Author | Braun, Martin. author |
---|---|
Title | Differential Equations and Their Applications [electronic resource] : An Introduction to Applied Mathematics / by Martin Braun |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1983 |
Edition | 3rd Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4684-0164-6 |
Descript | XIII, 546 p. online resource |
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; Eulerโs 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 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 -- 4.6 Qualitative properties of orbits -- 4.7 Phase portraits of linear systems -- 4.8 Long time behavior of solutions; the Poincarรฉ-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 Laplaceโs equation -- Appendix A -- Appendix B -- Appendix C -- Answers to odd-numbered exercises