This book has been written on the basis of lectures which I delivered at the department of mathematics and mechanics of Moscow State Uni­ versity. 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 tech­ nical 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 appli­ cations 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.

We received a mail with the title “Pontryagin – Ordinary Differential Equations (retyped in LaTeX)” from Althea Sindy

inside it was this gem of book typeset in LaTeX with this lovely message

“I have given this book the love it deserves. :3”

You sure have, thanks a ton!

All credits to Althea Sindy for reviving this gem of book and giving it a new life.

 

You can get the book here and here

 

Contents

1 INTRODUCTION 1

1 First-order differential equations . . . . . . . . . . . . . . 1

2 Some elementary integration methods . . . . . . . . . . . 6

3 Formulation of the existence and uniqueness theorem . . . 20

4 Reduction of a general system of differential equations to a

normal system . . . . . . . . . . . . . . . . . . . . . . . . 5 Complex differential equations . . . . . . . . . . . . . . . 6 Some properties of linear differential equations . . . . . . 28

36

42

2 LINEAR EQUATIONS WITH CONSTANT COEFFI-

CIENTS 45

7 The linear homogeneous equation with constant coefficients.

Case of simple roots . . . . . . . . . . . . . . . . . . . . . 46

8 The linear homogeneous equation with constant coefficients.

Case of multiple roots . . . . . . . . . . . . . . . . . . . . 9 Stable polynomials . . . . . . . . . . . . . . . . . . . . . . 55

62

10 The linear nonhomogeneous equation with constant coeffi-

cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 13 14 Method of elimination . . . . . . . . . . . . . . . . . . . . 12 The method of complex amplitudes . . . . . . . . . . . . . Electrical circuits . . . . . . . . . . . . . . . . . . . . . . . 68

73

82

87

The normal linear homogeneous system with constant coef-

ficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

15 Autonomous systems of differential equations and their

phase spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 113

16 The phase plane of a linear homogeneous system with con-

stant coefficients . . . . . . . . . . . . . . . . . . . . . . . 127

3 LINEAR EQUATIONS WITH VARIABLE COEFFI-

CIENTS 143

17 The normal system of linear equations . . . . . . . . . . . 143

vvi CONTENTS

18 19 The linear equation of nth order . . . . . . . . . . . . . . The normal linear homogeneous system with periodic coef-

ficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

161

4 EXISTENCE THEOREMS 169

20 Proof of the existence and uniqueness theorem for one

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

21 Proof of the existence and uniqueness theorem for a normal

system of equations . . . . . . . . . . . . . . . . . . . . . . 179

22 23 Local theorems of continuity and differentiability of solutions 190

First integrals . . . . . . . . . . . . . . . . . . . . . . . . . 203

24 Behavior of the trajectories on large time intervals . . . . 211

25 Global theorems of continuity and differentiability . . . . 214

5 STABILITY 223

26 27 28 29 30 Lyapunov’s theorem . . . . . . . . . . . . . . . . . . . . . 225

The centrifugal governor and the analysis of Vyshnegradskiy 237

Limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 245

The vacuum-tube oscillator . . . . . . . . . . . . . . . . . 265

The states of equilibrium of a second-order autonomous

system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Stability of periodic solutions . . . . . . . . . . . . . . . . 273

293

6 LINEAR ALGEBRA 311

32 33 34 The minimal annihilating polynomial . . . . . . . . . . . . Matrix functions . . . . . . . . . . . . . . . . . . . . . . . The Jordan form of a matrix . . . . . . . . . . . . . . . . 311

318

326