This book introduces the basic solutions and theories of difference and ordinary differential equations in detail,which meets the requirements of the relevant professional syllabus and is composed of 7 chapters,including the common practical models of difference and differential equations,the solutions of difference equations, the solutions of first order and higher order differential equations,the basic theory of differential equations and the solutions of linear differential equations,qualitative theory.This book starts with not the basic concepts and theories,but the practical model and pays more attention to the application of theoretical knowledge.
This book can be used as a reference for students,teachers and researchers of mathematics,physics,engineering and related majors in colleges and universities.
This book introduces the basic solutions and theories of difference and ordinary differential equations in detail,which meets the requirements of the relevant professional syllabus and is composed of 7 chapters,including the common practical models of difference and differential equations,the solutions of difference equations, the solutions of first order and higher order differential equations,the basic theory of differential equations and the solutions of linear differential equations,qualitative theory.This book starts with not the basic concepts and theories,but the practical model and pays more attention to the application of theoretical knowledge.
This book can be used as a reference for students,teachers and researchers of mathematics,physics,engineering and related majors in colleges and universities.
本书详细介绍了差分方程和常微分方程的基本解法和基本理论，其内容符合相关专业教学大纲的要求，共由七章组成，包括常见的差分和微分方程实际模型，差分方程的求解，一阶及高阶微分方程求解，微分方程组的基本理论及线性微分方程组的解法，定性理论初步。本书并没有以基本概念和理论作为开端，而是从实际模型出发，更加注重理论知识的应用。
本书可供高等学校数学、物理、工程及相关专业的学生、教师及研究人员参考使用。
目錄：
Chapter 1Basic difference equations models001
1.1Difference equations of financial mathematics001
1.1.1Compound interest and loan repayments001
1.1.2Some Money Related Models002
1.2Difference equations of population theory004
1.2.1Single equations for unstructured population models004
1.2.2Structured populations and linear systems of difference equations006
1.2.3Markov chain008
Chapter 2Basic differential equations models010
2.1Equations related to financial mathematics010
2.2Continuous population models011
2.3Equations of motion： second order equations015
2.4Modelling interacting quantities systems of differential equations018
Chapter 3Solution and applications of difference equations021
3.1Linear first-order difference equations021
3.2Difference calculus and general theory of linear difference equations024
3.2.1Difference calculus025
3.2.2General theory of linear difference equations027
3.3Linear Homogeneous equations with constant coefficients033
3.4Linear Nonhomogeneous equations037
3.5Limiting behavior of solution041
3.6Autonomous（Time-Invariant）Systems043
3.7Exercises043
Chapter 4Concepts and solutions of differential equations047
4.1Concepts047
4.2Existence and uniqueness of solutions052
4.3First-order linear differential equations056
4.4Exact equation and separation of variables062
4.5Integrating factors068
4.6Initial-value and two-point boundary-value071
4.7Exercises074
Chapter 5Second and higher order differential equations077
5.1Algebraic properties of solutions077
5.2Linear equations with constant coefficients085
5.3The non-homogeneous equation092
5.4Higher order differential equations096
5.5The Euler equation103
5.6Exercises105
Chapter 6Systems of differential equations106
6.1Existence and uniqueness theorem106
6.1.1Marks and definitions106
6.1.2Existence and uniqueness of solutions112
6.2General theory of linear differential systems117
6.2.1Linear homogeneous systems117
6.2.2Linear inhomogeneous systems123
6.3Linear differential systems with constant coefficients126
6.3.1Definition and properties of matrix exponent expA126
6.3.2Calculation of fundamental solution matrix129
6.4Exercises141
Chapter 7Qualitative and stability theories147
7.1Two-dimensional autonomous system and phase plane147
7.2Plane singularity155
7.2.1Trajectory distribution of two-dimensional linear systems156
7.2.2Distribution of orbits of two-dimensional nonlinear systems in the neighborhood of singularities165
7.3Limit cycle167
7.4Lyapunov stability169
7.4.1Stability169
7.4.2First approximation theory173
7.5Exercises178
Appendix182
A.1Solution of difference equations182
A.1.1First order linear constant coefficient difference equation182
A.1.2Higher order linear constant coefficient difference equation184
A.1.3Linear constant coefficient difference equations185
A.2Solutions of ordinary differential equations186
A.2.1Symbolic solutions186
A.2.2Numerical solutions189
A.3Exercises195
References196
內容試閱：
Preface
Ordinary differential equation is not only a compulsory basic course for mathematics, applied mathematics, financial mathematics and so on, but also an important theoretical basis for other disciplines.The purpose of this book is to provide students with the necessary theoretical knowledge and related skills.
This book introduces some commonly used practical difference and differential models to convey to readers the importance and interest of the knowledge, together with the idea of solving practical problems.The next part is used to expand the relevant theoretical system, including the basic concepts and solving methods of difference and differential equations, and the qualitative and stability theory of ordinary differential equations.In order to give students a deeper understanding of the basic theories and knowledge they have learned, and to improve their ability to solve practical problems, we provide mathematical experiments using MATLAB in the appendix.
Our book strives to be easy to understand and detailed, and the selected examples and models are as practical as possible, paying attention not only to the students understanding of the basic concepts and methods, but also to the application of knowledge; in addition, the discussion method of this book tries to be in line with the readers thinking habits, the mathematical expression is succinct and clear, and the professional nouns are refined and explained in the form of footnotes on each page.Easy for readers to read and find.
The Chapters 1-3 of the book are written by Dr.Dongmei Zheng, the Chapters 4-5 are written by Dr.Shunjun Jiang, and the Chapters 6-7 and Appendix are written by Dr.Yanqiu Li.Dr Yanqiu Li is responsible final compilation and editing.
Because of our limited ability, mistakes and shortcomings in the book are inevitable.I sincerely hope that readers will not be stingy with their advice.
Yanqiu Li,Dongmei Zheng,Shunjun Jiang
2019.10
前言
常微分方程是数学、应用数学、金融数学等专业的必修基础课程，同时也是其他学科所必需的重要理论基础。编写本书的目的是希望为学生提供必备的理论知识和相关技能。
本书为读者介绍了一些常用的微分和差分实际模型，向他们传递这部分知识的重要性和趣味性，同时也为他们提供了解决实际问题的思路；接下来依次展开相关理论体系，包括差分和微分方程、微分方程组的基本概念、求解方法，常微分方程的定性及稳定性理论初步；为了让学生更深刻地体会所学的基本理论和知识，并提高学生应用差分和微分方程解决实际问题的能力，我们在附录部分提供了利用MATLAB求解的数学实验。
本书在写法上努力做到通俗易懂，详略得当，所选实例及模型尽量做到理论联系实际，既关注学生对于基本概念和基本方法的理解，也注重知识的应用；另外本书的论述方法尽量做到符合读者的思维习惯，数学表达清晰明确，并在每一页以脚注的形式对专业名词进行了提炼和解释，便于读者阅读和查找。
本书的第一、二、三章由郑冬梅博士执笔，第四、五章由江舜君博士完成，第六、七章及附录部分由李艳秋博士负责撰写，全书由李艳秋博士统稿。
本书的编写获得江苏省第二批中外合作办学高水平示范性建设工程项目培育点：南京工业大学与英国谢菲尔德大学合作举办数学与应用数学（金融数学）专业本科教育项目 苏教办外［2017］14号经费支持。
由于作者水平有限，书中不足之处在所难免，诚恳希望读者不吝赐教。
李艳秋，郑冬梅，江舜君
2019年10月
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