Chapter 1 First-order differential equations 1.1 Introduction 1.2 First-orderlinear differential equations 1.3 The Van Meegeren art forgeries 1.4 Separableequations 1.5 Populationmodels 1.6 The spread of technological innovations 1.7 An atomic waste disposal problem 1.8 The dynamics of tumor growth， nuxing 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 Chapter 2 Second-order linear differential equations 2.1 Algebraic properties of solutions 2.2 Linear equations with constant coefficients 2.2.1 Complexroots 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 ofjudicious guessing 2.6 Mecharucalvibrations 2.6.1 The Tacoma Bridge disaster 2.6.2 Electricalnetworks 2.7 A model for the detection of diabetes 2.8 Series solutions 2.8.1 Singular points， Euler equations 2.8.2 Regular singular points， the method of Frobenius 2.8.3 Equal roots， and roots differing by an integer 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 Chapter 3 Systems of differential equations 3.1 Algebraic properties of solutions of linear systems 3.2 Vectorspaces 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 Complexroots 3.10 Equalroots 3.11 Fundamental matrix solutions； eAt 3.12 The nonhomogcneous equation； variation of parameters 3.13 Solving systems by Laplace transforms Chapter 4 Qualitative theory of differential equations 4.1 Introduction 4.2 Stability oflinear systems 4.3 Stability of equilibrium solutions 4.4 Thephase-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 Poincare?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 Chapter 5 Separation of variables and Fourier series 5.1 Two point boundary-value problems 5.2 Introduction to partialdifferentialequations 5.3 The heat equation； separation of variables 5.4 Fourierseries 5.5 Even and odd functions 5.6 Retum to the heat equation 5.7 The wave equation 5.8 Laplace'sequation Chapter 6 Sturm-Liouville boundary value problems 6.1 Introduction 6.2 Inner product spaces 6.3 Orthogonalbases， Hermitian operators 6.4 Sturm-'Liouvilletheory Appendix A Some simple facts concerning functions of several variables Appendix B Sequences and series Appendix C C Programs Answers to odd-numbered exercises Index
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