微分方程

內(nèi)容概要

本書(shū)是高等學(xué)校本科生“微分方程”課程雙語(yǔ)教學(xué)的教材，主要介紹各類微分方程的解法，全書(shū)共分6章，主要包括：微分方程模型與基本概念；一階常微分方程（包括一階顯式常微分方程和一階隱式常微分方程）的解法；常系數(shù)高階線性微分方程的解法、變系數(shù)微分方程的解法以及邊值問(wèn)題和可降階的高階微分方程的解法；線性方程組的基本原理、常系數(shù)齊次線性方程組的解法、常系數(shù)非齊次線性方程組的解法；首次積分；解的定性分析方法和穩(wěn)定性原理；一階和二階偏微分方程的解法。    全書(shū)各章均編寫了習(xí)題（答案附在全書(shū)的最后）。    本書(shū)除了適合作為高等學(xué)校本科生“微分方程”雙語(yǔ)課程教學(xué)使用外，也可作為自學(xué)讀本和研究生參考書(shū)。

書(shū)籍目錄

CHAPTER 1  INTRODUCTION TO DIFFERENTIAL EQUATIONS  1.1  MODELS ON DIFFERENTIAL EQUATIONS  1.2  BASIC CONCEPTS OF DIFFERENTIAL EQUATIONS    1.2.1  Classifications of Differential Equations    1.2.2  Solution of a Differential Equation    1.2.3  Initial-and Boundary-Value Problems  Summary  ExerciseCHAPTER 2  FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS  2.1  THREE BASIC TYPES OF FIRST-ORDER EXPLICIT EQUATIONS    2.1.1  Equations in Which the Variables Are Separable    2.1.2  First-Order Linear Differential Equations    2.1.3  Exact Differential Equations  2.2  TWO AVAILABLE TACTICS    2.2.1  Finding Integrating Factors    2.2.2  Use of Substitutions  2.3  FUNDAMENTAL THEORY OF INITIAL-VALUE PROBLEM    2.3.1  Geometric Interpretation of Solutions    2.3.2  Existence and Uniqueness of the Solutions    2.3.3 * Properties of the Solution on Initia-Value  2.4  METHODS OF APPROXIMATION    2.4.1  The Pieard Method    2.4.2  The Cauehy-Euler Method    2.4.3  Taylor Series Method  2.5  FIRST-ORDER IMPLICIT EQUATIONS    2.5.1  Special Methods for First-Order hnplicit Equations    2.5.2 * Singular Solutions and Envelopes  Summary  ExerciseCHAPTER 3  HIGH-ORDER ORDINARY DIFFERENTIAL EQUATIONS  3.1  FUNDAMENTAL THEORIES OF LINEAR EQUATIONS    3.1.1  Preliminary Knowledge for the Linear Equations    3.1.2  Properties of Solutions of the Linear Equations  3.2  LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS    3.2.1  Homogeneous Linear Equations with Constant Coefficients    3.2.2  Solution by Undetermined Coefficients    3.2.3  Solution by Laplace Transform  3.3  LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS    3.3.1  Euler' s Equation    3.3.2  Solution by Liouville' s Formula    3.3.3  Solution by Variation of Parameters  3.4  SOLUTION BY POWER SERIES    3.4.1  Ordinary and Singular Point of the Equation    3.4.2  Solution at an Ordinary Point of the Equation    3.4.3 * Soh:fion at a Regular Singular Point of the Equation  3.5  OTHER PROBLEMS OF nTH-ORDER EQUATIONS    3.5.1  Linear Boundary-Value Problems    3.5.2  Reducible nth-Order Differential Equations  Summary  ExerciseCHAPTER 4  FIRST-ORDER ORDINARY DIFFERENTIAL SYSTEMS  4.1  FUNDAMENTAL THEORIES OF LINEAR SYSTEMS    4.1.1  Preliminary Knowledge for the Linear Systems    4.1.2  Properties of Solutions of First-Order Linear Systems    4.1.3  Solution Matrix and General Solution Matrix  4.2  HOMOGENEOUS LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS    4.2.1  Solution by Finding a General Solution Matrix    4.2.2  Solution by Finding the Standard Solution Matrix  4.3  NON-HOMOGENEOUS LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS    4.3.1 * Solution by Undetermined Coefficients    4.3.2 * Solution by Variation of Parameters    4.3.3  Solution by Laplace Transform  4.4  THE FIRST INTEGRAL    4.4.1  Basic Concepts and Theories of the First Integral    4.4.2  Solution by Finding the First Integrals  Summary  ExerciseCHAPTER 5 "  QUALITATIVE ANALYSIS AND STABILITY OF SOLUTIONS  5.1  INTRODUCTION TO QUALITATIVE ANALYSIS    5.1.1  Differential Dynamic Systems    5.1.2  Equilibrium Point and Closed Trajectory  5.2  STABILITY OF THE TRIVIAL SOLUTION    5.2.1  Concepts of Stability of the Trivial Solution    5.2.2  The Trivial Solution of the Linear System    5.2.3  Method of Linear Approximation    5.2.4  Liapunov' s Second Method  5.3  LOCAL ANALYSIS OF TWO-DIMENSIONAL AUTONOMOUS SYSTEMS    5.3.1  Classification of the Equilibrium Points    5.3.2  Closed Trajectory and Limit Cycle  5.4  GLOBAL PHASE DIAGRAM OF TWO-DIMENSIONAL AUTONOMOUS SYSTEMS    5.4.1  Infinite Points of the System    5.4.2  Examples for the Global Phase Diagram  Summary  ExerciseCHAPTER 6  INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS  6.1  FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS    6.1.1  Basic Concept and Theory    6.1.2  First-Order Homogeneous Linear PDE    6.1.3  First-Order Quasi-Linear PDE    6.1.4  The Cauchy Problem  6.2  SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS    6.2.1  Simplification of Second-Order Quasi-Linear Equations    6.2.2  Second-Order Linear Partial Differential Equations  Summary  ExerciseKEYS OR HINTS

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