Science for Everyone Series 

Differential Equations in Applications 

by V. V. Amel'kin

This book is an easy-reading introduction to ordinary differential equations and their use in the study of real phenomena and processes. Problems taken from various fields of knowledge illustrate the tools used in setting up differential equations and the methods employed in their qualitative investigation. The book should be useful to  high-school students, teachers of science courses, college students, and specialists of, non-mathematical professions who use mathematics in their work.

The book discusses many interesting problems and has 2 chapters. The first one is dedicated to constructing the differential models and finding their solutions. The second chapter is more interesting, it shows you the way to deal with differential equations qualitatively.

The book was translated from the Russian by Eugene Yankovsky and was first published by Mir in 1990.


PDF | 7.8 MB | OCR | Bookmarked | 600 dpi | Paginated | Cover | Pages: 284


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Contents

Chapter 1. Construction of Differential Models and Their Solutions 11

1.1 Whose Coffee Was Hotter? 11
1.2 Steady-State Heat Flow 14
1.3 An Incident in a National Park 18
1.4 Liquid Flow Out of Vessels. The Water Clock 26
1.5 Effectiveness of Advertising 30
1.6 Supply and Demand 32
1.7 Chemical Reactions 34
1.8 Differential Models in Ecology 38
1.9 A Problem from the Mathematical Theory of Epidemics 44
1.10 The Pursuit Curve 51
1.11 Combat Models 55
1.12 Why Are Pendulum Clocks Inaccurate? 69
1.13 The Cycloidal Clock 73
1.14 The Brachistochrone Problem 81
1.15 The Arithmetic Mean, the Geometric Mean, and the Associated Differential Equation 88
1.16 On the Flight of an Object Thrown at an Angle to the Horizon 93
1.17 Weightlessness 96
1.18 Kepler’s Laws of Planetary Motion 100
1.19 Beam Deflection 112
1.20 Transportation of Logs 119
Chapter 2. Qualitative Methods of Studying Differential Models 136

2.1 Curves of Constant Direction of Magnetic Needle 136
2.2 Why Must an Engineer Know Existence and Uniqueness Theorems? 143
2.3 A Dynamical Interpretation of Second-Order Differential Equations  155
2.4 Conservative Systems in Mechanics 162
2.5 Stability of Equilibrium Points and of Periodic Motion 176
2.6 Lyapunov Functions 184
2.7 Simple States of Equilibrium  189
2.8 Motion of a Unit—Mass Object Under the Action of Linear Springs in a Medium with Linear Drag 195
2.9 Adiabatic Flow of a Perfect Gas Through a Nozzle of Varying Cross Section  203
2.10 Higher-Order Points of Equilibrium  211
2.11 Inversion with Respect to a Circle and Homogeneous Coordinates 218
2.12 Flow of a Perfect Gas Through a Rotating Tube of Uniform Cross Section 223
2.13 Isolated Closed Trajectories 236
2.14 Periodic Modes in Electric Circuits 250
2.15 Curves Without Contact 260
2.16 The Topographical System of Curves. The Contact Curve 263
2.17 The Divergence of a Vector Field and Limit Cycles 270

Selected Readings 274

Appendices 275

Appendix 1. Derivatives of Elementary Functions  275
Appendix 2. Basic Integrals 278