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Differential Equations: From Calculus to Dynamical Systems 2nd Revised edition [Hardback]

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  • Formāts: Hardback, 402 pages, height x width: 254x178 mm
  • Sērija : MAA Textbooks
  • Izdošanas datums: 01-Jan-2019
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470444003
  • ISBN-13: 9781470444006
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Differential Equations: From Calculus to Dynamical Systems 2nd Revised editionPalielināt
  • Formāts: Hardback, 402 pages, height x width: 254x178 mm
  • Sērija : MAA Textbooks
  • Izdošanas datums: 01-Jan-2019
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470444003
  • ISBN-13: 9781470444006
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470444006.html
Keywords:
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.
Preface ix
Acknowledgments ix
1 Introduction to Differential Equations
1(18)
1.1 Basic Terminology
2(4)
1.1.1 Ordinary vs. Partial Differential Equations
2(1)
1.1.2 Independent Variables, Dependent Variables, and Parameters
3(1)
1.1.3 Order of a Differential Equation
3(1)
1.1.4 What is a Solution?
3(2)
1.1.5 Systems of Differential Equations
5(1)
1.2 Families of Solutions, Initial-Value Problems
6(5)
1.3 Modeling with Differential Equations
11(8)
2 First-order Differential Equations
19(62)
2.1 Separable First-order Equations
20(8)
2.1.1 Application 1: Population Growth
23(2)
2.1.2 Application 2: Newton's Law of Cooling
25(3)
2.2 Graphical Methods, the Slope Field
28(8)
2.2.1 Using Graphical Methods to Visualize Solutions
32(4)
2.3 Linear First-order Differential Equations
36(8)
2.3.1 Application: Single-compartment Mixing Problem
41(3)
2.4 Existence and Uniqueness of Solutions
44(7)
2.5 More Analytic Methods for Nonlinear First-order Equations
51(8)
2.5.1 Exact Differential Equations
51(5)
2.5.2 Bernoulli Equations
56(2)
2.5.3 Using Symmetries of the Slope Field
58(1)
2.6 Numerical Methods
59(12)
2.6.1 Euler's Method
60(4)
2.6.2 Improved Euler Method
64(2)
2.6.3 Fourth-order Runge-Kutta Method
66(5)
2.7 Autonomous Equations, the Phase Line
71(10)
2.7.1 Stability---Sinks, Sources, and Nodes
73(1)
Bifurcation in Equations with Parameters
74(7)
3 Second-order Differential Equations
81(64)
3.1 General Theory of Homogeneous Linear Equations
82(6)
3.2 Homogeneous Linear Equations with Constant Coefficients
88(7)
3.2.1 Second-order Equation with Constant Coefficients
88(5)
3.2.2 Equations of Order Greater Than Two
93(2)
3.3 The Spring-mass Equation
95(7)
3.3.1 Derivation of the Spring-mass Equation
95(1)
3.3.2 The Unforced Spring-mass System*
96(6)
3.4 Nonhomogeneous Linear Equations
102(12)
3.4.1 Method of Undetermined Coefficients
102(7)
3.4.2 Variation of Parameters
109(5)
3.5 The Forced Spring-mass System
114(11)
Beats and Resonance
117(8)
3.6 Linear Second-order Equations with Nonconstant Coefficients
125(10)
3.6.1 The Cauchy-Euler Equation
125(2)
3.6.2 Series Solutions
127(8)
3.7 Autonomous Second-order Differential Equations
135(10)
3.7.1 Numerical Methods
136(1)
3.7.2 Autonomous Equations and the Phase Plane
137(8)
4 Linear Systems of First-order Differential Equations
145(38)
4.1 Introduction to Systems
146(4)
4.1.1 Writing Differential Equations as a First-order System
146(1)
4.1.2 Linear Systems
147(3)
4.2 Matrix Algebra
150(8)
4.3 Eigenvalues and Eigenvectors
158(7)
4.4 Analytic Solutions of the Linear System x = Ax
