Differential Equations
with Boundary
~ ~
Differential Equations
with Boundary
Value Problems
Second Edition
John Polking
Rice UniversityAlbert Boggess
Texas A&M UniversityDavid Arnold
College of the RedwoodsDifferential equations with boundary value problems / John Polking, Albert Boggess, David Arnold.—2nd ed.
p. cm. Includes index. ISBN 0-13-186236-7
1. Differential equations. 2. Boundary value problems. I. Boggess, Albert. II. Arnold, David. III. Title.
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Contents
Preface
ix
1 Introduction to Differential Equations
1
1.1 Differential Equation Models 2
1.2 The Derivative 6
1.3 Integration 10
2 First-Order Equations
16
2.1 Differential Equations and Solutions 16
2.2 Solutions to Separable Equations 27
2.3 Models of Motion 37
2.4 Linear Equations 47
2.5 Mixing Problems 56
2.6 Exact Differential Equations 63
2.7 Existence and Uniqueness of Solutions 77
2.8 Dependence of Solutions on Initial Conditions 88
2.9 Autonomous Equations and Stability 92
Project 2.10 The Daredevil Skydiver 102
3 Modeling and Applications
104
3.1 Modeling Population Growth 105
3.2 Models and the Real World 117
3.3 Personal Finance 121
3.4 Electrical Circuits 128
Project 3.5 The Spruce Budworm 132
Project 3.6 Social Security, Now or Later 134
4 Second-Order Equations
136
4.1 Definitions and Examples 136
4.2 Second-Order Equations and Systems 146
4.3 Linear, Homogeneous Equations with Constant Coefficients 150
4.4 Harmonic Motion 156
4.5 Inhomogeneous Equations; the Method of Undetermined
Coefficients 164
4.6 Variation of Parameters 173
4.7 Forced Harmonic Motion 177
Project 4.8 Nonlinear Oscillators 187
5 The Laplace Transform
189
5.1 The Definition of the Laplace Transform 190
5.2 Basic Properties of the Laplace Transform 197
5.3 The Inverse Laplace Transform 203
5.4 Using the Laplace Transform to Solve Differential Equations 209
5.5 Discontinuous Forcing Terms 215
5.6 The Delta Function 227
5.7 Convolutions 233
5.8 Summary 242
Project 5.9 Forced Harmonic Oscillators 243
6 Numerical Methods
245
6.1 Euler’s Method 246
6.2 Runge-Kutta Methods 255
6.3 Numerical Error Comparisons 261
6.4 Practical Use of Solvers 265
Project 6.5 Numerical Error Comparison 271
7 Matrix Algebra
272
7.1 Vectors and Matrices 272
7.2 Systems of Linear Equations with Two or Three Variables 283
7.3 Solving Systems of Equations 292
7.4 Homogeneous and Inhomogeneous Systems 300
7.5 Bases of a Subspace 307
7.6 Square Matrices 317
7.7 Determinants 322
8 An Introduction to Systems
331
8.1 Definitions and Examples 331
8.2 Geometric Interpretation of Solutions 338
8.3 Qualitative Analysis 347
8.4 Linear Systems 353
8.5 Properties of Linear Systems 362
Contents vii
9 Linear Systems with Constant Coefficients
372
9.1 Overview of the Technique 372
9.2 Planar Systems 378
9.3 Phase Plane Portraits 392
9.4 The Trace-Determinant Plane 402
9.5 Higher-Dimensional Systems 407
9.6 The Exponential of a Matrix 416
9.7 Qualitative Analysis of Linear Systems 429
9.8 Higher-Order Linear Equations 433
9.9 Inhomogeneous Linear Systems 444
Project 9.10 Phase Plane Portraits 454
Project 9.11 Oscillations of Linear Molecules 454
10 Nonlinear Systems
458
10.1 The Linearization of a Nonlinear System 458
10.2 Long-Term Behavior of Solutions 469
10.3 Invariant Sets and the Use of Nullclines 475
10.4 Long-Term Behavior of Solutions to Planar Systems 481
10.5 Conserved Quantities 490
10.6 Nonlinear Mechanics 495
10.7 The Method of Lyapunov 510
10.8 Predator–Prey Systems 517
Project 10.9 Human Immune Response to Infectious Disease 528
Project 10.10 Analysis of Competing Species 530
11 Series Solutions to Differential Equations
532
11.1 Review of Power Series 533
11.2 Series Solutions Near Ordinary Points 543
11.3 Legendre’s Equation 555
11.4 Types of Singular Points—Euler’s Equation 560
11.5 Series Solutions Near Regular Singular Points 566
11.6 Series Solutions Near Regular Singular
Points—The General Case 575
11.7 Bessel’s Equation and Bessel Functions 586
12 Fourier Series
595
12.1 Computation of Fourier Series 596
12.2 Convergence of Fourier Series 605
12.3 Fourier Cosine and Sine Series 613
12.4 The Complex Form of a Fourier Series 617
12.5 The Discrete Fourier Transform and the FFT 620
13 Partial Differential Equations
627
13.1 Derivation of the Heat Equation 627
13.2 Separation of Variables for the Heat Equation 634
13.3 The Wave Equation 644
13.5 Laplace’s Equation on a Disk 660
13.6 Sturm-Liouville Problems 666
13.7 Orthogonality and Generalized Fourier Series 675
13.8 Temperatures in a Ball—Legendre Polynomials 682
13.9 The Heat and Wave Equations in Higher Dimension 686
13.10 Domains with Circular Symmetry—Bessel Functions 692
Appendix
Complex Numbers and Matrices
699
Answers to Odd-Numbered Problems
A-1
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~
Preface
This book started in 1993, when the first author began to reorganize the teaching of ODEs at Rice University. It soon became apparent that a textbook was needed that brought to the students the expanded outlook that modern developments in the subject required, and the use of technology allowed. Over the ensuing years this book has evolved.
The mathematical subject matter of this book has not changed dramatically from that of many books published ten or even twenty years ago. The book strikes a balance between the traditional and the modern. It covers all of the traditional material and somewhat more. It does so in a way that makes it easily possible, but not necessary, to use modern technology, especially for the visualization of the ideas involved in ordinary differential equations. It offers flexibility of use that will allow instructors at a variety of institutions to use the book. In fact, this book could easily be used in a traditional differential equations course, provided the instructor care-fully chooses the exercises assigned. However, there are changes in our students, in our world, and in our mathematics that require some changes in the ODE course, and the way we teach it.
