## Contents

Chapter 1 Introduction 1

1.1 Applications Leading to Differential Equations

1.2 First Order Equations 5

1.3 Direction Fields for First Order Equations 16

Chapter 2 First Order Equations 30

2.1 Linear First Order Equations 30

2.2 Separable Equations 45

2.3 Existence and Uniqueness of Solutions of Nonlinear Equations 55

2.4 Transformation of Nonlinear Equations into Separable Equations 62

2.5 Exact Equations 73

2.6 Integrating Factors 82

Chapter 3 Numerical Methods

3.1 Euler’s Method 96

3.2 The Improved Euler Method and Related Methods 109

3.3 The Runge-Kutta Method 119

Chapter 4 Applications of First Order Equations1em 130

4.1 Growth and Decay 130

4.2 Cooling and Mixing 140

4.3 Elementary Mechanics 151

4.4 Autonomous Second Order Equations 162

4.5 Applications to Curves 179

Chapter 5 Linear Second Order Equations

5.1 Homogeneous Linear Equations 194

5.2 Constant Coefficient Homogeneous Equations 210

5.3 Nonhomgeneous Linear Equations 221

5.4 The Method of Undetermined Coefficients I 229

iv

5.5 The Method of Undetermined Coefficients II 238

5.6 Reduction of Order 248

5.7 Variation of Parameters 255

Chapter 6 Applcations of Linear Second Order Equations 268

6.1 Spring Problems I 268

6.2 Spring Problems II 279

6.3 The RLC Circuit 290

6.4 Motion Under a Central Force 296

Chapter 7 Series Solutions of Linear Second Order Equations

7.1 Review of Power Series 306

7.2 Series Solutions Near an Ordinary Point I 319

7.3 Series Solutions Near an Ordinary Point II 334

7.4 Regular Singular Points Euler Equations 342

7.5 The Method of Frobenius I 347

7.6 The Method of Frobenius II 364

7.7 The Method of Frobenius III 378

Chapter 8 Laplace Transforms

8.1 Introduction to the Laplace Transform 393

8.2 The Inverse Laplace Transform 405

8.3 Solution of Initial Value Problems 413

8.4 The Unit Step Function 419

8.5 Constant Coefficient Equations with Piecewise Continuous Forcing

Functions 430

8.6 Convolution 440

8.7 Constant Cofficient Equations with Impulses 452

8.8 A Brief Table of Laplace Transforms

Chapter 9 Linear Higher Order Equations

9.1 Introduction to Linear Higher Order Equations 465

9.2 Higher Order Constant Coefficient Homogeneous Equations 475

9.3 Undetermined Coefficients for Higher Order Equations 487

9.4 Variation of Parameters for Higher Order Equations 497

Chapter 10 Linear Systems of Differential Equations

10.1 Introduction to Systems of Differential Equations 507

10.2 Linear Systems of Differential Equations 515

10.3 Basic Theory of Homogeneous Linear Systems 521

10.4 Constant Coefficient Homogeneous Systems I 529

vi Contents

10.5 Constant Coefficient Homogeneous Systems II 542

10.6 Constant Coefficient Homogeneous Systems II 556

10.7 Variation of Parameters for Nonhomogeneous Linear Systems 568

Chapter 11 Boundary Value Problems and Fourier Expansions 580

11.1 Eigenvalue Problems for y00 + y = 0 580

11.2 Fourier Series I 586

11.3 Fourier Series II 603

Chapter 12 Fourier Solutions of Partial Differential Equations

12.1 The Heat Equation 618

12.2 The Wave Equation 630

12.3 Laplace’s Equation in Rectangular Coordinates 649

12.4 Laplace’s Equation in Polar Coordinates 666

Chapter 13 Boundary Value Problems for Second Order Linear Equations

13.1 Boundary Value Problems 676

13.2 Sturm–Liouville Problems 687

## Preface

Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering,

and mathematics who have completed calculus through partial differentiation. If your syllabus

includes Chapter 10 (Linear Systems of Differential Equations), your students should have some preparation

in linear algebra.

In writing this book I have been guided by the these principles:

• An elementary text should be written so the student can read it with comprehension without too

much pain. I have tried to put myself in the student’s place, and have chosen to err on the side of

too much detail rather than not enough.

