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