Our students are now as likely to be majoring in the biological sciences or economics as in the physical sciences or engineering. These students are more interested in systems of equations than they are in second-order equations. They are also more interested in applications to their own areas rather than to physics or engineering.
Our world is increasingly a technological world. In academia we are struggling with the problem of adapting to this new world. The easiest way to start a spirited discussion in a group of faculty is to raise the subject of the use of technology in our teaching. Regardless of one’s position on this subject, it is widely agreed that the course where the use of technology makes the most sense, and where the impact of computer visualization is the most beneficial, is in the study of ODEs. The use of computer visualization pervades this book. The degree to which the student and the instructor are involved is up to the instructor.
The subject of ordinary differential equations has progressed, as has all of math-ematics. To many it is now known by the new name, dynamical systems. Much of the progress, and many of the directions in which the research has gone, have been motivated by computer experiments. Much of the work is qualitative in nature. This is beautiful mathematics. Introducing some of these ideas to students at an early point is a move in the right direction. It gives them a better idea of what mathemat-ics is about than the standard way of discussing one solution method after another.
It should be emphasized that the introduction of qualitative methods is not, in itself, a move to less rigor.
New to This Edition
We are gratified with the success of the first edition. However, in the years since its first appearance, we have continued to teach ODEs and we have new, hopefully better ideas about the subject. We have learned from many comments from and conversations with our users. As a result many small parts have been rewritten to improve the exposition. In addition, this new edition incorporates the following more substantial changes.
• We have added a large number of new figures to help visualize the ideas in the text.
• More of the examples are now application based.
• Where appropriate we have highlighted methods of solution to make them easier to find.
• The section in Chapter 2 on exact first-order equations has been entirely rewrit-ten.
• Chapter 7 on matrix algebra has been rewritten. There is now a section devoted to the subject in two and three dimensions. For many instructors this is all that is needed. For the rest it is a good introduction to the subject.
• The introductory discussion of linear systems in Chapter 8 has been rewritten, to improve the exposition and to incorporate more applications.
• The section in Chapter 9 on phase plane portraits was too long, so it has now been split in two, resulting in a new section on the trace-determinant plane. • We have collected the material on complex numbers and matrices into an
ap-pendix to make it more easily found.
• We have added new exercises and eliminated old ones in many sections. Many of the new exercises are application oriented. This is especially true of the exercises on partial differential equations in Chapter 13.
The Use of Technology
The book covers the standard material with an appropriate level of rigor. However, it enables the instructor to be flexible in the use of modern technology. Available to all, without the use of any technology, is the large number of graphics in the book that display the ideas in ODEs. At the next level are a large number of exercises that require the student to compute and plot solutions. For these exercises, the student will have to have access to computer (or calculator) programs that will do this easily. The tools needed for most of these exercises are two. The student will need a program that will plot the direction field for a single differential equation, and superimpose the solution with given initial conditions. In addition, the student will need a program that will plot the vector field for an autonomous planar system of equations, and superimpose the solution with given initial conditions. Such tools are available in MATLABR, Maple, and Mathematica. For many purposes it will be useful for the students to have computer (or calculator) tools for graphing functions of a single variable.
The book can also be used to teach a course in which the students learn nu-merical methods early and are required to use them regularly throughout the course.
Preface xi
Students in such a course learn the valuable skill of solving equations and systems of equations numerically and interpreting the results using the subject matter of the course. The treatment of numerical methods is somewhat more substantial than in other books. However, just enough is covered so that readers get a sense of the complexity involved. Computational error is treated, but not so rigorously as to bog the reader down and interrupt the flow of the text. Students are encouraged to do some experimental analysis of computational error.
Modeling and Applications
It is becoming a common feature of mathematics books to include a large list of applications. Usually the students are presented with the mathematical model and they are required to apply it to a variety of cases. The derivation of the model is not done. There is some sense in this. After all, mathematics does not include all of the many application areas, and the derivation of the models is the subject of the application areas. Furthermore, the derivations are very time consuming.
However, mathematicians and mathematics are part of the modeling process. It should be a greater part of our teaching. This book takes a novel approach to the teaching of modeling. While a large number of applications are covered as examples, in some cases the applications are covered in more detail than is usual. There is a historical study of the models of motion, which demonstrates to students how models continue to evolve as knowledge increases. There is an in-depth study of several population models, including their derivation. Included are historical examples of how such models were applied both where they were appropriate and where they were not. This demonstrates to students that it is necessary to understand the assumptions that lie behind a model before using it, and that any model must be checked by experiments or observations before it is accepted.
In addition, models in personal finance are discussed. This is an area of po-tential interest to all students, but not one that is covered in any detail in college courses. Students majoring in almost all disciplines approach these problems on an even footing. As a result it is an area where students can be required to do some modeling on their own.
Linear Algebra and Systems
Most books at this level assume that students have an understanding of elementary matrix algebra, usually in two and three dimensions. In the experience of the authors this assumption is not valid. Accordingly, this book devotes a chapter to matrix al-gebra. The topics covered are carefully chosen to be those needed in the study of linear systems of ODEs. With this chapter behind them, the instructor can cover linear systems of ODEs in a more substantive way. On the other hand an instruc-tor who is confident in the knowledge of the students can skip the matrix algebra chapter.
Projects
There are a number of projects discussed in the book. These involve students in an in-depth study of either mathematics or an application that uses ODEs. The projects provide students with the opportunity to bring together much of what they have learned, including analytical, computational, and interpretative skills. The level of
difficulty of the projects varies. More projects will be made available to users of this book as they are developed.
Varied Approaches Possible
It should be noticed that the book has three authors from three very different schools. The ODE courses at these institutions are quite different. Indeed, there is no stan-dard ODE course across the country. The authors set the understandable goal of writing a book that could be used in the ODE courses at each of their own institu-tions. Meeting this goal required some compromises, but the result is a book that is flexible enough to allow its use in a variety of courses at a variety of institutions.
On one hand, it is possible to use the book and teach a more or less standard course. The standard material is covered in the standard order, with or without the use of technology.
However, at Rice University, after the first three chapters the class moves to numerical methods, and then to matrix algebra. This is followed by linear systems. Once this material is covered, higher-order equations, including the second-order equations that are important in science and engineering, are covered as examples of systems. This approach allows the students to use linear algebra throughout the course, thereby gaining a working knowledge of the subject. Technology is used throughout to enhance the students’ understanding of the mathematical ideas.
