## Ordinary Differential Equations – Pontryagin

In this post, we will see the book Ordinary Differential Equations by L. S. Pontryagin.

This book has been written on the basis of lectures which I delivered at the department of mathematics and mechanics of Moscow State Uni­versity. In drawing up the program for my lectures, I proceeded on the belief that the selection of material must not be random nor must it rest exclusively on established tradition. The most important and interesting applications of ordinary differential equations to engineering are found in the theory of oscillations and in the theory of automatic control. These applications were chosen to serve as guides in the selection of material. Since oscillation theory and automatic control theory without doubt also play a very important role in the development of our contemporary technical culture, my approach to the selection of material for the lecture course is, if not the only possible one, in any case a reasonable one. In attempting to give the students not only a purely mathematical tool suitable for engineering applications, but also to demonstrate the appli­cations themselves, I included certain engineering problems in the lectures. In the book they are presented in §13, 27, and 29. I consider that these problems constitute an integral organic part of the lecture course and, accordingly, of this book.

The book was translated from Russian by Leonas Kacinskas and Walter B. Counts and was published in  1962.

You can get the book here.

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# Contents

## CHAPTER 1. INTRODUCTION. 1

1. First-order differential equations 1
2. Some elementary integration methods 6
3. Formulation of the existence and uniqueness theorem 18
4. Reduction of a general system of differential equations to a normal system 25
5. Complex differential equations 33
6. Some properties of linear differential equations 39

## CHAPTER 2. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 41

7. Linear homogeneous equation with constant coefficients. The case of simple roots 42
8. The linear homogeneous equation with constant coefficients: Case of multiple roots 50
9. Stable polynomials 57
10. The linear nonhomogeneous equation with constant coefficients 62
11. Method of elimination 67
12. The method of complex amplitudes 76
13. Electrical circuits 80
14. The normal linear homogeneous system with constant coefficients 94
15. Autonomous systems of differential equations and their phase spaces 103
16. The phase plane of a linear homogeneous system with constant coefficients 115

## CHAPTER 3. LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS 127

17. The normal system of linear equations 127
18. The linear equation of nth order 137
19. The normal linear homogeneous system with periodic coefficients 144

## CHAPTER 4. EXISTENCE THEOREMS 150

20. Proof of the existence and uniqueness theorem for one equation 150
21. Proof of the existence and uniqueness theorem for a normal system of equations 159
22. Local theorems of continuity and differentiability of solutions 170
23. First integrals 181
24. Behavior of the trajectories on large time intervals 189
25. Global theorems of continuity and differentiability 192

## CHAPTER 5. STABILITY 200

26. Lyapunov’s theorem 201
27. The centrifugal governor and the analysis of Vyshnegradskiy 213
28. Limit cycles 220
29. The vacuum-tube oscillator 236
30. The states of equilibrium of a second-order 244
31. Stability of periodic solutions 261

## CHAPTER 6. LINEAR ALGEBRA 277

32. The minimal annihilating polynomial 277
33. Matrix functions 284
34. The Jordan form of a atone 291

INDEX 296