Differential Equations Techniques Theory and Applications 1st edition by Barbara MacCluer, Paul Bourdon, Thomas Kriete – Ebook PDF Instant Download/Delivery: 1470454388 , 9781470454388
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ISBN 10: 1470454388
ISBN 13: 9781470454388
Author: Barbara MacCluer, Paul Bourdon, Thomas Kriete
Differential Equations: Techniques, Theory, and Applications is designed for a modern first course in differential equations either one or two semesters in length. The organization of the book interweaves the three components in the subtitle, with each building on and supporting the others. Techniques include not just computational methods for producing solutions to differential equations, but also qualitative methods for extracting conceptual information about differential equations and the systems modeled by them. Theory is developed as a means of organizing, understanding, and codifying general principles. Applications show the usefulness of the subject as a whole and heighten interest in both solution techniques and theory. Formal proofs are included in cases where they enhance core understanding; otherwise, they are replaced by informal justifications containing key ideas of a proof in a more conversational format. Applications are drawn from a wide variety of fields: those in physical science and engineering are prominent, of course, but models from biology, medicine, ecology, economics, and sports are also featured. The 1,400+ exercises are especially compelling. They range from routine calculations to large-scale projects. The more difficult problems, both theoretical and applied, are typically presented in manageable steps. The hundreds of meticulously detailed modeling problems were deliberately designed along pedagogical principles found especially effective in the MAA study Characteristics of Successful Calculus Programs, namely, that asking students to work problems that require them to grapple with concepts (or even proofs) and do modeling activities is key to successful student experiences and retention in STEM programs. The exposition itself is exceptionally readable, rigorous yet conversational. Students will find it inviting and approachable. The text supports many different styles of pedagogy from traditional lecture to a flipped classroom model. The availability of a computer algebra system is not assumed, but there are many opportunities to incorporate the use of one.
Differential Equations Techniques Theory and Applications 1st Table of contents:
Chapter 1. Introduction
1.1. What is a differential equation?
1.2. What is a solution?
1.3. More on direction fields: Isoclines
Chapter 2. First-Order Equations
2.1. Linear equations
2.2. Separable equations
2.3. Applications: Time of death, time at depth, and ancient timekeeping
2.4. Existence and uniqueness theorems
2.5. Population and financial models
2.6. Qualitative solutions of autonomous equations
2.7. Change of variable
2.8. Exact equations
Chapter 3. Numerical Methods
3.1. Euler’s method
3.2. Improving Euler’s method: The Heun and Runge-Kutta Algorithms
3.3. Optical illusions and other applications
Chapter 4. Higher-Order Linear Homogeneous Equations
4.1. Introduction to second-order equations
4.2. Linear operators
4.3. Linear independence
4.4. Constant coefficient second-order equations
4.5. Repeated roots and reduction of order
4.6. Higher-order equations
4.7. Higher-order constant coefficient equations
4.8. Modeling with second-order equations
Chapter 5. Higher-Order Linear Nonhomogeneous Equations
5.1. Introduction to nonhomogeneous equations
5.2. Annihilating operators
5.3. Applications of nonhomogeneous equations
5.4. Electric circuits
Chapter 6. Laplace Transforms
6.1. Laplace transforms
6.2. The inverse Laplace transform
6.3. Solving initial value problems with Laplace transforms
6.4. Applications
6.5. Laplace transforms, simple systems, and Iwo Jima
6.6. Convolutions
6.7. The delta function
Chapter 7. Power Series Solutions
7.1. Motivation for the study of power series solutions
7.2. Review of power series
7.3. Series solutions
7.4. Nonpolynomial coefficients
7.5. Regular singular points
7.6. Bessel’s equation
Chapter 8. Linear Systems I
8.1. Nelson at Trafalgar and phase portraits
8.2. Vectors, vector fields, and matrices
8.3. Eigenvalues and eigenvectors
8.4. Solving linear systems
8.5. Phase portraits via ray solutions
8.6. More on phase portraits: Saddle points and nodes
8.7. Complex and repeated eigenvalues
8.8. Applications: Compartment models
8.9. Classifying equilibrium points
Chapter 9. Linear Systems II
9.1. The matrix exponential, Part I
9.2. A return to the Existence and Uniqueness Theorem
9.3. The matrix exponential, Part II
9.4. Nonhomogeneous constant coefficient systems
9.5. Periodic forcing and the steady-state solution
Chapter 10. Nonlinear Systems
10.1. Introduction: Darwin’s finches
10.2. Linear approximation: The major cases
10.3. Linear approximation: The borderline cases
10.4. More on interacting populations
10.5. Modeling the spread of disease
10.6. Hamiltonians, gradient systems, and Lyapunov functions
10.7. Pendulums
10.8. Cycles and limit cycles
Chapter 11. Partial Differential Equations and Fourier Series
11.1. Introduction: Three interesting partial differential equations
11.2. Boundary value problems
11.3. Partial differential equations: A first look
11.4. Advection and diffusion
11.5. Functions as vectors
11.6. Fourier series
11.7. The heat equation
11.8. The wave equation: Separation of variables
11.9. The wave equation: D’Alembert’s method
11.10. Laplace’s equation
Notes and Further Reading
Selected Answers to Exercises
Bibliography
Index
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Tags: Barbara MacCluer, Paul Bourdon, Thomas Kriete, Differential Equations, Techniques Theory