In another approach, used at College of the Redwoods, numerical methods is done early using Ordinary Differential Equations Using MATLABR, 3/E, while covering the first four chapters. Chapters 7, 8, and 9 are studied, emphasizing the material on planar systems. The course ends with nonlinear systems in Chapter 10. The goal is to locate and classify equilibrium points using the Jacobian, to locate nullclines and determine flow on the nullclines and in the regions determined by the nullclines, and then to draw the phase portrait by hand.
Mathematical Rigor
Mathematical ideas are not dodged. Motivated by a perceived lack of understanding on the part of our students, we have added material about the nature of theorems and of mathematics, and the importance of proof. Proofs are given when the proof will add to the students’ understanding of the material. Difficult proofs, or those that do not add to a student’s understanding, are avoided. Suggestions of how to proceed, and examples that use these suggestions, are usually offered as motivation before one has to wade through the abstraction of a proof. The authors believe that proof is fundamental to mathematics, and that students at this level should be introduced gently to proof as an integral part of their training in mathematics. This is true for the future engineer or doctor as well as for the math major.
Additional Material
The last two chapters of this version contain the solution of boundary value prob-lems and the material needed for that. In Chapter 12 we treat Fourier series. This is an expanded version of what appears in most books, including complex Fourier series and the discrete Fourier transform. Since this material is becoming of greater interest, some instructors might want to include it in their courses. In the final chap-ter, we treat boundary value problems. The way this material is taught is changing,
Preface xiii
and we have tried to make the treatment a little more modern, while not abandoning the traditional approach. We have added material on the d’Alembert solution to the wave equation, and put some emphasis on the eigenvalue problem for the Laplacian, and its importance in understanding the wave and heat equations in more than one space dimension.
Supplements
Instructors who use this book will have available a number of resources. There is an Instructor’s Solutions Manual, containing the complete solutions to all of the exercises. In addition there is a Student Solutions Manual with the solutions to the odd-numbered exercises. The Student Solutions Manual is available shrinkwrapped with this book at no extra cost (ISBN 0-13-155954-0).
One way to meet the software needs of the student is to use the programs
dfieldand pplane, written by the first author for use with MATLABR. These programs are described in the book Ordinary Differential Equations Using MATLABR, 3/E (ISBN 0-13-145679-2), written by two of the authors of this book. However, it should be emphasized that it is not necessary to use dfield and
pplanewith this book. There are many other possibilities. Several software man-uals are available shrinkwrapped with this book at no additional cost:
• Ordinary Differential Equations Using MATLABR, 3/E (ISBN 0-13-169834-6)
• Maple Projects for Differential Equations by Robert Gilbert and George Hsiao
(ISBN 0-13-169835-4)
• Mathematica for Differential Equations: Projects, Insights, Syntax, and Ani-mations by David Calvis (ISBN 0-13-169836-2)
• Mathematica Companion for Differential Equations by Selwyn Hollis (ISBN
0-13-169837-0)
It is also possible to get this book bundled with the Student Solutions Manual and
Ordinary Differential Equations Using MATLABR, 3/E (ISBN 0-13-155972-9). Contact www.prenhall.com to order any of these titles.
The Website http://www.prenhall.com/polking is a resource that is very valu-able to both instructors and students. Interactive java versions of the direction field program dfield and the phase plane program pplane are accessible from this site. It also provides animations of the examples in the book, links to other web re-sources involving differential equations, and true-false quizzes on the subject matter. As additional projects are developed for use with the book, they will be accessible from the Website.
Acknowledgments
The development of this book depended on the efforts of a large number of people. Not the least of these is the Prentice Hall editor, George Lobell. We would also like to thank Barbara Mack, the production editor, who so patiently worked with us. Our compositor, Dennis Kletzing, was the soul of patience and worked with us to solve the problems that inevitably arise.
The reviewers of the first drafts caused us to rethink many parts of the book and certainly deserve our thanks. They are Mark Cawood, Clemson University, Charles Li, University of Missouri at Columbia, Sophia Jang, University of Louisiana,
Mo-hammed Saleem, San Jose State University, David Russell, Virginia Tech, Moses Glasner, Penn State, Joan McCarter, Arizona State University, Gunther Uhlmann, University of Washington at Seattle, Joseph Biello, Courant Institute, New York University, Alexander Khapalov, Washington State University, Jiyuan Tao, Loyola College, and Yan Wu, Georgia Southern University.
Finally, and perhaps most important, we would like to thank the hundreds of students at Rice University, The College of the Redwoods, and Texas A&M Univer-sity who patiently worked with us on preliminary versions of the text. It was they who found many of the errors that are corrected in this edition.
John Polking [email protected] Albert Boggess [email protected] David Arnold [email protected]
1
~C H A P T E R
1
~Introduction to
Differential Equations
W
ith the systematic study of differential equations, the calculus of functions of a single variable reaches a state of completion. Modeling by differential equa-tions greatly expands the list of possible applicaequa-tions. The list continues to grow as we discover more differential equation models in old and in new areas of application. The use of differential equations makes available to us the full power of the calculus.When explicit solutions to differential equations are available, they can be used to predict a variety of phenomena. Whether explicit solutions are available or not, we can usually compute useful and very accurate approximate numerical solutions. The use of modern computer technology makes possible the visualization of the results. Furthermore, we continue to discover ways to analyze solutions without knowing the solutions explicitly.
The subject of differential equations is solving problems and making predic-tions. In this book, we will exhibit many examples of this—in physics, chemistry, and biology, and also in such areas as personal finance and forensics. This is the process of mathematical modeling. If it were not true that differential equations were so useful, we would not be studying them, so we will spend a lot of time on the modeling process and with specific models. In the first section of this chapter we will present some examples of the use of differential equations.
The study of differential equations, and their application, uses the derivative and the integral, the concepts that make up the calculus. We will review these ideas starting in Sections 1.2 and 1.3.
1.1
Differential
Equation Models
To start our study of differential equations, we will give a number of examples. This list is meant to be indicative of the many applications of the topic. It is far from being exhaustive. In each case, our discussion will be brief. Most of the examples will be discussed later in the book in greater detail. This section should be considered as advertising for what will be done in the rest of the book.
