There are three major changes in the Third Edition of Differential Equations and Their Applications. First, we have completely rewritten the section on singular solutions of differential equations. A new section, 2.8.1, dealing with Euler equations has been added, and this section is used to motivate a greatly expanded treatment of singular equations in sections 2.8.2 and 2.8.3. Our second major change is the addition of a new section, 4.9, dealing with bifurcation theory, a subject of much current interest. We felt it desirable to give the reader a brief but nontrivial introduction to this important topic. Our third major change is in Section 2.6, where we have switched to the metric system of units. This change was requested by many of our readers. In addition to the above changes, we have updated the material on population models, and have revised the exercises in this section. Minor editorial changes have also been made throughout the text. New York City November. 1982 Martin Braun Preface to the First Edition This textbook is a unique blend of the theory of differential equations and their exciting application to "real world" problems. First, and foremost, it is a rigorous study of ordinary differential equations and can be fully understood by anyone who has completed one year of calculus. However, in addition to the traditional applications, it also contains many exciting "real life" problems. These applications are completely self contained.
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ISBN-13
978-1-4684-0164-6 (9781468401646)
DOI
10.1007/978-1-4684-0164-6
Schweitzer Klassifikation
1 First-order differential equations.- 1.1 Introduction.- 1.2 First-order linear differential equations.- 1.3 The Van Meegeren art forgeries.- 1.4 Separable equations.- 1.5 Population models.- 1.6 The spread of technological innovations.- 1.7 An atomic waste disposal problem.- 1.8 The dynamics of tumor growth, mixing problems, and orthogonal trajectories.- 1.9 Exact equations, and why we cannot solve very many differential equations.- 1.10 The existence-uniqueness theorem; Picard iteration.- 1.11 Finding roots of equations by iteration.- 1.12 Difference equations, and how to compute the interest due on your student loans.- 1.13 Numerical approximations; Euler's method.- 1.14 The three term Taylor series method.- 1.15 An improved Euler method.- 1.16 The Runge-Kutta method.- 1.17 What to do in practice.- 2 Second-order linear differential equations.- 2.1 Algebraic properties of solutions.- 2.2 Linear equations with constant coefficients.- 2.3 The nonhomogeneous equation.- 2.4 The method of variation of parameters.- 2.5 The method of judicious guessing.- 2.6 Mechanical vibrations.- 2.7 A model for the detection of diabetes.- 2.8 Series solutions.- 2.9 The method of Laplace transforms.- 2.10 Some useful properties of Laplace transforms.- 2.11 Differential equations with discontinuous right-hand sides.- 2.12 The Dirac delta function.- 2.13 The convolution integral.- 2.14 The method of elimination for systems.- 2.15 Higher-order equations.- 3 Systems of differential equations.- 3.1 Algebraic properties of solutions of linear systems.- 3.2 Vector spaces.- 3.3 Dimension of a vector space.- 3.4 Applications of linear algebra to differential equations.- 3.5 The theory of determinants.- 3.6 Solutions of simultaneous linear equations.- 3.7 Linear transformations.- 3.8 Theeigenvalue-eigenvector method of finding solutions.- 3.9 Complex roots.- 3.10 Equal roots.- 3.11 Fundamental matrix solutions; eAt.- 3.12 The nonhomogeneous equation; variation of parameters.- 3.13 Solving systems by Laplace transforms.- 4 Qualitative theory of differential equations.- 4.1 Introduction.- 4.2 Stability of linear systems.- 4.3 Stability of equilibrium solutions.- 4.4 The phase-plane.- 4.5 Mathematical theories of war.- 4.6 Qualitative properties of orbits.- 4.7 Phase portraits of linear systems.- 4.8 Long time behavior of solutions; the Poincaré-Bendixson Theorem.- 4.9 Introduction to bifurcation theory.- 4.10 Predator-prey problems; or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I.- 4.11 The principle of competitive exclusion in population biology.- 4.12 The Threshold Theorem of epidemiology.- 4.13 A model for the spread of gonorrhea.- 5 Separation of variables and Fourier series.- 5.1 Two point boundary-value problems.- 5.2 Introduction to partial differential equations.- 5.3 The heat equation; separation of variables.- 5.4 Fourier series.- 5.5 Even and odd functions.- 5.6 Return to the heat equation.- 5.7 The wave equation.- 5.8 Laplace's equation.- Appendix A.- Appendix B.- Appendix C.- Answers to odd-numbered exercises.
