Suitable for undergraduate students, this book provides an introduction to differential equations. Highlights include material on bifurcation theory, Sturm-Liouville boundary value problems and computer programs in C, Pascal and Fortran. First-order differential equations 1(126) Introduction 1(1) First-order linear differential equations 2(9) The Van Meegeren art forgeries 11(9) Separable equations 20(6) Population models 26(13) The spread of technological innovations 39(7) An atomic waste disposal problem 46(6) The dynamics of tumor growth, mixing problems, and orthogonal trajectories 52(6) Exact equations, and why we cannot solve very many differential equations 58(9) The existence-uniqueness theorem; Picard iteration 67(14) Finding roots of equations by iteration 81(10) Newton's method 87(4) Difference equations, and how to compute the interest due on your student loans 91(5) Numerical approximations; Euler's method 96(11) Error analysis for Euler's method 100(7) The three term Taylor series method 107(2) An improved Euler method 109(3) The Runge-Kutta method 112(4) What to do in practice 116(11) Second-order linear differential equations 127(137) Algebraic properties of solutions 127(11) Linear equations with constant coefficients 138(13) Complex roots 141(4) Equal roots; reduction of order 145(6) The nonhomogeneous equation 151(2) The method of variation of parameters 153(4) The method of judicious guessing 157(8) Mechanical vibrations 165(13) The Tacoma Bridge disaster 173(2) Electrical networks 175(3) A model for the detection of diabetes 178(7) Series solutions 185(40) Singular points, Euler equations 198(5) Regular singular points, the method of Frobenius 203(16) Equal roots, and roots differing by an integer 219(6) The method of Laplace transforms 225(8) Some useful properties of Laplace transforms 233(5) Differential equations with discontinuous right-hand sides 238(5) The Dirac delta function 243(8) The convolution integral 251(6) The method of elimination for systems 257(2) Higher-order equations 259(5) Systems of differential equations 264(108) Algebraic properties of solutions of linear systems 264(9) Vector spaces 273(6) Dimension of a vector space 279(12) Applications of linear algebra to differential equations 291(6) The theory of determinants 297(13) Solutions of simultaneous linear equations 310(10) Linear transformations 320(13) The eigenvalue-eigenvector method of finding solutions 333(8) Complex roots 341(4) Equal roots 345(10) Fundamental matrix solutions; eAt 355(5) The nonhomogeneous equation; variation of parameters 360(8) Solving systems by Laplace transforms 368(4) Qualitative theory of differential equations 372(104) Introduction 372(6) Stability of linear systems 378(7) Stability of equilibrium solutions 385(9) The phase-plane 394(4) Mathematical theories of war 398(16) L.F. Richardson's theory of conflict 398(7) Lanchester's combat models and the battle of Iwo Jima 405(9) Qualitative properties of orbits 414(4) Phase portraits of linear systems 418(10) Long time behavior of solutions; the Poincare-Bendixson Theorem 428(9) Introduction to bifurcation theory 437(6) Predator-prey problems; or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I 443(8) The principle of competitive exclusion in population biology 451(7) The Threshold Theorem of epidemiology 458(7) A model for the spread of gonorrhea 465(11) Separation of variables and Fourier series 476(38) Two point boundary-value problems 476(5) Introduction to partial differential equations 481(2) The heat equation; separation of variables 483(4) Fourier series 487(6) Even and odd functions 493(5) Return to the heat equation 498(5) The wave equation 503(5) Laplace's equation 508(6) Sturm---Liouville boundary value problems 514(31) Introduction 514(1) Inner product spaces 515(11) Orthogonal bases, Hermitian operators 526(7) Sturm-Liouville theory 533(12) Appendix A Some simple facts concerning functions of several variables 545(2) Appendix B Sequences and series 547(2) Appendix C C Programs 549(8) Answers to odd-numbered exercises 557(18) Index 575