Ordinary Differential Equations

Chapter 1. Introduction

1. First-Order Differential Equations

2. Some Elementary Integration Methods

3. Formulation of the Existence and Uniqueness Theorem

4. Reduction of a General System of Differential Equations to a Normal System

5. Complex Differential Equations

6. Some Properties of Linear Differential Equations

Chapter 2. Linear Equations with Constant Coefficients

7. Linear Homogeneous Equation with Constant Coefficients. The Case of Simple Roots

8. The Linear Homogeneous Equation with Constant Coefficients. Case of Multiple Roots

9. Stable Polynomials

10. The Linear Nonhomogeneous Equation with Constant Coefficients

11. Method of Elimination

12. The Method of Complex Amplitudes

13. Electrical Circuits

14. The Normal Linear Homogeneous System with Constant Coefficients

15. Autonomous Systems of Differential Equations and Their Phase Spaces

16. The Phase Plane of a Linear Homogeneous System with Constant Coefficients

Chapter 3. Linear Equations with Variable Coefficients

17. The Normal System of Linear Equations

18. The Linear Equation of nth Order

19. The Normal Linear Homogeneous System with Periodic Coefficients

Chapter 4. Existence Theorems

20. Proof of the Existence and Uniqueness Theorem for One Equation

21. Proof of the Existence and Uniqueness Theorem for a Normal System of Equations

22. Local Theorems of Continuity and Differentiability of Solutions

23. First Integrals

24. Behavior of the Trajectories on Large Time Intervals

25. Global Theorems of Continuity and Differentiability

Chapter 5. Stability

26. Lyapunov's Theorem

27. The Centrifugal Governor and the Analysis of Vyshnegradskiy

28. Limit Cycles

29. The Vacuum-Tube Oscillator

30. The States of Equilibrium of a Second-Order Autonomous System

31. Stability of Periodic Solutions

Chapter 6. Linear Algebra

32. The Minimal Annihilating Polynomial

33. Matrix Functions

34. The Jordan Form of a Matrix

Index