A thoroughly modern textbook for the sophomore-level differential equations course. The examples and exercises emphasize modeling not only in engineering and physics but also in applied mathematics and biology. There is an early introduction to numerical methods and, throughout, a strong emphasis on the qualitative viewpoint of dynamical systems. Bifurcations and analysis of parameter variation is a persistent theme. Presuming previous exposure to only two semesters of calculus, necessary linear algebra is developed as needed. The exposition is very clear and inviting. The book would serve well for use in a flipped-classroom pedagogical approach or for self-study for an advanced undergraduate or beginning graduate student. This second edition of Noonburg's best-selling textbook includes two new chapters on partial differential equations, making the book usable for a two-semester sequence in differential equations. It includes exercises, examples, and extensive student projects taken from the current mathematical and scientific literature. An instructor's manual for this title is available electronically to those instructors who have adopted the textbook for classroom use. Please send email to
[email protected] for more information.
Author(s): V W Noonburg
Publisher: Maa Press
Year: 2019
Cover
Title Page
Copyright Page
Table of Contents
Preface
1 Introduction to Differential Equations
1.1 Basic Terminology
1.1.1 Ordinary vs. Partial Differential Equations
1.1.2 Independent Variables, Dependent Variables, and Parameters
1.1.3 Order of a Differential Equation
1.1.4 What is a Solution?
1.1.5 Systems of Differential Equations
1.2 Families of Solutions, Initial-Value Problems
1.3 Modeling with Differential Equations
2 First-order Differential Equations
2.1 Separable First-order Equations
2.1.1 Application 1: Population Growth
2.1.2 Application 2: Newton’s Law of Cooling
2.2 Graphical Methods, the Slope Field
2.2.1 Using Graphical Methods to Visualize Solutions
2.3 Linear First-order Differential Equations
2.3.1 Application: Single-compartment Mixing Problem
2.4 Existence and Uniqueness of Solutions
2.5 More Analytic Methods for Nonlinear First-order Equations
2.5.1 Exact Differential Equations
2.5.2 Bernoulli Equations
2.5.3 Using Symmetries of the Slope Field
2.6 Numerical Methods
2.6.1 Euler’s Method
2.6.2 Improved Euler Method
2.6.3 Fourth-order Runge-Kutta Method
2.7 Autonomous Equations, the Phase Line
2.7.1 Stability—Sinks, Sources, and Nodes
Bifurcation in Equations with Parameters
3 Second-order Differential Equations
3.1 General Theory of Homogeneous Linear Equations
3.2 Homogeneous Linear Equations with Constant Coefficients
3.2.1 Second-order Equation with Constant Coefficients
3.2.2 Equations of Order Greater Than Two
3.3 The Spring-mass Equation
3.3.1 Derivation of the Spring-mass Equation
3.3.2 The Unforced Spring-mass System
3.4 Nonhomogeneous Linear Equations
3.4.1 Method of Undetermined Coefficients
3.4.2 Variation of Parameters
3.5 The Forced Spring-mass System
Beats and Resonance
3.6 Linear Second-order Equations with Nonconstant Coefficients
3.6.1 The Cauchy-Euler Equation
3.6.2 Series Solutions
3.7 Autonomous Second-order Differential Equations
3.7.1 Numerical Methods
3.7.2 Autonomous Equations and the Phase Plane
4 Linear Systems of First-order Differential Equations
4.1 Introduction to Systems
4.1.1 Writing Differential Equations as a First-order System
4.1.2 Linear Systems
4.2 Matrix Algebra
4.3 Eigenvalues and Eigenvectors
4.4 Analytic Solutions of the Linear System ⃗?′ = ??⃗?
4.4.1 Application 1: Mixing Problem with Two Compartments
4.4.2 Application 2: Double Spring-mass System
4.5 Large Linear Systems; the Matrix Exponential
4.5.1 Definition and Properties of the Matrix Exponential
4.5.2 Using the Matrix Exponential to Solve a Nonhomogeneous System
4.5.3 Application: Mixing Problem with Three Compartments
5 Geometry of Autonomous Systems
5.1 The Phase Plane for Autonomous Systems
5.2 Geometric Behavior of Linear Autonomous Systems
5.2.1 Linear Systems with Real (Distinct, Nonzero) Eigenvalues
5.2.2 Linear Systems with Complex Eigenvalues
5.2.3 The Trace-determinant Plane
5.2.4 The Special Cases
5.3 Geometric Behavior of Nonlinear Autonomous Systems
5.3.1 Finding the Equilibrium Points
5.3.2 Determining the Type of an Equilibrium
5.3.3 A Limit Cycle—the Van der Pol Equation
5.4 Bifurcations for Systems
5.4.1 Bifurcation in a Spring-mass Model
5.4.2 Bifurcation of a Predator-prey Model
5.4.3 Bifurcation Analysis Applied to a Competing Species Model
5.5 Student Projects
5.5.1 The Wilson-Cowan Equations
5.5.2 A New Predator-prey Equation—Putting It All Together
6 Laplace Transforms
6.1 Definition and Some Simple Laplace Transforms
6.1.1 Four Simple Laplace Transforms
6.1.2 Linearity of the Laplace Transform
6.1.3 Transforming the Derivative of ?(?)
6.2 Solving Equations, the Inverse Laplace Transform
6.2.1 Partial Fraction Expansions
6.3 Extending the Table
6.3.1 Inverting a Term with an Irreducible Quadratic Denominator
6.3.2 Solving Linear Systems with Laplace Transforms
6.4 The Unit Step Function
6.5 Convolution and the Impulse Function
6.5.1 The Convolution Integral
6.5.2 The Impulse Function
6.5.3 Impulse Response of a Linear, Time-invariant System
7 Introduction to Partial Differential Equations
7.1 Solving Partial Differential Equations
7.1.1 An Overview of the Method of Separation of Variables
7.2 Orthogonal Functions and Trigonometric Fourier Series
7.2.1 Orthogonal Families of Functions
7.2.2 Properties of Fourier Series, Cosine and Sine Series
7.3 Boundary-Value Problems: Sturm-Liouville Equations
8 Solving Second-order Partial Differential Equations
8.1 Classification of Linear Second-order Partial Differential Equations
8.2 The 1-dimensional Heat Equation
8.2.1 Solution of the Heat Equation by Separation of Variables
8.2.2 Other Boundary Conditions for the Heat Equation
8.3 The 1-dimensional Wave Equation
8.3.1 Solution of the Wave Equation by Separation of Variables
8.3.2 D’Alembert’s Solution of the Wave Equation on an Infinite Interval
8.4 Numerical Solution of Parabolic and Hyperbolic Equations
8.5 Laplace’s Equation
8.6 Student Project: Harvested Diffusive Logistic Equation
Appendix
A Answers to Odd-numbered Exercises
B Derivative and Integral Formulas
C Cofactor Method for Determinants
D Cramer’s Rule for Solving Systems of Linear Equations
E The Wronskian
F Table of Laplace Transforms
G Review of Partial Derivatives
Index