• An elementary text can’t be better than its exercises. This text includes 2041 numbered exercises,

many with several parts. They range in difficulty from routine to very challenging.

• An elementary text should be written in an informal but mathematically accurate way, illustrated

by appropriate graphics. I have tried to formulate mathematical concepts succinctly in language

that students can understand. I have minimized the number of explicitly stated theorems and definitions,

preferring to deal with concepts in a more conversational way, copiously illustrated by

299 completely worked out examples. Where appropriate, concepts and results are depicted in 188

figures.

Although I believe that the computer is an immensely valuable tool for learning, doing, and writing

mathematics, the selection and treatment of topics in this text reflects my pedagogical orientation along

traditional lines. However, I have incorporated what I believe to be the best use of modern technology,

so you can select the level of technology that you want to include in your course. The text includes 414

exercises – identified by the symbols C and C/G – that call for graphics or computation and graphics.

There are also 79 laboratory exercises – identified by L – that require extensive use of technology. In

addition, several sections include informal advice on the use of technology. If you prefer not to emphasize

technology, simply ignore these exercises and the advice.

There are two schools of thought on whether techniques and applications should be treated together or

separately. I have chosen to separate them; thus, Chapter 2 deals with techniques for solving first order

equations, and Chapter 4 deals with applications. Similarly, Chapter 5 deals with techniques for solving

second order equations, and Chapter 6 deals with applications. However, the exercise sets of the sections

dealing with techniques include some applied problems.

Traditionally oriented elementary differential equations texts are occasionally criticized as being collections

of unrelated methods for solving miscellaneous problems. To some extent this is true; after all,

no single method applies to all situations. Nevertheless, I believe that one idea can go a long way toward

unifying some of the techniques for solving diverse problems: variation of parameters. I use variation of

parameters at the earliest opportunity in Section 2.1, to solve the nonhomogeneous linear equation, given

a nontrivial solution of the complementary equation. You may find this annoying, since most of us learned

that one should use integrating factors for this task, while perhaps mentioning the variation of parameters

option in an exercise. However, there’s little difference between the two approaches, since an integrating

factor is nothing more than the reciprocal of a nontrivial solution of the complementary equation. The

advantage of using variation of parameters here is that it introduces the concept in its simplest form and

vii

viii Preface

focuses the student’s attention on the idea of seeking a solution y of a differential equation by writing it

as y = uy1, where y1 is a known solution of related equation and u is a function to be determined. I use

this idea in nonstandard ways, as follows:

• In Section 2.4 to solve nonlinear first order equations, such as Bernoulli equations and nonlinear

homogeneous equations.

• In Chapter 3 for numerical solution of semilinear first order equations.

• In Section 5.2 to avoid the necessity of introducing complex exponentials in solving a second order

constant coefficient homogeneous equation with characteristic polynomials that have complex

zeros.

• In Sections 5.4, 5.5, and 9.3 for the method of undetermined coefficients. (If the method of annihilators

is your preferred approach to this problem, compare the labor involved in solving, for

example, y00 + y0 + y = x4ex by the method of annihilators and the method used in Section 5.4.)

Introducing variation of parameters as early as possible (Section 2.1) prepares the student for the concept

when it appears again in more complex forms in Section 5.6, where reduction of order is used not

merely to find a second solution of the complementary equation, but also to find the general solution of the

nonhomogeneous equation, and in Sections 5.7, 9.4, and 10.7, that treat the usual variation of parameters

problem for second and higher order linear equations and for linear systems.

Chapter 11 develops the theory of Fourier series. Section 11.1 discusses the five main eigenvalue problems

that arise in connection with the method of separation of variables for the heat and wave equations

and for Laplace’s equation over a rectangular domain:

Problem 1: y00 + y = 0, y(0) = 0, y(L) = 0

Problem 2: y00 + y = 0, y0(0) = 0, y0(L) = 0

Problem 3: y00 + y = 0, y(0) = 0, y0(L) = 0

Problem 4: y00 + y = 0, y0(0) = 0, y(L) = 0

Problem 5: y00 + y = 0, y(−L) = y(L), y0(−L) = y0(L)

These problems are handled in a unified way for example, a single theorem shows that the eigenvalues

of all five problems are nonnegative.

