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Differential Equations and Their Applications An Introduction to Applied Mathematics

There are three major changes in the Third Edition of Differential Equations and Their Applications. First, we have completely rewritten the section on singular solutions of differential equations. A new section, 2.8.1, dealing with Euler equations has been added, and this section is used to motivat...

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Bibliographic Details
Main Author: Braun, M.
Format: eBook
Language:English
Published: New York, NY Springer New York 1983, 1983
Edition:3rd ed. 1983
Series:Applied Mathematical Sciences
Subjects:
Mathematical Analysis
Analysis
Applications Of Mathematics
Mathematics
Differential Equations
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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; 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
  • 2.3 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 Theeigenvalue-eigenvector method of finding solutions
  • 3.9 Complex roots
  • 3.10 Equal roots
  • 3.11 Fundamental matrix solutions; eAt
  • 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
  • Appendix B
  • Appendix C
  • Answers to odd-numbered exercises
  • 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 Poincaré-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

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