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#
Applied Differential Equations

The Primary Course

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## Book Description

This book started as a collection of lecture notes for a course in differential equations taught by the Division of Applied Mathematics at Brown University. To some extent, it is a result of collective insights given by almost every instructor who taught such a course over the last 15 years. Therefore, the material and its presentation covered in this book were practically tested for many years.

This text is designed for a two-semester sophomore or junior level course in differential equations. It offers novel approaches in presentation and utilization of computer capabilities. This text intends to provide a solid background in differential equations for students majoring in a breadth of fields.

Differential equations are described in the context of applications. The author stresses differential equations constitute an essential part of modeling by showing their applications, including numerical algorithms and syntax of the four most popular software packages. Students learn how to formulate a mathematical model, how to solve differential equations (analytically or numerically), how to analyze them qualitatively, and how to interpret the results.

In writing this textbook, the author aims to assist instructors and students through:

- Showing a course in differential equations is essential for modeling real-life phenomena
- Stressing the mastery of traditional solution techniques and presenting effective methods, including reliable numerical approximations
- Providing qualitative analysis of ordinary differential equations. The reader should get an idea of how all solutions to the given problem behave, what are their validity intervals, whether there are oscillations, vertical or horizontal asymptotes, and what is their long-term behavior
- The reader will learn various methods of solving, analysis, visualization, and approximation, exploiting the capabilities of computers
- Introduces and employs Maple™, Mathematica®, MatLab®, and Maxima
- This textbook facilitates the development of the student’s skills to model real-world problems

Ordinary and partial differential equations is a classical subject that has been studied for about 300 years. The beauty and utility of differential equations and their application in mathematics, biology, chemistry, computer science, economics, engineering, geology, neuroscience, physics, the life sciences, and other fields reaffirm their inclusion in myriad curricula.

A great number of examples and exercises make this text well suited for self-study or for traditional use by a lecturer in class. Therefore, this textbook addresses the needs of two levels of audience, the beginning and the advanced.