The theme that you will see in the examples is that in every case we compute the rate of change of a variable in two different ways. First there is the mathematical way. In mathematics, the rate at which a quantity changes is the derivative of that quantity. This is the same for each example. The second way of computing the rate of change comes from the application itself and is different from one application to another. When these two ways of expressing the rate of change are equated, we get a differential equation, the subject we will be studying.
Mechanics
Isaac Newton was responsible for a large number of discoveries in physics and math-ematics, but perhaps the three most important are the following:
• The systematic development of the calculus. Newton’s achievement was the realization and utilization of the fact that integration and differentiation are operations inverse to each other.
F m
Figure 1. The force F results in the acceleration of the mass m according to equation (1.1).
• The discovery of the laws of mechanics. Principal among these was Newton’s second law, which says that force acting on a mass (see Figure 1) is equal to the rate of change of momentum with respect to time. Momentum is defined to be the product of mass and velocity, or mv. Thus the force is equal to the derivative of the momentum. If the mass is constant,
d
dtmv = m dv
dt = ma,
where a is the acceleration. Newton’s second law says that the rate of change of momentum is equal to the force F. Expressing the equality of these two ways of looking at the rate of change, we get the equation
F= ma, (1.1)
the standard expression for Newton’s second law.
m m
r
M M
Figure 2. The magnitude of the force attracting two masses is given by equation (1.2).
• The discovery of the universal law of gravitation. This law says that any body with mass M attracts any other body with mass m directly toward the mass M, with a magnitude proportional to the product of the two masses and inversely proportional to the square of the distance separating them. This means that there is a constant G, which is universal, such that the magnitude of the force is
G Mm
r2 , (1.2)
where r is the distance between the centers of mass of the two bodies. See Figure 2.
All of these discoveries were made in the period between 1665 and 1671. The discoveries were presented originally in Newton’s Philosophiae Naturalis Principia
Mathematica, better known as Principia Mathematica, published in 1687.
Newton’s development of the calculus is what makes the theory and use of differential equations possible. His laws of mechanics create a template for a model
1.1 Differential Equation Models 3
for motion in almost complete generality. It is necessary in each case to figure out what forces are acting on a body. His law of gravitation does just that in one very important case.
m
x
Figure 3. x is the height of the ball above the surface of the earth.
The simplest example is the motion of a ball thrown into the air near the surface of the earth. See Figure 3. If x measures the distance the ball is above the earth, then the velocity and acceleration of the ball are
v = d x dt and a= dv dt = d2x dt2.
Since the ball is assumed to move only a short distance in comparison to the radius of the earth, the force given by (1.2) may be assumed to be constant. Notice that m, the mass of the ball, occurs in (1.2). We can write the force as F = −mg, where
g = G M/r2 and r is the radius of the earth. The constant g is called the earth’s
acceleration due to gravity. The minus sign reflects the fact that the displacement x is measured positively above the surface of the earth, and the force of gravity tends to decrease x. Newton’s second law, (1.1), becomes
−mg = ma = mdv
dt = m d2x dt2.
The masses cancel, and we get the differential equation
d2x
dt2 = −g, (1.3)
which is our mathematical model for the motion of the ball.
The equation in (1.3) is called a differential equation because it involves an unknown function x(t) and at least one of its derivatives. In this case the highest derivative occurring is the second order, so this is called a differential equation of second order.
A more interesting example of the application of Newton’s ideas has to do with planetary motion. For this case, we will assume that the sun with mass M is fixed and put the origin of our coordinate system at the center of the sun. We will denote by x(t) the vector that gives the location of a planet relative to the sun. See Figure 4. The vector x(t) has three components. Its derivative is
v(t) = dx
dt,
which is the vector-valued velocity of the planet. For this example, Newton’s second
M
m
X
Figure 4. The vector x gives the location of the planet m relative to the sun M.
law and his law of gravitation become
md 2x dt2 = − G Mm |x|2 x |x|.
This system of three second-order differential equations is Newton’s model of planetary motion. Newton solved these and verified that the three laws observed by Kepler follow from his model.
Population models
Consider a population P(t) that is varying with time.1 A mathematician will say
that the rate at which the population is changing with respect to time is given by the derivative
d P dt .
On the other hand, a population biologist will say that the rate of change is roughly proportional to the population. This means that there is a constant r , called the reproductive rate, such that the rate of change is equal to r P. Putting together the ideas of the mathematician and the biologist, we get the equation
d P
dt = r P. (1.4)
This is an equation for the function P(t). It involves both P and its derivative, so it is a differential equation. It is not difficult to show by direct substitution into (1.4) that the exponential function
P(t) = P0er t,
where P0is a constant representing the initial population, is a solution. Thus,
assum-ing that the reproductive rate r is positive, our population will grow exponentially. If at this point you go back to the biologist he or she will undoubtedly say that the reproductive rate is not really a constant. While that assumption works for small populations, over the long term you have to take into account the fact that resources of food and space are limited. When you do, a better model for the the reproductive rate is the function r(1 − P/K ), and then the rate at which the population changes is better modeled by r(1 − P/K )P. Here both r and K are constants.
When we equate our two ideas about the rate at which the population changes, we get the equation
d P
dt = r(1 − P/K )P. (1.5)
This differential equation for the function P(t) is called the logistic equation. It is much harder to solve than (1.4), but it does a creditable job of predicting how single populations grow in isolated circumstances.
Pollution
Consider a lake that has a volume of V = 100 km3. It is fed by an input river, and
there is another river which is fed by the lake at a rate that keeps the volume of the lake constant. The flow of the input river varies with the season, and assuming that
t = 0 corresponds to January 1 of the first year of the study, the input rate is r(t) = 50 + 20 cos(2π(t − 1/4)).
Notice that we are measuring time in years. Thus the maximum flow into the lake occurs when t = 1/4, or at the beginning of April.
In addition, there is a factory on the lake that introduces a pollutant into the lake at the rate of 2 km3/year. Let x(t) denote the total amount of pollution in the lake
1For the time being, the population can be anything—humans, paramecia, butterflies, and so on. We will
1.1 Differential Equation Models 5
at time t. If we make the assumption that the pollutant is rapidly mixed throughout the lake, then we can show that x(t) satisfies the differential equation
d x dt = 2 −
52+ 20 cos(2π(t − 1/4)) x 100.
This equation can be solved and we can then answer questions about how dan-gerous the pollution problem really is. For example, if we know that a concentration of less than 2% is safe, will there there be a problem? The solution will tell us.
