind. The book is also intended for the teacher of the courses of differential equations. The time that he has to prepare the course is not enough to study several books of physics and of other fields of the human knowledge. He can do that in a second stage, later. Our sections devoted to the applications give to that teacher, in the language with which he is accustomed, the basic principles of these other areas. This will allow him to talk, without fear, to his students on these applications.
Question: So is the text completely different from the others on Differential Equations?
Answer: No. Most of the books on Differential Equations have applications, but in general the applications are presented very concisely, which makes it impossible for students to really appreciate the role of differential equations in problems. But of course there are good books with the same line as ours. G. F. Simmons, M. Braun, among others. We believe that no one can claim to be original in the subject presented here. Some problems come from Newton himself! Originality in this type of work resides only in the arrangement of subjects, in the choice of problems and in the style of commenting the results. In fact, we try to emphasize, throughout the text, the extremely important attitude of interpreting the results obtained. We think this is essential. It is important to solve equations. But it is equally important to interpret the solutions obtained.
Question: So, one should not give a course on differential equations with an exclusivelly mathematical approach?
Answer: Here, we must separate the levels and objectives of the courses. We believe that, in a first course, at undergraduate level, great attention should be paid to the applications, regardless of the student’s future specialization, Mathematics, Physics or Engineering. Differential Equations were created to solve problems of other sciences. How to forget this? We are also not advocating that, in this first course on Differential Equations, Mathematics be left in the background. It is essential to have both things on equal terms. One justifies and values the other. In more advanced courses on Differential Equations, Mathematics becomes necessarily more sophisticated, and then purely mathematical questions require elaborate treatment and dominate the scene.
Question: Well, but if the student is going to be a researcher on these sophisticated Differential Equations, what was the use of that first half-applied course?
Answer: As a rule, the researcher is a teacher. And being a teacher, he should be prepared to teach students from other areas. We feel that there has been an increasing tendency to remove the application problems from the courses of Differential Equations (and other Mathematics courses). This is not good. Our colleagues from other areas complain and criticize this approach. And what is more serious, they begin to develop within their own departments the Mathematics they need, thus removing from the Departments of Mathematics one of its primary functions.
Question: So does it mean that those students of mathematics that do not intend to follow a career in teaching can do without this course?
Answer: No. Our view is that the researcher in Differential Equations who knows the applications can, occasionally use these applications as a sort of beacon to guide him. How many times a general and abstract result is discovered after analyzing an example from the applications?
Question: Do you, then, believe that every mathematician should know applications? How do you position the pure mathematician within that conception?
Answer: Mathematics is a very extensive science. Obviously there are other areas besides Differential Equations. And notice, that even within these, there are several different lines. Some very far from the applications. These, like other branches of Mathematics, should develop by their own problems and motivations. Mathematics develops by the needs and the problems coming from other sciences, but it is not determined solely by this. And it is good that it be so, because needs change over time, and the richer the Mathematics, the better it can help man. So there is a lot of fields for mathematical research independent of immediate applications. One can think of mathematical knowledge as an account in the bank. The reserves can be used when needed. And this has happened in the past with several branches of mathematics, developed independently of immediate necessity, which were later used in an essential way in other sciences.
ABOUT THE AUTHORS
Djairo Guedes de Figueiredo
Djairo Guedes de Figueiredo, was born in Limoeiro do Norte, Ceará, is a Civil Engineer (UFRJ, 1956), Master of Science (NYU, 1958) and Doctor of Philosophy (NYU, 1961). For several years he was Professor at the universities of Illinois and Brasília. He is currently a Professor at UNICAMP. In 1965 and 1984 was awarded a fellowship from the Guggenheim Foundation. He is a full member of the Brazilian Academy of Sciences and since 1985 a category 1A researcher of CNPq. In 1992 he was awarded the Scholarship of Academic Recognition “Zeferino Vaz”, by the University Council of UNICAMP and in 1995 with the Grã Cruz da Ordem do Mérito Científico.
His research field is the theory of partial differential equations, having written several monographs and research papers published in specialized journals in Brazil and abroad.
Aloísio Freiria Neves
Aloísio Freiria Neves joined the course in Mathematics at PUC motivated by Mathematics itself, and by the possibility of learning how to solve problems. He graduated at UNICAMP, where he obtained his Masters and Doctor degrees. He works at UNICAMP since 1973, when he was invited to be Teacher of Integral Differential Calculus. Currently he is Professor “Livre Docente” at UNICAMP and his area of greatest interest and research is the Partial Differential Equations of Evolution.
