關于常微分方程方面的教科書有許多種，但本書卻獨具特物色，書中強調常微分方程的定性性質和幾何性質及其它們的解，全書有272個幾何插圖，卻沒有一個復雜的數(shù)學公式。全書分為5章36節(jié)。本書是俄羅斯數(shù)學家(1937-2010)，1974年菲爾茲獎得主，他的許多優(yōu)秀作品都被翻譯為英文，本書是其中的一本，其簡明的寫作風格、嚴謹?shù)臄?shù)學基礎結合物理直覺，給人一種很輕松漫談式的教學特點，被評為優(yōu)秀的常微分教材。
俄羅斯數(shù)學家(1937-2010)，1974年菲爾茲獎得主，他的許多優(yōu)秀作品都被翻譯為英文，本書是其中的一本。本書被評為優(yōu)秀的常微分教材，簡明的寫作風格、嚴謹?shù)臄?shù)學基礎結合物理直覺，給人一種很輕松漫談式的教學特點。
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
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
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
2. Vector Fields on the Line
1. Existence and Uniqueness of Solutions
2. A Counterexample
3. Proof of Uniqueness
4. Direct Products
5. Examples of Direct Products
6. Equations with Separable Variables
7. An Example: The Lotka-Volterra Model
3. Linear Equations
1. Homogeneous Linear Equations
2. First-order Homogeneous Linear Equations with Periodic Coefficients
3. Inhomogeneous Linear Equations
4. The Influence Function and b-shaped Inhomogeneities
5. Inhomogeneous Linear Equations with Periodic Coefficients
4. Phase Flows
1. The Action of a Group on a Set
2. One-parameter Transformation Groups
3. One-parameter Diffeomorphism Groups
4. The Phase Velocity Vector Field
5. The Action of Diffeomorphisms on Vector Fields and Direction Fields
1. The Action of Smooth Mappings on Vectors
2. The Action of Diffeomorphisms on Vector Fields
3. Change of Variables in an Equation
4. The Action of a Diffeomorphism on a Direction Field
5. The Action of a Diffeomorphism on a Phase Flow
6. Symmetries
1. Symmetry Groups
2. Application of a One-parameter Symmetry Group to Integrate an Equation
3. Homogeneous Equations
4. Quasi-homogeneous Equations
5. Similarity and Dimensional Considerations
6. Methods of Integrating Differential Equations
Chapter 2. Basic Theorems
7. Rectification Theorems
1. Rectification of a Direction Field
2. Existence and Uniqueness Theorems
3. Theorems on Continuous and Differentiable Dependence of the Solutions on the Initial Condition
4. Transformation over the Time Interval from to to t
5. Theorems on Continuous and Differentiable Dependence on a Parameter
6. Extension Theorems
7. Rectification of a Vector Field
8. Applications to Equations of Higher Order than First
1. The Equivalence of an Equation of Order n and a System of n First-order Equations
2. Existence and Uniqueness Theorems
3. Differentiability and Extension Theorems
Chapter 3. Linear Systems
Chapter 4. Proofs of the Main Theorems
Chapter 5. Differential Equations on Manifolds
Examination Topics
Sample Examination Problems
Subject Index
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