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William F. Trench
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For more information, please contact [email protected] ELEMENTARY
DIFFERENTIAL EQUATIONS WITH
BOUNDARY VALUE PROBLEMS William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus
Department of Mathematics
San Antonio, Texas, USA
This book has been judged to meet the evaluation criteria set by the Editorial Board of the American Institute of Mathematics in connection with
the Institute’s Open Textbook Initiative. It may be copied, modified, redistributed, translated, and built upon subject to the Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 Unported License.
FREE DOWNLOAD: STUDENT SOLUTIONS MANUAL Free Edition 1.01 (December 2013)
This book was published previously by Brooks/Cole Thomson Learning, 2001. This free edition is made
available in the hope that it will be useful as a textbook or reference. Reproduction is permitted for
any valid noncommercial educational, mathematical, or scientific purpose. However, charges for profit
beyond reasonable printing costs are prohibited. TO BEVERLY Contents
Chapter 1 Introduction 1 1.1 Applications Leading to Differential Equations
1.2 First Order Equations
1.3 Direction Fields for First Order Equations 5
16 Chapter 2 First Order Equations 30 2.1
82 Linear First Order Equations
Existence and Uniqueness of Solutions of Nonlinear Equations
Transformation of Nonlinear Equations into Separable Equations
Integrating Factors Chapter 3 Numerical Methods
3.1 Euler’s Method
3.2 The Improved Euler Method and Related Methods
3.3 The Runge-Kutta Method 96
119 Chapter 4 Applications of First Order Equations1em 130 4.1
179 Growth and Decay
Cooling and Mixing
Autonomous Second Order Equations
Applications to Curves Chapter 5 Linear Second Order Equations
5.4 Homogeneous Linear Equations
Constant Coefficient Homogeneous Equations
Nonhomgeneous Linear Equations
The Method of Undetermined Coefficients I iv 194
229 5.5 The Method of Undetermined Coefficients II
5.6 Reduction of Order
5.7 Variation of Parameters 238
255 Chapter 6 Applcations of Linear Second Order Equations 268 6.1
296 Spring Problems I
Spring Problems II
The RLC Circuit
Motion Under a Central Force Chapter 7 Series Solutions of Linear Second Order Equations
7.7 Review of Power Series
Series Solutions Near an Ordinary Point I
Series Solutions Near an Ordinary Point II
Regular Singular Points Euler Equations
The Method of Frobenius I
The Method of Frobenius II
The Method of Frobenius III 306
378 Chapter 8 Laplace Transforms
8.1 Introduction to the Laplace Transform
8.2 The Inverse Laplace Transform
8.3 Solution of Initial Value Problems
8.4 The Unit Step Function
8.5 Constant Coefficient Equations with Piecewise Continuous Forcing
8.7 Constant Cofficient Equations with Impulses
8.8 A Brief Table of Laplace Transforms 393
452 Chapter 9 Linear Higher Order Equations
9.4 Introduction to Linear Higher Order Equations
Higher Order Constant Coefficient Homogeneous Equations
Undetermined Coefficients for Higher Order Equations
Variation of Parameters for Higher Order Equations 465
497 Chapter 10 Linear Systems of Differential Equations
10.4 Introduction to Systems of Differential Equations
Linear Systems of Differential Equations
Basic Theory of Homogeneous Linear Systems
Constant Coefficient Homogeneous Systems I 507
529 vi Contents
10.5 Constant Coefficient Homogeneous Systems II
10.6 Constant Coefficient Homogeneous Systems II
10.7 Variation of Parameters for Nonhomogeneous Linear Systems 542
568 Chapter 11 Boundary Value Problems and Fourier Expansions 580 11.1 Eigenvalue Problems for y00 + λy = 0
11.2 Fourier Series I
11.3 Fourier Series II 580
603 Chapter 12 Fourier Solutions of Partial Differential Equations
12.4 The Heat Equation
The Wave Equation
Laplace’s Equation in Rectangular Coordinates
Laplace’s Equation in Polar Coordinates 618
666 Chapter 13 Boundary Value Problems for Second Order Linear Equations
13.1 Boundary Value Problems
13.2 Sturm–Liouville Problems 676
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
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
• 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
• 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 = x4 ex 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:
y00 + λy = 0, y(0) = 0, y(L) = 0
Problem 2: y00 + λy = 0, y0 (0) = 0, y0 (L) = 0 Problem 3: y00 + λy = 0, y(0) = 0, y0 (L) = 0 Problem 4: y00 + λy = 0, y0 (0) = 0, y(L) = 0 Problem 5: y00 + λy = 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 the mixed 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
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 Solutions Manual contains solutions of most of the even-numbered exercises.
