Differential equations and their applications an introduction to applied mathematics 4th edition by Martin Braun ISBN 0387978941 978-0387978949

First-order differential equations 1

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 problern

1.8 The dynamics of tumor growth, mixing problemsand 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 jor 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

Second -order linear differential equations 2

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

2.6.1 The Tacoma Bridgedisaster

2.7 A model for the detection of diabetes

2.8 Series solutions

2.8.1 Singular points, Euler equations

2.8.2 Regularsingular 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 discontinuousright-hand sides

2.12 The Dirac delta function

Systems of differential equations 3

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; eA1

3.12 The nonhomogeneous equation;variation of parameters

3.13 Solving systems by Laplace transforms

Qualitative theory of differential equations 4

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

4.5.1. L. F. Richardson’s theory of conflict

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 dr

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

Separation of variables and Fourier series 5

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 Laplace’s equation

Sturm-Liouville boundary value problems 6

6.1 Introduction

6.2 Inner product spaces

6.3 Orthogonal bases, Hermitian operators

6.4 Sturm-Liouville theory

Appendix A

Some simple facts concerning functionsof several variables

Appendix B

Sequences and series

Appendix C

C Programs