Section 11.2 presents the Fourier series expansion of functions defined on on [−L, L], interpreting it

as an expansion in terms of the eigenfunctions of Problem 5.

Section 11.3 presents the Fourier sine and cosine expansions of functions defined on [0, L], interpreting

them as expansions in terms of the eigenfunctions of Problems 1 and 2, respectively. In addition, Section

11.2 includes what I call themixed Fourier sine and cosine expansions, in terms of the eigenfunctions

of Problems 4 and 5, respectively. In all cases, the convergence properties of these series are deduced

from the convergence properties of the Fourier series discussed in Section 11.1.

Chapter 12 consists of four sections devoted to the heat equation, the wave equation, and Laplace’s

equation in rectangular and polar coordinates. For all three, I consider homogeneous boundary conditions

of the four types occurring in Problems 1-4. I present the method of separation of variables as a way of

choosing the appropriate form for the series expansion of the solution of the given problem, stating—

without belaboring the point—that the expansion may fall short of being an actual solution, and giving

an indication of conditions under which the formal solution is an actual solution. In particular, I found it

necessary to devote some detail to this question in connection with the wave equation in Section 12.2.

In Sections 12.1 (The Heat Equation) and 12.2 (The Wave Equation) I devote considerable effort to

devising examples and numerous exercises where the functions defining the initial conditions satisfy

Preface ix

the homogeneous boundary conditions. Similarly, in most of the examples and exercises Section 12.3

(Laplace’s Equation), the functions defining the boundary conditions on a given side of the rectangular

domain satisfy homogeneous boundary conditions at the endpoints of the same type (Dirichlet or Neumann)

as the boundary conditions imposed on adjacent sides of the region. Therefore the formal solutions

obtained in many of the examples and exercises are actual solutions.

Section 13.1 deals with two-point value problems for a second order ordinary differential equation.

Conditions for existence and uniqueness of solutions are given, and the construction of Green’s functions

is included.

Section 13.2 presents the elementary aspects of Sturm-Liouville theory.

You may also find the following to be of interest:

• Section 2.6 deals with integrating factors of the form μ = p(x)q(y), in addition to those of the

form μ = p(x) and μ = q(y) discussed in most texts.

• Section 4.4 makes phase plane analysis of nonlinear second order autonomous equations accessible

to students who have not taken linear algebra, since eigenvalues and eigenvectors do not enter

into the treatment. Phase plane analysis of constant coefficient linear systems is included in Sections

10.4-6.

• Section 4.5 presents an extensive discussion of applications of differential equations to curves.

• Section 6.4 studies motion under a central force, which may be useful to students interested in the

mathematics of satellite orbits.

• Sections 7.5-7 present the method of Frobenius in more detail than in most texts. The approach

is to systematize the computations in a way that avoids the necessity of substituting the unknown

Frobenius series into each equation. This leads to efficiency in the computation of the coefficients

of the Frobenius solution. It also clarifies the case where the roots of the indicial equation differ by

an integer (Section 7.7).

• The free Student SolutionsManual contains solutions of most of the even-numbered exercises.

• The free Instructor’s Solutions Manual is available by email to wtrench@trinity.edu, subject to

verification of the requestor’s faculty status.

The following observations may be helpful as you choose your syllabus:

• Section 2.3 is the only specific prerequisite for Chapter 3. To accomodate institutions that offer a

separate course in numerical analysis, Chapter 3 is not a prerequisite for any other section in the

text.

• The sections in Chapter 4 are independent of each other, and are not prerequisites for any of the

later chapters. This is also true of the sections in Chapter 6, except that Section 6.1 is a prerequisite

for Section 6.2.

• Chapters 7, 8, and 9 can be covered in any order after the topics selected from Chapter 5. For

example, you can proceed directly from Chapter 5 to Chapter 9.

• The second order Euler equation is discussed in Section 7.4, where it sets the stage for the method

of Frobenius. As noted at the beginning of Section 7.4, if you want to include Euler equations in

your syllabus while omitting the method of Frobenius, you can skip the introductory paragraphs

in Section 7.4 and begin with Definition 7.4.2. You can then cover Section 7.4 immediately after

Section 5.2.