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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