The assumption that the pollutant is rapidly mixed into the lake is not very re-alistic. We know that this does not happen, especially in this situation, where there is a flow of water through the lake. This assumption can be removed, but to do so, we need to allow the concentration of the pollutant to vary with position in the lake as well as with time. Thus the concentration is a function c(t, x, y, z), where
(x, y, z) represents a position in the three-dimensional lake. Instead of assuming
perfect mixing, we will assume that the pollutant diffuses through water at a certain rate. Once again we can construct a mathematical model. Again it will be a differ-ential equation, but now it will involve partial derivatives with respect to the spatial coordinates x, y, and z, as well as the time t.
Personal finance
How much does a person need to save during his or her work life in order to be sure of a retirement without money worries? How much is it necessary to save each year in order to accumulate these assets? Suppose one’s salary increases over time. What percent of one’s salary should be saved to reach one’s retirement goal?
All of these questions, and many more like them, can be modeled using dif-ferential equations. Then, assuming particular values for important parameters like return on investment and rate of increase of one’s salary, answers can be found.
Other examples
We have given four examples. We could have given a hundred more. We could talk about electrical circuits, the behavior of musical instruments, the shortest paths on a complicated-looking surface, finding a family of curves that are orthogonal to a given family, discovering how two coexisting species interact, and many others.
All of these examples use ordinary differential equations. The applications of partial differential equations go much farther. We can include electricity and mag-netism; quantum chromodynamics, which unifies electricity and magnetism with the weak and strong nuclear forces, the flow of heat, oscillations of many kinds, such as vibrating strings, the fair pricing of stock options, and many more.
The use of differential equations provides a way to reduce many areas of appli-cation to mathematical analysis. In this book, we will learn how to do the modeling and how to use the models after we make them.
EXERCISES
The phrase “y is proportional to x” implies that y is related to x via the equation y = kx, where k is a constant. In a similar manner, “y is proportional to the square of x” implies
y = kx2, “y is proportional to the product of x and z” implies y= kxz, and “y is inversely proportional to the cube of x”
im-plies y= k/x3. For example, when Newton proposed that the
force of attraction of one body on another is proportional to the
product of the masses and inversely proportional to the square of the distance between them, we can immediately write
F= G Mm r2 ,
where G is the constant of proportionality, usually known as the universal gravitational constant. In Exercises 1–11, use
these ideas to model each application with a differential equa-tion. All rates are assumed to be with respect to time.
1. The rate of growth of bacteria in a petri dish is propor-tional to the number of bacteria in the dish.
2. The rate of growth of a population of field mice is in-versely proportional to the square root of the population. 3. A certain area can sustain a maximum population of 100
ferrets. The rate of growth of a population of ferrets in this area is proportional to the product of the population and the difference between the actual population and the maximum sustainable population.
4. The rate of decay of a given radioactive substance is pro-portional to the amount of substance remaining.
5. The rate of decay of a certain substance is inversely pro-portional to the amount of substance remaining.
6. A potato that has been cooking for some time is removed from a heated oven. The room temperature of the kitchen is 65◦F. The rate at which the potato cools is proportional to the difference between the room temperature and the temperature of the potato.
7. A thermometer is placed in a glass of ice water and al-lowed to cool for an extended period of time. The ther-mometer is removed from the ice water and placed in a room having temperature 77◦F. The rate at which the thermometer warms is proportional to the difference in the room temperature and the temperature of the thermometer. 8. A particle moves along the x-axis, its position from the origin at time t given by x(t). A single force acts on the particle that is proportional to, but opposite the object’s displacement. Use Newton’s law to derive a differential equation for the object’s motion.
9. Use Newton’s law to develop the equation of motion for the particle in Exercise 8 if the force is proportional to, but opposite the square of the particle’s velocity.
10. Use Newton’s law to develop the equation of motion for the particle in Exercise 8 if the force is inversely propor-tional to, but opposite the square of the particle’s displace-ment from the origin.
11. The voltage drop across an inductor is proportional to the rate at which the current is changing with respect to time.
1.2
The Derivative
Before reading this section, ask yourself, “What is the derivative?” Several answers may come to mind, but remember your first answer.Chances are very good that your answer was one of the following five:
1. The rate of change of a function
2. The slope of the tangent line to the graph of a function 3. The best linear approximation of a function
4. The limit of difference quotients,
f(x0) = lim
x→x0
f(x) − f (x0)
x− x0 5. A table containing items such as we see in Table 1
All of these answers are correct. Each of them provides a different way of looking at the derivative. The best answer to the question is “all of the above.” Since we will be using all five ways of looking at the derivative, let’s spend a little time discussing each.
Table 1 A table of derivatives f(x) = f(x) = C 0 x 1 xn nxn−1 cos(x) − sin(x) sin(x) cos(x) ex ex ln(|x|) 1/x
The rate of change
In calculus, we learn that a function has an instantaneous rate of change, and this rate is equal to the derivative. For example, if we have a distance x(t) measured from a fixed point on a line, then the rate at which x changes with respect to time is the velocityv. We know that
v = x= d x
dt.
Similarly, the acceleration a is the rate of change of the velocity, so
a= v= dv dt =
d2x
1.2 The Derivative 7
These facts about linear motion are reflected in many other fields. For example, in economics, the law of supply and demand says that the price of a product is determined by the supply of that product and the demand for it. If we assume that the demand is constant, then the price P is a function of the supply S, or P =
P(S). The rate at which P changes with the supply is called the marginal price. In
mathematical terms, the marginal price is simply the derivative P = d P/dS. We can also talk about the rate of change of the mass of a radioactive material, of the size of population, of the charge on a capacitor, of the amount of money in a savings account or an investment account, or of many more quantities.2
We will see all of these examples and more in this book. The point is that when any quantity changes, the rate at which it changes is the derivative of that quantity. It is this fact that starts the modeling process and makes the study of differential equations so useful. For this reason we will refer to the statement that the derivative is the rate of change as the modeling definition of the derivative.
The slope of the tangent line
This provides a good way to visualize the derivative. Look at Figure 1. There you see the graph of a function f , and the tangent line to the graph of f at the point
(x0, f (x0)). The equation of the tangent line is
y= f (x0) + f(x0)(x − x0).
From this formula, it is easily seen that the slope of the tangent line is f(x0).
y f (x0) f'(x0)(xx0) (x 0, f (x0))
y f (x)
Figure 1. The derivative is the slope of the tangent line to the graph of the function.