• The free Instructor’s Solutions Manual is available by email to [email protected], 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
• 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
• Chapters 11, 12, and 13 can be covered at any time after the completion of Chapter 5.
William F. Trench CHAPTER 1
Introduction IN THIS CHAPTER we begin our study of differential equations.
SECTION 1.1 presents examples of applications that lead to differential equations.
SECTION 1.2 introduces basic concepts and definitions concerning differential equations.
SECTION 1.3 presents a geometric method for dealing with differential equations that has been known
for a very long time, but has become particularly useful and important with the proliferation of readily
available differential equations software. 1 2 Chapter 1 Introduction 1.1 APPLICATIONS LEADING TO DIFFERENTIAL EQUATIONS
In order to apply mathematical methods to a physical or “real life” problem, we must formulate the problem in mathematical terms; that is, we must construct a mathematical model for the problem. Many
physical problems concern relationships between changing quantities. Since rates of change are represented mathematically by derivatives, mathematical models often involve equations relating an unknown
function and one or more of its derivatives. Such equations are differential equations. They are the subject
of this book.
Much of calculus is devoted to learning mathematical techniques that are applied in later courses in
mathematics and the sciences; you wouldn’t have time to learn much calculus if you insisted on seeing
a specific application of every topic covered in the course. Similarly, much of this book is devoted to
methods that can be applied in later courses. Only a relatively small part of the book is devoted to
the derivation of specific differential equations from mathematical models, or relating the differential
equations that we study to specific applications. In this section we mention a few such applications.
The mathematical model for an applied problem is almost always simpler than the actual situation
being studied, since simplifying assumptions are usually required to obtain a mathematical problem that
can be solved. For example, in modeling the motion of a falling object, we might neglect air resistance
and the gravitational pull of celestial bodies other than Earth, or in modeling population growth we might
assume that the population grows continuously rather than in discrete steps.
A good mathematical model has two important properties:
• It’s sufficiently simple so that the mathematical problem can be solved.
• It represents the actual situation sufficiently well so that the solution to the mathematical problem
predicts the outcome of the real problem to within a useful degree of accuracy. If results predicted
by the model don’t agree with physical observations, the underlying assumptions of the model must
be revised until satisfactory agreement is obtained.
We’ll now give examples of mathematical models involving differential equations. We’ll return to these
problems at the appropriate times, as we learn how to solve the various types of differential equations that
occur in the models.
All the examples in this section deal with functions of time, which we denote by t. If y is a function of
t, y0 denotes the derivative of y with respect to t; thus,
y0 = dy
dt Population Growth and Decay
Although the number of members of a population (people in a given country, bacteria in a laboratory culture, wildflowers in a forest, etc.) at any given time t is necessarily an integer, models that use differential
equations to describe the growth and decay of populations usually rest on the simplifying assumption that
the number of members of the population can be regarded as a differentiable function P = P (t). In most
models it is assumed that the differential equation takes the form
P 0 = a(P )P, (1.1.1) where a is a continuous function of P that represents the rate of change of population per unit time per
individual. In the Malthusian model, it is assumed that a(P ) is a constant, so (1.1.1) becomes
P 0 = aP. (1.1.2) Section 1.1 Applications Leading to Differential Equations 3 (When you see a name in blue italics, just click on it for information about the person.) This model
assumes that the numbers of births and deaths per unit time are both proportional to the population. The
constants of proportionality are the birth rate (births per unit time per individual) and the death rate
(deaths per unit time per individual); a is the birth rate...
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