Many textbooks on differential equations are written to be interesting to the teacher rather than the student. Introduction to Differential Equations with Dynamical Systems is directed toward students. This concise and up-to-date textbook addresses the challenges that undergraduate mathematics, engineering, and science students experience during a first course on differential equations. And, while covering all the standard parts of the subject, the book emphasizes linear constant coefficient equations and applications, including the topics essential to engineering students. Stephen Campbell and Richard Haberman--using carefully worded derivations, elementary explanations, and examples, exercises, and figures rather than theorems and proofs--have written a book that makes learning and teaching differential equations easier and more relevant. The book also presents elementary dynamical systems in a unique and flexible way that is suitable for all courses, regardless of length.
Autorentext
Stephen L. Campbell is professor of mathematics and director of the graduate program in mathematics at North Carolina State University. Richard Haberman is professor of mathematics at Southern Methodist University. Campbell and Haberman are prolific researchers in applied mathematics and the authors of a number of textbooks.
Zusammenfassung
Some groups participate in politics more than others. Why? And does it matter for policy outcomes? In this richly detailed and fluidly written book, Andrea Campbell argues that democratic participation and public policy powerfully reinforce each other. Through a case study of senior citizens in the United States and their political activity around Social Security, she shows how highly participatory groups get their policy preferences fulfilled, and how public policy itself helps create political inequality. Using a wealth of unique survey and historical data, Campbell shows how the development of Social Security helped transform seniors from the most beleaguered to the most politically active age group. Thus empowered, seniors actively defend their programs from proposed threats, shaping policy outcomes. The participatory effects are strongest for low-income seniors, who are most dependent on Social Security. The program thus reduces political inequality within the senior population--a laudable effect--while increasing inequality between seniors and younger citizens. A brief look across policies shows that program effects are not always positive. Welfare recipients are even less participatory than their modest socioeconomic backgrounds would imply, because of the demeaning and disenfranchising process of proving eligibility. Campbell concludes that program design profoundly shapes the nature of democratic citizenship. And proposed policies--such as Social Security privatization--must be evaluated for both their economic and political effects, because the very quality of democratic government is influenced by the kinds of policies it chooses.
Inhalt
Preface ix
CHAPTER 1: First-Order Differential Equations and Their Applications 1
1.1 Introduction to Ordinary Differential Equations 1
1.2 The Definite Integral and the Initial Value Problem 4
1.2.1 The Initial Value Problem and the Indefinite Integral 5
1.2.2 The Initial Value Problem and the Definite Integral 6
1.2.3 Mechanics I: Elementary Motion of a Particle with Gravity Only 8
1.3 First-Order Separable Differential Equations 13
1.3.1 Using Definite Integrals for Separable Differential Equations 16
1.4 Direction Fields 19
1.4.1 Existence and Uniqueness 25
1.5 Euler's Numerical Method (optional) 31
1.6 First-Order Linear Differential Equations 37
1.6.1 Form of the General Solution 37
1.6.2 Solutions of Homogeneous First-Order Linear Differential Equations 39
1.6.3 Integrating Factors for First-Order Linear Differential Equations 42
1.7 Linear First-Order Differential Equations with Constant Coefficients and Constant Input 48
1.7.1 Homogeneous Linear Differential Equations with Constant Coefficients 48
1.7.2 Constant Coefficient Linear Differential Equations with Constant
Input 50
1.7.3 Constant Coefficient Differential Equations with Exponential Input 52
1.7.4 Constant Coefficient Differential Equations with Discontinuous Input 52
1.8 Growth and Decay Problems 59
1.8.1 A First Model of Population Growth 59
1.8.2 Radioactive Decay 65
1.8.3 Thermal Cooling 68
1.9 Mixture Problems 74
1.9.1 Mixture Problems with a Fixed Volume 74
1.9.2 Mixture Problems with Variable Volumes 77
1.10 Electronic Circuits 82
1.11 Mechanics II: Including Air Resistance 88
1.12 Orthogonal Trajectories (optional) 92
CHAPTER 2: Linear Second- and Higher-Order Differential Equations 96
2.1 General Solution of Second-Order Linear Differential Equations 96
2.2 Initial Value Problem (for Homogeneous Equations) 100
2.3 Reduction of Order 107
2.4 Homogeneous Linear Constant Coefficient Differential Equations (Second Order) 112
2.4.1 Homogeneous Linear Constant Coefficient Differential Equations (nth-Order) 122
2.5 Mechanical Vibrations I: Formulation and Free Response 124
2.5.1 Formulation of Equations 124
2.5.2 Simple Harmonic Motion (No Damping, d =0) 128
2.5.3 Free Response with Friction (d > 0) 135
2.6 The Method of Undetermined Coefficients 142
2.7 Mechanical Vibrations II: Forced Response 159
2.7.1 Friction is Absent (d = 0) 159
2.7.2 Friction is Present (d > 0) (Damped Forced Oscillations) 168
2.8 Linear Electric Circuits 174
2.9 Euler Equation 179
2.10 Variation of Parameters (Second-Order) 185
2.11 Variation of Parameters (nth-Order) 193
CHAPTER 3: The Laplace Transform 197
3.1 Definition and Basic Properties 197
3.1.1 The Shifting Theorem (Multiplying by an Exponential) 205
3.1.2 Derivative Theorem (Multiplying by t ) 210
3.2 Inverse Laplace Transforms (Roots, Quadratics, and Partial Fractions) 213
3.3 Initial Value Problems for Differential Equations 225
3.4 Discontinuous Forcing Functions 234
3.4.1 Solution of Differential Equations 239
3.5 Periodic Functions 248
3.6 Integrals and the Convolution Theorem 253
3.6.1 Derivation of the Convolution Theorem (optional) 256
3.7 Impulses and Distributions 260
CHAPTER 4: An Introduction to Linear Systems of Differential Equations and Their Phase Plane 265
4.1 Introduction 265
4.2 Introduction to Linear Systems of Differential Equations 268
4.2.1 Solving Linear Systems Using Eigenvalues and Eigenvectors of the Matrix 269
4.2.2 Solving Linear Systems if the Eigenvalues are Real and Unequal 272
4.2.3 Finding General Solutions of Linear Systems in the Case of Complex Eigenvalues 276
4.2.4 Special Systems with Complex Eigenvalues (optional) 279
4.2.5 General Solution of a Linear System if the Two Real Eigenvalues are Equal (Repeated) Roots 281
4.2.6 Eigenvalues and Trace and Determinant (optional) 283
4.3 The Phase Plane for Linear Systems of Differential Equations 287
4.3.1 Introduction to the Phase Plane for Linear Systems of Differential Equations 287
4.3.2 Phase Plane for Linear Systems of Differential Equations 295
4.3.3 Real Eigenvalues 296
4.3.4 Complex Eigenvalues 304
4.3.5 General Theorems 310
CHAPTER 5: Mostly Nonlinear First-Order Differential Equations 315
5.1 First-Order Differential Equations 315
5.2 Equilibria and Stability 316
5.2.1 Equilibrium 316
5.2.2 Stability 317
5.2.3 Review of Linearization 318
5.2.4 Linear Stability Analysis 318
5.3 One-Dimensional Phase Lines 322
5.4 Application to Population Dynamics: The Logistic Equation 327
CHAPTER 6: Nonlinear Systems of Differential Equati…