A Workbook for Differential Equations
An accessible and hands-on approach to modeling and predicting real-world phenomena using differential equations
A Workbook for Differential Equations presents an interactive introduction to fundamental solution methods for ordinary differential equations. The author emphasizes the importance of manually working through computations and models, rather than simply reading or memorizing formulas.
Utilizing real-world applications from spring-mass systems and circuits to vibrating strings and an overview of the hydrogen atom, the book connects modern research with the presented topics, including first order equations, constant coefficient equations, Laplace transforms, partial differential equations, series solutions, systems, and numerical methods. The result is a unique guide to understanding the significance of differential equations in mathematics, science, and engineering.
The workbook contains modules that involve readers in as many ways as possible, and each module begins with "Prerequisites" and "Learning Objectives" sections that outline both the skills needed to understand the presented material and what new skills will be obtained by the conclusion of the module. Detailed applications are intertwined in the discussion, motivating the investigation of new classes of differential equations and their accompanying techniques. Introductory modeling sections discuss applications and why certain known solution techniques may not be enough to successfully analyze certain situations. Almost every module concludes with a section that contains various projects, ranging from programming tasks to theoretical investigations.
The book is specifically designed to promote the development of effective mathematical reading habits such as double-checking results and filling in omitted steps in a computation. Rather than provide lengthy explanations of what readers should do, good habits are demonstrated in short sections, and a wide range of exercises provide the opportunity to test reader comprehension of the concepts and techniques. Rich illustrations, highlighted notes, and boxed comments offer illuminating explanations of the computations. The material is not specific to any one particular software package, and as a result, necessary algorithms can be implemented in various programs, including Mathematica®, Maple, and Mathcad®. The book's related Web site features supplemental slides as well as videos that discuss additional topics such as homogeneous first order equations, the general solution of separable differential equations, and the derivation of the differential equations for a multi-loop circuit. In addition, twenty activities are included at the back of the book, allowing for further practice of discussed topics whether in the classroom or for self-study.
With its numerous pedagogical features that consistently engage readers, A Workbook for Differential Equations is an excellent book for introductory courses in differential equations and applied mathematics at the undergraduate level. It is also a suitable reference for professionals in all areas of science, physics, and engineering.
1 Modeling with Differential Equations.
1.2 Differential Equations Describing Populations.
1.3 Remarks on Modeling with Differential Equations.
1.4 Newton's Law of Cooling.
1.5 Loaded Horizontal Beams.
2 Some Special First Order Ordinary Differential Equations.
2.1 Separable Differential Equations.
2.2 Linear First Order Differential Equations.
2.3 Bernoulli Equations.
2.4 Homogeneous Equations.
2.5 Exact Differential Equations.
2.7 How to Review and Remember.
2.8 Review of First Order Differential Equations.
Before Module 3 Oscillating Systems and Hanging Cables.
B3.2 LRC Circuits.
B3.3 The Simple Pendulum.
B3.4 Suspended Cables.
3 Linear Differential Equations with Constant Coefficients.
3.1 Homogeneous Linear Differential Equations with Constant Coefficients.
3.2 Solving Initial and Boundary Value Problems.
3.3 Designing Oscillating Systems.
3.4 The Method of Undetermined Coefficients.
3.5 Variation of Parameters.
3.6 Cauchy-Euler Equations.
3.7 Some Results on Boundary Value Problems.
4 Qualitative and Numerical Analysis of Differential Equations.
4.1 Direction Fields and Autonomous Equations.
4.2 From Visualization to Algorithm: Euler's Method.
4.3 Runge-Kutta Methods.
4.4 Finite Difference Methods for Second Order Boundary Value Problems.
5 Linear Differential Equations-Theory.
5.1 Existence and Uniqueness of Solutions.
5.2 Linear Independence for Vectors.
5.3 Matrices and Determinants.
5.4 Linear Independence for Functions.
5.5 The General Solution of Homogeneous Equations.
Before Module 6 Coupled Electrical and Mechanical Systems.
B6.1 Multi-Loop Circuits and Kirchhoff's Laws.
B6.2 Coupled Spring-Mass-Systems.
6 Laplace Transforms.
6.1 Introducing the Laplace Transform.
6.2 Solving Differential Equations with Laplace Transforms.
6.3 Systems of Linear Differential Equations.
6.4 Expanding the Transform Table.
6.5 Discontinuous Forcing Terms.
6.6 Complicated Forcing Functions and Convolutions.
Before Module 7 Vibration and Heat.
B7.1 Vibrating Strings.
B7.2 The Heat Equation.
B7.3 The Schrodinger Equation.
7 Introduction to Partial Differential Equations.
7.1 Separation of Variables.
7.2 Fourier Polynomials and Fourier Series.
7.3 Fourier Series and Separation of Variables.
7.4 Bessel and Legendre Equations.
8 Series Solutions of Differential Equations.
8.1 Expansions About Ordinary Points.
8.2 Legendre Polynomials.
8.3 Expansions about Singular Points.
8.4 Bessel Functions.
8.5 Reduction of Order.
9 Systems of Linear Differential Equations.
9.1 Existence and Uniqueness of Solutions.
9.2 Matrix Algebra.
9.3 Diagonalizable Systems with Constant Coefficients.
9.4 Non-Diagonalizable Systems with Constant Coefficients.
9.5 Qualitative Analysis.
9.6 Variation of Parameters.
9.7 Outlook on the Theory: Matrix Exponentials and the Jordan Normal Form.
C Hints and Solutions for Selected Problems.
Includes 20 student activities at the back of the book on perforated paper, allowing for classroom convenience as well as accessible homework problems
Includes programming projects with detailed specifications that require readers to manually write computer code to solve certain problems. The book is not specific to one computer program and can be used with Mathematica, Maple, R, etc.
Supplemented with online PowerPoint slides for classroom use as well as videos featuring discussions of various topics including homogeneous first order equations, the general solution of separable differential equations, the derivation of the differential equations for a multi-loop circuit and discussion of Kirchhoff's laws, step functions and Laplace transforms, etc. (http://www2.latech.edu/~schroder/videos.htm)
Contains solutions and hints throughout the book
An Instructor's Guide is available via the author's website. The Instructor's Guide is a collection of the author's class preparations.
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