Difference equations and ordinary differential equations

Author: Ruan Jiong
Publisher:
Publish Date: 2002-08-01
Features: Over 10 years ago, several teachers in our department compiled a series of books titled "Guidance for University Mathematics Learning." After its publication, the series was very well-received and quickly sold out. Subsequently, colleagues from sister institutions and many young students wrote to request purchases, and the publisher contacted us multiple times about reissuing the books. However, the authors, who have long been burdened with heavy teaching and research tasks, have had little time to revise the work. In recent years, with the development of the discipline, curriculum construction has once again been placed on the agenda. New textbooks for some important foundational courses in our department have been published, while at the same time, many teachers have once again expressed the desire to organize and summarize their accumulated teaching experiences. With the encouragement of Fudan University Press, it has become timely and natural to launch this brand-new series. The development of mathematical science is currently in an extraordinary period. Advances in science and technology, increased practical applications, the influence of computers, and the progress of mathematical science itself have greatly expanded the scope and fields of mathematical science. In many contexts, mathematics has moved from the background of scientific research to the forefront of technological application, becoming the key to unlocking numerous opportunities. This has led to ever-increasing societal demands for mathematical proficiency, expecting the emergence of more mathematicians with solid foundations, strong innovative abilities, broad knowledge, and outstanding comprehensive qualities. Correspondingly, the goals of mathematical education have evolved beyond merely imparting professional knowledge to guiding students in mastering a scientific language, learning a rational mode of thinking, and receiving training in mathematical qualities such as deduction, induction, analysis, and analogy. Effective mathematical training will prepare students to fully participate in future competition. The theory of mathematics is wonderful and fascinating; its methods are exquisite and diverse; but mastering mathematics requires arduous effort. In teaching, we often encounter students who can recite basic formulas but struggle with slightly modified calculations, or who can remember basic theorems but fail to provide well-structured reasoning. Some students still cling to the "pattern-based teaching" they received in early years, which was inappropriately exaggerated for exam purposes, and they do not understand why this teaching method has disappeared from university classrooms. These students' approaches to learning mathematics are often immature, and their understanding of mathematical knowledge remains superficial rather than substantial. In fact, compared to most other disciplines, mathematics offers students more opportunities for independent thinking. In the learning process of any mathematics course, the primary role is not the teacher but the student. The process of learning mathematics should be one of re-creation. Students should interpret and analyze the content based on their own understanding and transform their original understanding with new perspectives, thereby imprinting their unique mark on their personal repository of mathematical knowledge. Only through deep reflection, by organically integrating newly acquired knowledge into their existing knowledge structure and experiencing mathematics through the creativity of their minds, can they truly master abstract concepts and develop intuitive understanding. We hope this series can provide students with valuable references and inspiration in their mathematical learning. Although methods of learning mathematics vary from person to person, some fundamental aspects of mathematics courses are worth common attention. First, to excel in a mathematics course, it is undeniable that one must master its core mathematical ideas. This means not only deeply understanding the concepts and properties of the main objects but also scientifically integrating a series of definitions and theorems, grasping the overall development, progression, and advancement of the knowledge system, and comprehensively the mathematical ideas and spirit throughout the course. Second, mathematical ideas are realized through specific mathematical methods. The mathematical methods embedded in each course provide the primary tools for constructing corresponding theoretical frameworks and also serve as the basis for making analytical, judgmental, transformative, and solution-oriented strategies. From the formation of conjectures to the analysis, calculation, reasoning, refinement, and expansion of solutions, mathematical methods demonstrate their value in the process of problem-solving. Third, each mathematics course contains numerous special mathematical techniques. These not only showcase the flexibility of operations and arguments but are also essential elements of successful mathematical methods. A profound technique often arises from rich imagination and keen observation. The introduction and training of mathematical techniques are of great significance for stimulating, broadening, and deepening students' thinking. In summary, mathematical ideas, methods, and techniques form an integral whole, collectively creating the rich and vibrant essence of each mathematics course. Therefore, to excel in mathematics, one must focus on grasping ideas, mastering methods, and becoming proficient in techniques. Based on this understanding, each book in this series generally includes brief introductions and summaries of concepts and properties, analyses and discussions of main methods and typical examples, as well as a certain number of exercises. We hope readers can refer to this tripartite structure—thinking about, digesting, and applying mathematical ideas, methods, and techniques—to elevate their problem-solving abilities to a new level. The authors of this series all possess extensive teaching experience and have taken care to address the diverse needs of readers: whether used synchronously while studying a course, as a reference for review after completing a course, or as preparation for postgraduate entrance exams, readers will gain significant benefits. We are also willing to use this series to engage in broad teaching exchanges with colleagues from sister institutions. Fudan University's Department of Mathematics has included the compilation of this series in its plan to strengthen undergraduate teaching. Many professors from the department and research institutes have offered valuable suggestions on how to compile the series, creating favorable conditions for improving its quality. Ms. Fan Renmei of Fudan University Press has devoted considerable effort to the planning and editing of this series. We extend our sincere gratitude to all of them. Given our limitations, it is inevitable that this series contains errors and shortcomings, and we sincerely hope that readers will kindly point them out. Through the joint efforts of authors and readers, we hope that subsequent revisions will make this series increasingly mature. This book systematically introduces various problems and methods for solving them in difference equations and ordinary differential equations. The first part introduces the reduction, solution methods, and properties of solutions for difference equations; the second part covers various solvable types of ordinary differential equations and methods for analytical and numerical solutions; the third part presents the fundamental theory and qualitative theory and methods of solutions for ordinary differential equations; the fourth part discusses how to model practical problems as ordinary differential equations. Each chapter includes examples and exercises, with answers or hints provided for the exercises. This book can serve as a reference for science and engineering students, self-learners, and teachers of difference equations and ordinary differential equations courses, as well as a study guide for readers preparing for postgraduate entrance exams.