导语

  

    The first two chapters of this book have been thoroughly revised and significantly expanded. Sections have been added on elementary methods of integration (on homogeneous and inhomogeneous first-order linear equations and on homogeneous and quasi-homogeneous equations), on first-order linear and quasi-linear partial differential equations, on equations not solved for the derivative, and on Sturm's theorems on the zeros of second-order linear equations. Thus the new edition contains all the questions of the current syllabus in the theory of ordinary differential equations.

内容提要

  

Chapter 1.Basic Concepts
1.Phase Spaces
1.Examples of Evolutionary Processes
2.Phase Spaces
3.The Integral Curves of a Direction Field
4.A Differential Equation and its Solutions
5.The Evolutionary Equation with a One-dimensional Phase Space
6.Example: The Equation of Normal Reproduction
Chapter 2.Basic Theorems
7.Example: The Explosion Equation
8.Example: The Logistic Curve
9.Example: Harvest Quotas
10.Example: Harvesting with a Relative Quota
11.Equations with a Multidimensional Phase Space
12.Example: The Differential Equation of a Predator-Prey System
Chapter 3.Linear Systems
13.Example: A Free Particle on a Line
14.Example: Free Fall
15.Example: Small Oscillations
16.Example: The Mathematical Pendulum
17.Example: The Inverted Pendulum
18.Example: Small Oscillations of a Spherical Pendulum
19.The Linear Equation with a Complex Phase Space
20.The Complexification of a Real Linear Equation
21.The Classification of Singular Points of Linear Systems
22.The Topological Classification of Singular Points
23.Stability of Equilibrium Positions
24.The Case of Purely Imaginary Eigenvalues
25.The Case of Multiple Eigenvalues
26.Quasi-polynomials
27.Nonautonomous Linear Equations
28.Linear Equations with Periodic Confficients
29.Variation of Constants
Chapter 4.Proofs of the Main Theorems
30.Contraction Mappings
31.Proof of the Theorems on Existence and Continuous Dependence on the Initial Conditions
32.The Theorem on Differentiability
Chapter 5.Differential Equations on Manifolds
33.Differentiable Manifolds
34.The Tangent Bundle.Vector Fields on a Maniflod
35.The Phase Flow Defined by a Vector Field
36.The Indices of the Singular Points of a Vector Field
Examination Topics
Sample Examination Problems
Subject Index