Differential Equations Graphics, Models, Data

by Lomen, David O.; Lovelock, David

• ISBN13: 9780471076483
• ISBN10: 0471076481
• Edition: 1st
• Format: Paperback
• Copyright: 1998-09-01
• Publisher: Wiley

Summary

Viewing stained glass from different angles or in various lights is necessary to discover its many qualities. Likewise, viewing solutions of differential equations from several points of view is essential to fully understand their behavior. Lomen and Lovelock provide an active environment for students to explore differential equations by using analytical, numerical, graphical, and descriptive techniques, and for students to use ODEs as a natural tool for modeling many interesting processes in science and engineering.

Author Biography

David O. Lomen is the author of Differential Equations: Graphics, Models, Data, published by Wiley. David Lovelock is a British theoretical physicist and mathematician. He is known for Lovelock theory of gravity and the Lovelock's theorem.

Table of Contents

PREFACE	v
CHAPTER 1 BASIC CONCEPTS	1	(26)
1.1 Simple Differential Equations and Explicit Solutions	1	(6)
1.2 Graphical Solutions Using Calculus	7	(4)
1.3 Slope Fields and Isoclines	11	(11)
1.4 Functions and Power Series Expansions	22	(5)
CHAPTER 2 AUTONOMOUS DIFFERENTIAL EQUATIONS	27	(54)
2.1 Autonomous Equations	28	(8)
2.2 Simple Models	36	(6)
2.3 The Logistic Equation	42	(11)
2.4 Existence and Uniqueness of Solutions, and Words of Caution	53	(12)
2.5 Qualitative Behavior of Solutions Using Phases Lines	65	(10)
2.6 Bifurcation Diagrams	75	(6)
CHAPTER 3 FIRST ORDER DIFFERENTIAL EQUATIONS -- QUALITATIVE AND QUANTITATIVE ASPECTS	81	(38)
3.1 Graphical Solutions Using Calculus	81	(13)
3.2 Symmetry of Slope Fields	94	(3)
3.3 Numerical Solutions and Chaos	97	(15)
3.4 Comparing Solutions of Differential Equations	112	(3)
3.5 Finding Power Series Solutions	115	(4)
CHAPTER 4 MODELS AND APPLICATIONS LEADING TO NEW TECHNIQUES	119	(64)
4.1 Solving Separable Differential Equations	119	(17)
4.2 Solving Differential Equations with Homogeneous Coefficients	136	(9)
4.3 Models: Deriving Differential Equations From Data	145	(13)
4.4 Models: Objects in Motion	158	(9)
4.5 Application: Orthogonal Trajectories	167	(8)
4.6 Piecing Together Differential Equations	175	(8)
CHAPTER 5 FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS AND MODELS	183	(40)
5.1 Solving Linear Differential Equations	183	(13)
5.2 Models That Use Linear Equations	196	(15)
5.3 Models That Use Bernoulli's Equation	211	(12)
CHAPTER 6 INTERPLAY BETWEEN FIRST ORDER SYSTEMS AND SECOND ORDER EQUATIONS	223	(56)
6.1 Simple Models	223	(8)
6.2 How First Order Systems and Second Order Equations Are Related	231	(10)
6.3 Second Order Linear Differential Equations with Constant Coefficients	241	(9)
6.4 Modeling Physical Situations	250	(9)
6.5 Interpreting the Phase Plane	259	(7)
6.6 How Explicit Solutions Are Related to Orbits	266	(6)
6.7 The Motion of a Nonlinear Pendulum	272	(7)
CHAPTER 7 SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH FORCING FUNCTIONS	279	(36)
7.1 The General Solution	279	(4)
7.2 Finding Solutions by the Method of Undetermined Coefficients	283	(11)
7.3 Application and Models	294	(21)
CHAPTER 8 SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS -- QUALITATIVE AND QUANTITATIVE ASPECTS	315	(64)
8.1 Qualitative Behavior of Solutions	315	(8)
8.2 Finding Solutions by Reduction of Order	323	(7)
8.3 Finding Solutions by Variation of Parameters	330	(6)
8.4 The Importance of Linear Independence and Dependence	336	(7)
8.5 Solving Cauchy -- Euler Equations	343	(7)
8.6 Boundary Value Problems and the Shooting Method	350	(5)
8.7 Solving Higher Order Homogeneous Differential Equations	355	(13)
8.8 Solving Higher Order Nonhomogeneous Differential Equations	368	(11)
CHAPTER 9 LINEAR AUTONOMOUS SYSTEMS	379
9.1 Solving Linear Autonomous Systems	379	(9)
9.2 Classification of Solutions via Stability	388	(15)
9.3 When Do Straight -- Line Orbits Exist?	403	(5)
9.4 Qualitative Behavior Using Nullclines	408	(5)
9.5 Matrix Formulation of Solutions	413	(17)
9.6 Compartmental Models	430
CHAPTER 10 NONLINEAR AUTONOMOUS SYSTEMS	315
10.1 Introduction to Nonlinear Autonomous Systems	437	(11)
10.2 Qualitative Behavior Using Nullcline Analysis	448	(8)
10.3 Qualitative Behavior Using Linearization	456	(11)
10.4 Models Involving Nonlinear Autonomous Equations	467	(18)
10.5 Bungee Jumping	485	(7)
10.6 Linear Versus Nonlinear Differential Equations	492	(6)
10.7 Autonomous Versus Nonautonomous Differential Equations	498	(7)
CHAPTER 11 USING LAPLACE TRANSFORMS	505	(70)
11.1 Motivation	505	(11)
11.2 Constructing New Laplace Transforms from Old	516	(7)
11.3 The Inverse Laplace Transform and the Convolution Theorem	523	(8)
11.4 Functions That Jump	531	(14)
11.5 Models Involving First Order Linear Differential Equations	545	(8)
11.6 Models Involving Higher Order Linear Differential Equations	553	(7)
11.7 Applications to Systems of Linear Differential Equations	560	(6)
11.8 When Do Laplace Transforms Exist?	566	(9)
CHAPTER 12 USING POWER SERIES	575	(56)
12.1 Solutions Using Taylor Series	576	(5)
12.2 Solutions Using Power Series	581	(18)
12.3 What To Do When Power Series Fail	599	(10)
12.4 Solutions Using the Method of Frobenius	609	(22)
APPENDICES	631	(28)
A.1 Background Material	631	(6)
A.2 Partial Fractions	637	(3)
A.3 Infinite Series, Power Series, and Taylor Series	640	(2)
A.4 Complex Numbers	642	(3)
A.5 Elementary Matrix Operations	645	(6)
A.6 Least Squares Approximation	651	(3)
A.7 Proofs of the Oscillation Theorems	654	(5)
ANSWERS	659	(13)
INDEX	672