## Table of Contents

Preface

1 Introduction

1.1 Motivation

1.2 Classification of Differential Equations

1.3 Solutions to Differential Equations

1.4 Particular and Singular Solutions

1.5 Direction Fields

1.6 Existence and Uniqueness

Review Questions for Chapter 1

2 First Order Equations

2.1 Separable Equations

2.1.1 Autonomous Equations

2.2 Equations Reducible to Separable Equations

2.2.1 Equations with Homogeneous Coefficients

2.2.2 Equations with Homogeneous Fractions

2.2.3 Equations with Linear Coefficients

2.3 Exact Differential Equations

2.4 Simple Integrating Factors

2.5 First-Order Linear Differential Equations

2.6 Special Classes of Equations

2.6.1 The Bernoulli Equation

2.6.2 The Riccati Equation

2.6.3 Equations with the Dependent or Independent Variable Missing

2.6.4 Equations Homogeneous with Respect to Their Dependent Variable

2.6.5 Equations Solvable for a Variable

2.7 Qualitative Analysis

2.7.1 Bifurcation Points

2.7.2 Validity Intervals of Autonomous Equations

Summary for Chapter 2

Review Questions for Chapter 2

3 Numerical Methods

3.1 Difference Equations

3.2 Euler’s Methods

3.3 The Polynomial Approximation

3.4 Error Estimates

3.5 The Runge–Kutta Methods

Summary for Chapter 3

Review Questions for Chapter 3

4 Second and Higher Order Linear Differential Equations

4.1 Second and Higher Order Differential Equations

4.1.1 Linear Operators

4.1.2 Exact Equations and Integrating Factors

4.1.3 Change of Variables

4.2 Linear Independence and Wronskians

4.3 The Fundamental Set of Solutions

4.4 Equations with Constant Coefficients

4.5 Complex Roots

4.6 Repeated Roots. Reduction of Order

4.6.1 Reduction of Order

4.6.2 Euler’s Equations

4.7 Nonhomogeneous Equations

4.7.1 The Annihilator

4.7.2 The Method of Undetermined Coefficients

4.8 Variation of Parameters

Summary for Chapter 4

Review Questions for Chapter 4

5 Laplace Transforms

5.1 The Laplace Transform

5.2 Properties of the Laplace Transform

5.3 Discontinuous and Impulse Functions

5.4 The Inverse Laplace Transform

5.4.1 Partial Fraction Decomposition

5.4.2 Convolution Theorem

5.4.3 The Residue Method

5.5 Homogeneous Differential Equations

5.5.1 Equations with Variable Coefficients

5.6 Nonhomogeneous Differential Equations

5.6.1 Differential Equations with Intermittent Forcing Functions

Summary for Chapter 5

Review Questions for Chapter 5

6 Series Solutions of Differential Equations 335

6.1 Power Series Solutions

6.2 Picard’s Iterations

6.3 Adomian Decomposition Method

6.4 Power Series Solutions to Equations with Analytic Coefficients

6.4.1 The Ordinary Point at Infinity

6.5 Euler Equations

6.6 Series Solutions Near a Regular Singular Point

6.6.1 Regular Singular Point at Infinity

6.6.2 Inhomogeneous Equations

6.7 Bessel Equations

6.7.1 Parametric Bessel Equation

6.7.2 Bessel Functions of Half-Integer Order

6.7.3 Related Differential Equations

Summary for Chapter 6

Review Questions for Chapter 6

7 Introduction to Systems of ODEs

7.1 Some ODE Models

7.1.1 RLC-circuits

7.1.2 Spring-Mass Systems

7.1.3 The Euler–Lagrange Equation

7.1.4 Pendulum

7.1.5 Laminated Material

7.1.6 Flow Problems

7.2 Matrices

7.3 Linear Systems of First Order ODEs

7.4 Reduction to a Single ODE

7.5 Existence and Uniqueness

Summary for Chapter 7

Review Questions for Chapter 7

8 Topics from Linear Algebra

8.1 The Calculus of Matrix Functions

8.2 Inverses and Determinants

8.2.1 Solving Linear Equations

8.3 Eigenvalues and Eigenvectors

8.4 Diagonalization

8.5 Sylvester’s Formula

8.6 The Resolvent Method

8.7 The Spectral Decomposition Method

Summary for Chapter 8

Review Questions for Chapter 8

9 Systems of Linear Differential Equations

9.1 Systems of Linear Equations

9.1.1 The Euler Vector Equations

9.2 Constant Coefficient Homogeneous Systems

9.2.1 Simple Real Eigenvalues

9.2.2 Complex Eigenvalues

9.2.3 Repeated Eigenvalues

9.2.4 Qualitative Analysis of Linear Systems

9.3 Variation of Parameters

9.3.1 Equations with Constant Coefficients

9.4 Method of Undetermined Coefficients

9.5 The Laplace Transformation

9.6 Second Order Linear Systems

Summary for Chapter 9

Review Questions for Chapter 9

10 Qualitative Theory of Differential Equations

10.1 Autonomous Systems

10.1.1 Two-Dimensional Autonomous Equations

10.2 Linearization

10.2.1 Two-Dimensional Autonomous Equations

10.2.2 Scalar Equations

10.3 Population Models

10.3.1 Competing Species

10.3.2 Predator-Prey Equations

10.3.3 Other Population Models

10.4 Conservative Systems

10.4.1 Hamiltonian Systems

10.5 Lyapunov’s Second Method

10.6 Periodic Solutions

10.6.1 Equations with Periodic Coefficients

Summary for Chapter 10

Review Questions for Chapter 10

11 Orthogonal Expansions

11.1 Sturm–Liouville Problems

11.2 Orthogonal Expansions

11.3 Fourier Series

11.3.1 Music as Motivation

11.3.2 Sturm–Liouville Periodic Problem

11.3.3 Fourier Series

11.4 Convergence of Fourier Series

11.4.1 Complex Fourier Series

11.4.2 The Gibbs Phenomenon

11.5 Even and Odd Functions

Summary for Chapter 11

Review Questions for Chapter 11

12 Partial Differential Equations

12.1 Separation of Variables for the Heat Equation

12.1.1 Two-Dimensional Heat Equation

12.2 Other Heat Conduction Problems

12.3 Wave Equation

12.3.1 Transverse Vibrations of Beams

12.4 Laplace Equation

12.4.1 Laplace Equation in Polar Coordinates

Summary for Chapter 12

Review Questions for Chapter 12

Bibliography

Index

## Author(s)

### Biography

**Vladimir A. Dobrushkin **is a Professor at the Division of Applied Mathematics, Brown University. He holds a Ph.D. in Applied mathematics and Dr.Sc. in mechanical engineering. He is the author of three books for CRC Press, including* Applied Differential Equations with Boundary Value Problems*, and *Methods in Algorithmic Analysis*. He is the co-author of *Handbook of Differential Equations,* Fourth Edition with Daniel Zwillinger.

## Reviews

As the author states in his preface, the Ordinary Differential Equations course is taught to undergraduates since some 200 years. If however a comparison is made between the book of Coddington and Levinson or the one of Hartman - two rather standard textbooks for at least half of century - and this one, the differences may be striking. The book aims to lay down a bridge between calculus,modeling and advanced topics (resulting from application requirements). Next the book emphasizes the role and the applications of qualitative theory. Among other aspects differentiating this book from others there can be mentioned: insertion of the topics from linear algebra as pre-requisite for the study of the systems of linear differential equations; discussion of the Sturm Liouville problems in two contexts: the orthogonal expansions and the boundary value problems occurring within the method of the separationof the variables for second order partial differential equations. Summarizing, this highly original textbook can introduce a new way of teaching ordinary differential equations.

~Vladimir Rasvan (

Craiova) Sept. 2018