Differential Equations and Their Applications

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 An atomic waste disposal problem.- 1.7 The dynamics of tumor growth, mixing problems, and orthogonal trajectories.- 1.8 Exact equations, and why we cannot solve very many differential equations.- 1.9 The existence-uniqueness theorem; Picard iteration.- 1.10 Difference equations, and how to compute the interest due on your student loans.- 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 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 The eigenvalue-eigenvector method of finding solutions.- 3.7 Complex roots.- 3.8 Equal roots.- 3.9 Fundamental matrix solutions; eAt.- 3.10 The nonhomogeneous equation; variation of parameters.- 3.11 Solving systems by Laplace transforms.- 4 Qualitative theory of differential equations.- 4.1 Introduction.- 4.2 The phase-plane.- 4.3 Lanchester's combat models and the battle of Iwo Jima.- Appendix A.- Some simple facts concerning functions of several variables.- Appendix B.- Sequences and series.- Answers to odd-numbered exercises.