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 un derstood 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. First, the problem to be solved is outlined clearly, and one or more differential equa tions are derived as a model for this problem. These equations are then solved, and the results are compared with real world data. The following applications are covered in this text. I. In Section 1.3 we prove that the beautiful painting "Disciples of Emmaus" which was bought by the Rembrandt Society of Belgium for $170,000 was a modem forgery. 2. In Section 1.5 we derive differential equations which govern the population growth of various species, and compare the results predicted by our models with the known values of the populations. 3. In Section 1.6 we derive differential equations which govern the rate at which farmers adopt new innovations. Surprisingly, these same differen tial equations govern the rate at which technological innovations are adopted in such diverse industries as coal, iron and steel, brewing, and railroads.
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ISBN-13
978-1-4684-9360-3 (9781468493603)
DOI
10.1007/978-1-4684-9360-3
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.- 1.11.1 Newton's method.- 1.12 Difference equations, and how to compute the interest due on your student loans.- 1.13 Numerical approximations; Euler's method.- 1.13.1 Error analysis for 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.- 2.2.1 Complex roots.- 2.2.2 Equal roots; reduction of order.- 2.3 The nonhomogeneous equation.- 2.4 The method of variation of parameters.- 2.5 The method of judicious guessing.- 2.6 Mechanical vibrations.- 2.6.1 The Tacoma Bridge disaster.- 2.6.2 Electrical networks.- 2.7 A model for the detection of diabetes.- 2.8 Series solutions.- 2.8.1 Singular points; the method of Frobenius.- 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 A few words about higher-order equations.- 3 Systems of differential equations.- 3.1 Algebraic properties of solutionsof 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 The eigenvalue-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.- 4.5.1 L. F. Richardson's theory of conflict.- 4.5.2 Lanchester's combat models and the battle of Iwo Jima.- 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 Predator-prey problems; or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I.- 4.10 The principle of competitive exclusion in population biology.- 4.11 The Threshold Theorem of epidemiology.- 4.12 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 Some simple facts concerning functions of several variables.- Appendix B Sequences and series.- Appendix C Introduction to APL.- Answers to odd-numbered exercises.