Again looking at Figure 1, we can visualize the rate at which the function f is changing as x changes near the point x0. It is the same as the slope of the tangent
line.
We will refer to this characterization of the derivative as the geometric
defini-tion of the derivative.
2In all but one of the mentioned examples, the quantity changes with respect to time. Most of the
applications of ordinary differential equations involve rates of change with respect to time. For this reason, t is usually used as the independent variable. However, there are cases where things change depending on other parameters, as we will see. Where appropriate, we will use other letters to denote the independent variable. Sometimes we will do so just for practice.
The best linear approximation
Let
L(x) = f (x0) + f(x0)(x − x0). (2.1)
L is a linear (or affine) function of x. Taylor’s theorem says there is a remainder
function R(x), such that
f(x) = L(x) + R(x) and lim
x→x0
R(x) x− x0 = 0.
(2.2)
The limit in (2.2) means that R(x) gets small as x → x0. In fact, it gets enough
smaller than x− x0that the ratio goes to 0. It turns out that the function L defined in
(2.1) is the only linear function with this property. This is what we mean when we say that L is the best linear approximation to the nonlinear function f . You will also notice that the straight line in Figure 1 is the graph of L. In fact, Figure 1 provides a pictorial demonstration that L(x) is a good approximation for f (x) for x near x0.
The formula in (2.1) defines L(x) in terms of the derivative of f . In this sense, the derivative gives us the best linear approximation to the nonlinear function f near x = x0. [Actually (2.1) contains three important pieces of data, x0, f(x0), and
f(x0). We are perhaps stretching the point when we say that it is the derivative
alone that enables us to find a linear approximation to f , but it is clear that the derivative is the most important of these three.]
Since the linear approximation is an algebraic object, we will refer to this as the
algebraic definition of derivative.
The limit of difference quotients
Consider the difference quotient
m = f(x) − f (x0) x− x0
. (2.3)
This is equal to the slope of the line through the two points(x0, f (x0)) and (x, f (x))
as illustrated in Figure 2. We will refer to this line as a secant line. As x approaches
x0, the secant line approaches the tangent line shown in Figure 1. This is reflected
in the fact that
f(x0) = lim
x→x0
f(x) − f (x0)
x− x0
. (2.4)
Thus the slope of the tangent line, f(x0), is the limit of the slopes of secant lines.
The difference quotient in (2.3) is also the average rate of change of the function
f between x0and x. As the interval between x0and x is made smaller, these average
rates approach the instantaneous rate of change of f . Thus we see the connection with our modeling definition.
The definition of the derivative given in (2.4) will be called the limit quotient
definition. This is the definition that most mathematicians think of when asked to
define the derivative. However, as we will see, it is also very useful, even when attempting to find mathematical models.
1.2 The Derivative 9
y f(x0) m(xx0)
(x, f (x))
y f (x)
(x0, f (x0))
Figure 2. The secant line with slope m given by the difference quotient in (2.3).
The table of formulas
By memorizing a table of derivatives and a few formulas (especially the chain rule), we can learn the skill of differentiation. It isn’t hard to be confident that you can compute the derivative of any given function. This skill is important. However, it is clear that this formulaic definition of derivative is quite different from those given previously.
A complete understanding of the formulaic definition is important, but it does not provide any information about the other definitions we have examined. There-fore, it helps us neither to apply the derivative in modeling nature nor to understand its properties. For that reason, the formulaic definition is incomplete. This is not true of the other definitions. Starting with one of them, it is possible to construct a table that will give us the formulaic finesse we need. Admittedly that is a big task. That was what was done (or should have been done) in your first calculus course.
To sum up, we have examined five definitions of the derivative. Each of these emphasizes a different aspect or property of the derivative. All of them are impor-tant. We will see this as we progress through the study of differential equations. If your answer to the question at the beginning of the section was any of these five, your answer is correct. However, a complete understanding of the derivative requires the understanding of all five definitions.
Even if your answer was not on the list of five, it may be correct. The famous mathematician William Thurston once compiled a list of over 40 “definitions” of the derivative. Of course many of these appear only in more advanced parts of math-ematics, but the point is made that the derivative appears in many ways in mathemat-ics and in its applications. It is one of the most fundamental ideas in mathematmathemat-ics and in its application to science and technology.
EXERCISES
You might recall the following rules of differentiation from your calculus class. Let f and g be differentiable functions of
x. Then (cf )= cf ( f ± g)= f± g ( f g)= fg+ f g f g = fg− f g g2 .
Also, the chain rule is essential.
( f ◦ g)(x) = f(g(x))g(x)
Use these rules, plus the table of derivatives in Table 1, to find the derivative of each of the functions in Exercises 1–12.
1. f(x) = 3x − 5 2. f(x) = 5x2− 4x − 8
3. f(x) = 3 sin 5x 4. f(x) = cos 2πx 5. f(x) = e3x 6. f(x) = 5ex2 7. f(x) = ln |5x| 8. f(x) = ln(cos 2x) 9. f(x) = x ln x 10. f(x) = exsinπx
11. f(x) = x
2
ln x 12. f(x) =
x ln x
cos x
13. Suppose that f is differentiable at x0. Let L be the
“best linear approximation” defined by L(x) = f (x0) + f(x0)(x − x0). Given that R(x) = f (x) − L(x), show
that lim x→x0 R(x) x− x0 = 0.
For each of the functions given in Exercises 14–17, sketch the function f and its linearization L(x) = f (x0)+ f(x0)(x −x0)
at the given point x0on the same set of coordinate axes.
14. f(x) = ex, at x
0= 0
15. f(x) = cos x, at x0= π/4
16. f(x) =√x, at x0= 1
17. f(x) = ln(1 + x), at x0= 0
In order that R(x)/(x − x0) of equation (2.2) approach zero as x → x0, the numerator R(x) must approach zero at a faster rate
than does the denominator x− x0. For each of Exercises 18–
21, sketch the graph of y= x − x0and R(x) = f (x) − L(x)
on the same set of coordinate axes. Do both x− x0and R(x)
approach zero as x → x0? Which approaches zero at a faster
rate, R(x) or x − x0? 18. f(x) = x3/2, at x 0= 1 19. f(x) = sin 2x, at x = π/8 20. f(x) =√x+ 1, at x = 0 21. f(x) = xex−1, at x = 1
1.3
Integration
We can start once more by asking the question, “What is the integral?” This time our list of possible answers is not so long.1. The area under the graph of a function 2. The antiderivative
3. A table containing items such as we see in Table 1
Table 1 A table of integrals
f(x) = f(x) dx = f(x) = f(x) dx = 0 C cos(x) sin(x) + C 1 x+ C sin(x) − cos(x) + C x x 2 2 + C e x ex+ C xn x n+1 n+ 1+ C 1 x ln(|x|) + C
Let’s look at each of them briefly.
