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.

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