Ordinary Differential Equations
豆瓣
L.S. Pontryagin
简介
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.
目录
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 autonomous system . 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 matrix 291
Index 296