The area under the graph
The first answer emphasizes the definite integral. The definite integral b
a
f(x) dx (3.1) is interpreted as the area under the graph of the function f between x = a and
x= b. It represents the area of the shaded region in Figure 1.
This is the most fundamental definition of the integral. The integral was in-vented to solve the problem of finding the area of regions that are not simple rect-angles or circles. Despite its origin as a method to use in this one application, it has found numerous other applications.
1.3 Integration 11
a b
y f (x)
Figure 1. The area of the shaded region is the integral in (3.1).
The antiderivative
This answer emphasizes the indefinite integral. In fact, the phrase indefinite
inte-gral is a synonym for antiderivative. The definition is summed up in the following
equivalence. If the function g is continuous, then
f= g if and only if
g(x) dx = f (x) + C. (3.2) In (3.2), C refers to the arbitrary constant of integration. Thus the process of indefi-nite integration involves finding antiderivatives. Given a function g, we want to find a function f such that f= g.
The connection between the definite and the indefinite integral is found in the
fundamental theorem of calculus. This says that if f= g, then
b a
g(x) dx = f (b) − f (a).
The table of formulas
This formulaic approach to the integral has the same features and failures as the formulaic approach to the derivative. It leads to the handy skill of integration, but it does not lead to any deep understanding of the integral.
All of these approaches to the integral are important. It is very important to understand the first two and how they are connected by the fundamental theorem. However, for the elementary part of the study of ordinary differential equations, it is really the second and third approaches that are most important. In other words, it is important to be able to find antiderivatives.
Solution by integration
The solution of an important class of differential equations amounts to finding an-tiderivatives. A first-order differential equation can be written as
where the right-hand side is a function of the independent variable t and the un-known function y. Suppose that the right-hand side is a function only of t and does not depend on y. Then equation (3.3) becomes
y= f (t).
Comparing this with (3.2), we see immediately that the solution is
y(t) =
f(t) dt. (3.4) Let’s look at an example.
E x a m p l e 3 . 5 Solve the differential equation
y= cos t. (3.6)
According to (3.4), the solution is
y(t) =
cos(t) dt = sin t + C, (3.7) where C is an arbitrary constant. That’s pretty easy. It is just the process of integra-tion. It’s old hat to you by now. Solving the more general equation in (3.3) is not so easy, as we will see.
The constant of integration C makes (3.7) a one-parameter family of solutions of (3.6) defined on(−∞, ∞). This is an example of a general solution to a differ-ential equation. Some of these solutions are drawn in Figure 2. ●
0 5 0 2 4 t y 5 4 2
Figure 2. Several solutions to (3.6).
It is significant that the solution curves of equation (3.6) shown in Figure 2 are vertical translates of one another. That is to say, any solution curve can be obtained from any other by a vertical translation. This is always the case for solution curves of an equation of the form y = f (t). According to (3.2), if y(t) = F(t) is one solution to the equation, then all others are of the form y(t) = F(t) + C for some constant C. The graphs of such functions are vertical translates of the graph of
y(t) = F(t).
The constant of integration allows us to put an extra condition on a solution. This is illustrated in the next example.
E x a m p l e 3 . 8 Find the solution to y(t) = tet that satisfies y(0) = 2.
This is an example of an initial value problem. It requires finding the particular solution that satisfies the initial condition y(0) = 2. According to (3.2), the general solution to the differential equation is given by
y(t) =
tetdt. (3.9)
This integral can be evaluated using integration by parts. Since this method is so useful, we will briefly review it. In general, it says
u dv = uv −
v du, (3.10) where u andv are functions. If they are functions of t, then du = u(t) dt and
dv = v(t) dt. For the integral in equation (3.9), we let u(t) = t, and dv = v(t) dt = etdt. Then du = dt and v(t) = et, and equation (3.10) gives
tetdt = u dv = uv − v du = tet− etdt.
1.3 Integration 13
After evaluating the last integral, we see that
y(t) = tet− et+ C = et(t − 1) + C. (3.11) This one-parameter family of solutions is the general solution to the equation
y= tet. Each member of the family exists on the interval(−∞, ∞). The condition
y(0) = 2 can be used to determine the constant C.
2= y(0) = e0(0 − 1) + C = −1 + C Therefore, C = 3 and the solution of the initial value problem is
y(t) = et(t − 1) + 3. (3.12) It is important to note that the solution curve defined by equation (3.12) is the member of the family of solution curves defined by (3.11) that passes through the
point(0, 2), as shown in Figure 3. ●
0 1 t x 4 2 0 2 4 1
Figure 3. The solution of the initial value problem in Example 3.8 passes through the point(0, 2).
The use of initial conditions to determine a particular solution can be affected from the beginning of the solution process by using definite integrals instead of indefinite integrals. For example, in Example 3.8, we can proceed using the funda-mental theorem of calculus:
y(t) − y(0) = t 0 y(u) du. Hence, y(t) = y(0) + t 0 ueudu = 2 + ueu− eut 0 = et(t − 1) + 3.
We will not always use the letter t to designate the independent variable. Any letter will do, as long as we are consistent. The same is true of the dependent variable.
E x a m p l e 3 . 1 3 Find the solution to the initial value problem
y= 1
x with y(1) = 3.
Here we are using x as the independent variable. By integration, we find that
y(x) = ln(|x|) + C.
We are asked for the solution that satisfies the initial condition 3= y(1) = ln(1) + C = C. Thus, C= 3.
A solution to a differential equation has to have a derivative at every point. Therefore, it is also continuous. However, the function y(x) = ln(|x|) + 3 is not defined for x = 0. To get a continuous function from y, we have to limit its domain to(0, ∞) or (−∞, 0). Since we want a solution that is defined at x = 1, we must choose(0, ∞). Thus, our solution is
The motion of a ball
In Section 1.1, we talked about the application of Newton’s laws to the motion of a ball near the surface of the earth. See Figure 4. The model we derived [in equation (1.3)] was
d2x
dt2 = −g,
where x(t) is the height of the ball above the surface of the earth and g is the acceleration due to gravity. If we measure x in feet and time in seconds, g = 32 ft/s2.
