导语

  

内容提要

  

    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。

Chapter 1Basic difference equations models
  1.1  Difference equations of financial mathematics
    1.1.1  Compound interest and loan repayments
    1.1.2  Some Money Related Models
  1.2  Difference equations of population theory
    1.2.1  Single equations for unstructured population models
    1.2.2  Structured populations and linear systems of difference equations
    1.2.3  Markov chain
Chapter 2Basic differential equations models
  2.1  Equations related to financial mathematics
  2.2  Continuous population models
  2.3  Equations of motion: second order equations
  2.4  Modelling interacting quantities systems of differential equations
Chapter 3Solution and applications of difference equations
  3.1  Linear first-order difference equations
  3.2  Difference calculus and general theory of linear difference equations
    3.2.1  Difference calculus
    3.2.2  General theory of linear difference equations
  3.3  Linear Homogeneous equations with constant coefficients
  3.4  Linear Nonhomogeneous equations
  3.5  Limiting behavior of solution
  3.6  Autonomous（Time-Invariant）Systems
  3.7  Exercises
Chapter 4Concepts and solutions of differential equations
  4.1  Concepts
  4.2  Existence and uniqueness of solutions
  4.3  First-order linear differential equations
  4.4  Exact equation and separation of variables
  4.5  Integrating factors
  4.6  Initial-value and two-point boundary-value
  4.7  Exercises
Chapter 5Second and higher order differential equations
  5.1  Algebraic properties of solutions
  5.2  Linear equations with constant coefficients
  5.3  The non-homogeneous equation
  5.4  Higher order differential equations
  5.5  The Euler equation
  5.6  Exercises
Chapter 6Systems of differential equations
  6.1  Existence and uniqueness theorem
    6.1.1  Marks and definitions
    6.1.2  Existence and uniqueness of solutions
  6.2  General theory of linear differential systems
    6.2.1  Linear homogeneous systems
    6.2.2  Linear inhomogeneous systems
  6.3  Linear differential systems with constant coefficients
    6.3.1  Definition and properties of matrix exponent expA
    6.3.2  Calculation of fundamental solution matrix
  6.4  Exercises
Chapter 7Qualitative and stability theories
  7.1  Two-dimensional autonomous system and phase plane
  7.2  Plane singularity
    7.2.1  Trajectory distribution of two-dimensional linear systems
    7.2.2  Distribution of orbits of two-dimensional nonlinear systems in the neighborhood of singularities
  7.3  Limit cycle
  7.4  Lyapunov stability
    7.4.1  Stability
    7.4.2  First approximation theory
  7.5  Exercises
Appendix
  A.1  Solution of difference equations
    A.1.1  First order linear constant coefficient difference equation
    A.1.2  Higher order linear constant coefficient difference equation
    A.1.3  Linear constant coefficient difference equations
  A.2  Solutions of ordinary differential equations
    A.2.1  Symbolic solutions
    A.2.2  Numerical solutions
  A.3  Exercises
References