165(11)
4.4.1 Application 1: Mixing Problem with Two Compartments
169(2)
4.4.2 Application 2: Double Spring-mass System
171(5)
4.5 Large Linear Systems; the Matrix Exponential
176(7)
4.5.1 Definition and Properties of the Matrix Exponential
176(2)
4.5.2 Using the Matrix Exponential to Solve a Nonhomogeneous System
178(2)
4.5.3 Application: Mixing Problem with Three Compartments
180(3)
5 Geometry of Autonomous Systems
183(38)
5.1 The Phase Plane for Autonomous Systems
184(3)
5.2 Geometric Behavior of Linear Autonomous Systems
187(11)
5.2.1 Linear Systems with Real (Distinct, Nonzero) Eigenvalues
188(2)
5.2.2 Linear Systems with Complex Eigenvalues
190(1)
5.2.3 The Trace-determinant Plane
191(2)
5.2.4 The Special Cases
193(5)
5.3 Geometric Behavior of Nonlinear Autonomous Systems
198(9)
5.3.1 Finding the Equilibrium Points
199(1)
5.3.2 Determining the Type of an Equilibrium
200(4)
5.3.3 A Limit Cycle---the Van der Pol Equation
204(3)
5.4 Bifurcations for Systems
207(7)
5.4.1 Bifurcation in a Spring-mass Model
208(1)
5.4.2 Bifurcation of a Predator-prey Model
208(3)
5.4.3 Bifurcation Analysis Applied to a Competing Species Model
211(3)
5.5 Student Projects
214(7)
5.5.1 The Wilson-Cowan Equations
214(4)
5.5.2 A New Predator-prey Equation---Putting It All Together
218(3)
6 Laplace Transforms
221(40)
6.1 Definition and Some Simple Laplace Transforms
221(6)
6.1.1 Four Simple Laplace Transforms
223(1)
6.1.2 Linearity of the Laplace Transform
224(1)
6.1.3 Transforming the Derivative of f(t)
225(2)
6.2 Solving Equations, the Inverse Laplace Transform
227(5)
6.2.1 Partial Fraction Expansions
228(4)
6.3 Extending the Table
232(7)
6.3.1 Inverting a Term with an Irreducible Quadratic Denominator
233(3)
6.3.2 Solving Linear Systems with Laplace Transforms
236(3)
6.4 The Unit Step Function
239(12)
6.5 Convolution and the Impulse Function
251(10)
6.5.1 The Convolution Integral
251(2)
6.5.2 The Impulse Function
253(3)
6.5.3 Impulse Response of a Linear, Time-invariant System
256(5)
7 Introduction to Partial Differential Equations
261(24)
7.1 Solving Partial Differential Equations
261(5)
7.1.1 An Overview of the Method of Separation of Variables
263(3)
7.2 Orthogonal Functions and Trigonometric Fourier Series
266(8)
7.2.1 Orthogonal Families of Functions
266(3)
7.2.2 Properties of Fourier Series, Cosine and Sine Series
269(5)
7.3 Boundary-Value Problems: Sturm-Liouville Equations
274(11)
8 Solving Second-order Partial Differential Equations
285(44)
8.1 Classification of Linear Second-order Partial Differential Equations
285(4)
8.2 The 1-dimensional Heat Equation
289(10)
8.2.1 Solution of the Heat Equation by Separation of Variables
291(3)
8.2.2 Other Boundary Conditions for the Heat Equation
294(5)
8.3 The 1-dimensional Wave Equation
299(10)
8.3.1 Solution of the Wave Equation by Separation of Variables
300(5)
8.3.2 D'Alembert's Solution of the Wave Equation on an Infinite Interval
305(4)
8.4 Numerical Solution of Parabolic and Hyperbolic Equations
309(9)
8.5 Laplace's Equation
318(6)
8.6 Student Project: Harvested Diffusive Logistic Equation
324(5)
Appendix
329(68)
A Answers to Odd-numbered Exercises
331(54)
B Derivative and Integral Formulas
385(2)
C Cofactor Method for Determinants
387(2)
D Cramer's Rule for Solving Systems of Linear Equations
389(2)
E The Wronskian
391(2)
F Table of Laplace Transforms
393(2)
G Review of Partial Derivatives
395(2)
Index 397
Virginia W. Noonburg, University of Hartford, West Hartford, CT.