E x a m p l e 3 . 1 4 Suppose a ball is thrown into the air with initial velocityv0= 20 ft/s. Assuming the
ball is thrown from a height of x0 = 6 feet, how long does it take for the ball to hit
the ground?
We can solve this equation using the methods of this section. First we
intro-m
x
Figure 4. x is the height of the ball above the surface of the earth.
duce the velocity to reduce the second-order equation to a system of two first-order equations:
d x
dt = v, and dv
dt = −g. (3.15)
Solving the second equation by integration, we get
v(t) = −gt + C1.
Evaluating this at t = 0, we see that the constant of integration is C1 = v(0) =
v0 = 20, the initial velocity. Hence, the velocity is v(t) = −gt + v0 = −32t + 20,
and the first equation in (3.15) becomes
d x
dt = −gt + v0 = −32t + 20.
Solving by integration, we get
x(t) = −1
2gt
2+ v
0t+ C2= −16t2+ 20t + C2.
Once more we evaluate this at t = 0 to show that C2 = x(0) = x0 = 6, the initial
elevation of the ball. Hence, our final solution is
x(t) = −1
2gt
2+ v
0t+ x0= −16t2+ 20t + 6. (3.16)
The ball hits the ground when x(t) = 0. By solving the quadratic equation, and using the positive solution, we see that the ball hits the ground after 1.5 seconds. ●
EXERCISES
In Exercises 1–8, find the general solution of the given differ-ential equation. In each case, sketch at least six members of the family of solution curves.
1. y= 2t + 3 2. y= 3t2+ 2t + 3
3. y= sin 2t + 2 cos 3t 4. y= 2 sin 3t − cos 5t
5. y= t
1+ t2 6. y
= 3t
1+ 2t2
1.3 Integration 15
In Exercises 1–8 above, each equation has the form y =
f(t, y), the goal being to find a solution y = y(t); that is,
find y as a function of t. Of course, you are free to choose different letters, both for the dependent and independent vari-ables. For example, in the differential equation s = xex, it
is understood that s = ds/dx, and the goal is to find a so-lution s as a function of x; that is, s = s(x). In Exercises 9–14, find the general solution of the given differential equa-tion. In each case, sketch at least six members of the family of solution curves. 9. s= e−2ωsinω 10. y= x sin 3x 11. x= s2e−s 12. s= e−ucos u 13. r= 1 u(1 − u) 14. y= 3 x(4 − x)
Note: Exercises 13 and 14 require a partial fraction
decom-position. If you have forgotten this technique, you can find extensive explanation in Section 5.3 of this text. In particular, see Example 3.6 in that section.
In Exercises 15–24, find the solution of each initial value prob-lem. In each case, sketch the solution.
15. y= 4t − 6, y(0) = 1 16. y= x2+ 4, y(0) = −2 17. x= te−t2, x(0) = 1 18. r= t/(1 + t2), r(0) = 1 19. s= r2cos 2r , s(0) = 1 20. P= e−tcos 4t, P(0) = 1 21. x=√4− t, x(0) = 1 22. u= 1/(x − 5), u(0) = −1 23. y= t+ 1 t(t + 4), y(−1) = 0 24. v= r 2 r+ 1, v(0) = 0
In Exercises 25–28, assume that the motion of a ball takes place in the absence of friction. That is, the only force act-ing on the ball is the force due to gravity.
25. A ball is thrown into the air from an initial height of 3 m with an initial velocity of 50 m/s. What is the position and velocity of the ball after 3 s?
26. A ball is dropped from rest from a height of 200 m. What is the velocity and position of the ball 3 seconds later? 27. A ball is thrown into the air from an initial height of 6 m
with an initial velocity of 120 m/s. What will be the max-imum height of the ball and at what time will this event occur?
28. A ball is propelled downward from an initial height of 1000 m with an initial speed of 25 m/s. Calculate the time that the ball hits the ground.
2
~C H A P T E R
2
~
First-Order Equations
I
n this chapter, we will study first-order equations. We will begin in Section 2.1 by making some definitions and presenting an overview of what we will cover in this chapter. We will then alternate between methods of finding exact solutions and some applications that can be studied using those methods. For each application, we will carefully derive the mathematical models and explore the existence of exact solutions. We will end by showing how qualitative methods can be used to derive useful information about the solutions.2.1
Differential
Equations and
Solutions
In this section, we will give an overview of what we want to learn in this chapter. We will visit each topic briefly to give a flavor of what will follow in succeeding sections.
Ordinary differential equations
An ordinary differential equation is an equation involving an unknown function of a single variable together with one or more of its derivatives. For example, the equation
d y
dt = y − t (1.1)
is an ordinary differential equation. Here y= y(t) is the unknown function and t is the independent variable.
2.1 Differential Equations and Solutions 17
Some other examples of ordinary differential equations are
y= y2− t t y= y
y+ 4y = e−3t y= cos(ty) (1.2)
yy+ t2y= cos(t) y= y2.
The order of a differential equation is the order of the highest derivative that occurs in the equation. Thus the equation in (1.1) is a first-order equation since it involves only the first derivative of the unknown function. All of the equations listed in the first two rows of (1.2) are first order. Those in the third row are second order because they involve the second derivative of y.
The equation
∂2w
∂t2 = c 2∂2w
∂x2 (1.3)
is not an ordinary differential equation, since the unknown functionw is a function of the two independent variables t and x. Because it involves partial derivatives of an unknown function of more than one independent variable, equation (1.3) is called a partial differential equation. For the time being we are interested only in ordinary differential equations.
Normal form
Any first order equation can be put into the form
φ(t, y, y) = 0, (1.4)
whereφ is a function of three variables. For example, the equation in (1.1) can be written as
y− y − t = 0.
This equation has the form in (1.4) withφ(t, y, z) = z− y−t. Similarly, the general equation of order n can be written as
φ(t, y, y, . . . , y(n)) = 0, (1.5)
whereφ is a function of n + 1 variables. Notice that all of the equations in (1.2) can be put into this form.
The general forms in (1.4) and (1.5) are too general to deal with in many in-stances. Frequently we will find it useful to solve for the highest derivative. We will give the result a name.
DEFINITION 1.6
A first-order differential equation of the formy= f (t, y)
is said to be in normal form. Similarly, an equation of order n having the form
y(n)= f (t, y, y, . . . , y(n−1))
is said to be in normal form.