微分方程

CHAPTER1INTRODUCTIONTODIFFERENTIALEQUATIONS

1.1MODELSONDIFFERENTIALEQUATIONS

1.2BASICCONCEPTSOFDIFFERENTIALEQUATIONS

1.2.1ClassificationsofDifferentialEquations

1.2.2SolutionofaDifferentialEquation

1.2.3Initial-andBoundary-ValueProblems

Summary

Exercise

CHAPTER2FIRST-ORDERORDINARYDIFFERENTIALEQUATIONS

2.1THREEBASICTYPESOFFIRST-ORDEREXPLICITEQUATIONS

2.1.1EquationsinWhichtheVariablesAreSeparable

2.1.2First-OrderLinearDifferentialEquations

2.1.3ExactDifferentialEquations

2.2TWOAVAILABLETACTICS

2.2.1FindingIntegratingFactors

2.2.2UseofSubstitutions

2.3FUNDAMENTALTHEORYOFINITIAL-VALUEPROBLEM

2.3.1GeometricInterpretationofSolutions

2.3.2ExistenceandUniquenessoftheSolutions

2.3.3*PropertiesoftheSolutiononInitia-Value

2.4METHODSOFAPPROXIMATION

2.4.1ThePieardMethod

2.4.2TheCauehy-EulerMethod

2.4.3TaylorSeriesMethod

2.5FIRST-ORDERIMPLICITEQUATIONS

2.5.1SpecialMethodsforFirst-OrderhnplicitEquations

2.5.2*SingularSolutionsandEnvelopes

Summary

Exercise

CHAPTER3HIGH-ORDERORDINARYDIFFERENTIALEQUATIONS

3.1FUNDAMENTALTHEORIESOFLINEAREQUATIONS

3.1.1PreliminaryKnowledgefortheLinearEquations

3.1.2PropertiesofSolutionsoftheLinearEquations

3.2LINEAREQUATIONSWITHCONSTANTCOEFFICIENTS

3.2.1HomogeneousLinearEquationswithConstantCoefficients

3.2.2SolutionbyUndeterminedCoefficients

3.2.3SolutionbyLaplaceTransform

3.3LINEAREQUATIONSWITHVARIABLECOEFFICIENTS

3.3.1EulersEquation

3.3.2SolutionbyLiouvillesFormula

3.3.3SolutionbyVariationofParameters

3.4SOLUTIONBYPOWERSERIES

3.4.1OrdinaryandSingularPointoftheEquation

3.4.2SolutionatanOrdinaryPointoftheEquation

3.4.3*Soh:fionataRegularSingularPointoftheEquation

3.5OTHERPROBLEMSOFnTH-ORDEREQUATIONS

3.5.1LinearBoundary-ValueProblems

3.5.2Reduciblenth-OrderDifferentialEquations

Summary

Exercise

CHAPTER4FIRST-ORDERORDINARYDIFFERENTIALSYSTEMS

4.1FUNDAMENTALTHEORIESOFLINEARSYSTEMS

4.1.1PreliminaryKnowledgefortheLinearSystems

4.1.2PropertiesofSolutionsofFirst-OrderLinearSystems

4.1.3SolutionMatrixandGeneralSolutionMatrix

4.2HOMOGENEOUSLINEARSYSTEMSWITHCONSTANTCOEFFICIENTS

4.2.1SolutionbyFindingaGeneralSolutionMatrix

4.2.2SolutionbyFindingtheStandardSolutionMatrix

4.3NON-HOMOGENEOUSLINEARSYSTEMSWITHCONSTANTCOEFFICIENTS

4.3.1*SolutionbyUndeterminedCoefficients

4.3.2*SolutionbyVariationofParameters

4.3.3SolutionbyLaplaceTransform

4.4THEFIRSTINTEGRAL

4.4.1BasicConceptsandTheoriesoftheFirstIntegral

4.4.2SolutionbyFindingtheFirstIntegrals

Summary

Exercise

CHAPTER5"QUALITATIVEANALYSISANDSTABILITYOFSOLUTIONS

5.1INTRODUCTIONTOQUALITATIVEANALYSIS

5.1.1DifferentialDynamicSystems

5.1.2EquilibriumPointandClosedTrajectory

5.2STABILITYOFTHETRIVIALSOLUTION

5.2.1ConceptsofStabilityoftheTrivialSolution

5.2.2TheTrivialSolutionoftheLinearSystem

5.2.3MethodofLinearApproximation

5.2.4LiapunovsSecondMethod

5.3LOCALANALYSISOFTWO-DIMENSIONALAUTONOMOUSSYSTEMS

5.3.1ClassificationoftheEquilibriumPoints

5.3.2ClosedTrajectoryandLimitCycle

5.4GLOBALPHASEDIAGRAMOFTWO-DIMENSIONALAUTONOMOUSSYSTEMS

5.4.1InfinitePointsoftheSystem

5.4.2ExamplesfortheGlobalPhaseDiagram

Summary

Exercise

CHAPTER6INTRODUCTIONTOPARTIALDIFFERENTIALEQUATIONS

6.1FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS

6.1.1BasicConceptandTheory

6.1.2First-OrderHomogeneousLinearPDE

6.1.3First-OrderQuasi-LinearPDE

6.1.4TheCauchyProblem

6.2SECOND-ORDERPARTIALDIFFERENTIALEQUATIONS

6.2.1SimplificationofSecond-OrderQuasi-LinearEquations

6.2.2Second-OrderLinearPartialDifferentialEquations

Summary

Exercise

KEYSORHINTS

　　本书是高等学校本科生“微分方程”课程双语教学的教材，主要介绍各类微分方程的解法，全书共分6章，主要包括：微分方程模型与基本概念；一阶常微分方程(包括一阶显式常微分方程和一阶隐式常微分方程)的解法；常系数高阶线性微分方程的解法、变系数微分方程的解法以及边值问题和可降阶的高阶微分方程的解法；线性方程组的基本原理、常系数齐次线性方程组的解法、常系数非齐次线性方程组的解法；首次积分；解的定性分析方法和稳定性原理；一阶和二阶偏微分方程的解法。　　本书是高等学校本科生“微分方程”课程双语教学的教材，主要介绍各类微分方程的解法，全书共分6章，主要包括：微分方程模型与基本概念；一阶常微分方程(包括一阶显式常微分方程和一阶隐式常微分方程)的解法；常系数高阶线性微分方程的解法、变系数微分方程的解法以及边值问题和可降阶的高阶微分方程的解法；线性方程组的基本原理、常系数齐次线性方程组的解法、常系数非齐次线性方程组的解法；首次积分；解的定性分析方法和稳定性原理；一阶和二阶偏微分方程的解法。　　全书各章均编写了习题(答案附在全书的最后)。　　本书除了适合作为高等学校本科生“微分方程”双语课程教学使用外，也可作为自学读本和研究生参考书。