Differential equations with applications

The main changes in this new edition are as follows: the number of problems in the first part of the book has been more than doubled; there are two new chapters, on Fourier Series and on

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The sole aim of science is the honor of the human mind,

and from this point of view

a question about numbers

is as important

as a question about the system of the world

-C G J Jacobi

DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND

HISTORICAL NOTES

Second Edition

George F Simmons

Professor of Mathematics Colorado College

with a new chapter on numerical methods by

JohnS Robertson Department of Mathematical Sciences United States Military Academy

McGraw-Hill, Inc

New York St Louis San Francisco Auckland Bogota Caracas Hamburg Lisbon London Madrid Mexico Milan Montreal New Delhi Paris San Juan Sao Paulo Singapore Sydney

Tokyo Toronto

This book was set in Times Roman

The editors were Richard Wallis and John M Morriss ;

the production supervisor was Louise Karam

The cover was designed by Carla B auer

Project supervision was done by The Universities Press

R R Donnelley & Sons Company was printer and binder

DIFFERENTIAL EQUATIONS WITH APPLICATIONS

AND HISTORICAL NOTES

Copyright© 1991 , 1972 by McGraw-Hill , Inc All rights reserved Printed in the United States of America Except as permitted under the United States Copyright Act of 1976,

no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, w:thout the prior written permission

of the publisher

2 3 4 5 6 7 8 9 0 DOC DOC 9 5 4 3 2 1

ISBN 0-07-057540-1

Library of Congress Cataloging-in-Publication Data

Simmons, George Finlay, (date)

Differential equations with applications and historical notes I

George F Simmons.-2nd ed

ABOUT THE AUTHOR

George Simmons has academic degrees from the California Institute of Technology, the University of Chicago, and Yale University He taught

at several colleges and universities before joining the faculty of Colorado College in 1962, where he is Professor of Mathematics He is also the author of Introduction to Topology and Modern Analysis (McGraw-Hill , 1963), Precalculus Mathematics in a Nutshell (Janson Publications, 1981), and Calculus with Analytic Geometry (McGraw-Hill, 1985)

When not working or talking or eating or drinking or cooking, Professor Simmons is likely to be traveling (Western and Southern Europe, Turkey, Israel , Egypt, Russia, China, Southeast Asia) , trout fishing (Rocky Mountain states), playing pocket billiards, or reading (literature, history, biography and autobiography, science , and enough thrillers to achieve enjoyment without guilt)

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FOR HOPE AND NANCY

my wife and daughter who still make it all worthwhile

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1

2

3

CONTENTS

Preface to the Second Edition

Preface to the First Edition

Suggestions for the Instructor

The Nature of Differential Equations Separable Equations

1 Introduction

2 General Remarks on Solutions

3 Families of Curves Orthogonal Trajectories

4 Growth, Decay, Chemical Reactions, and Mixing

5 Falling Bodies and Other Motion Problems

6 The Brachistochrone Fermat and the Bernoullis

First Order Equations

12 The Hanging Chain Pursuit Curves

13 Simple Electric Circuits

Second Order Linear Equations

14 Introduction

15 The General Solution of the Homogeneous Equation

16 The Use of a Known Solution to Find Another

17 The Homogeneous Equation with Constant Coefficients

18 The Method of Undetermined Coefficients

XV xvii xxi

xii CONTENTS

20 Vibrations in Mechanical and Electrical Systems 106

21 Newton's Law of Gravitation and the Motion of the Planets 115

22 Higher Order Linear Equations Coupled Harmonic

23 Operator Methods for Finding Particular Solutions 128

5 Power Series Solutions and Special Functions 165

Appendix B Hermite Polynomials and Quantum

Appendix D Chebyshev Polynomials and the Minimax

7 Partial Differential Equations and Boundary

40 Eigenvalues, Eigenfunctions, and the Vibrating String 302

42 The Dirichlet Problem for a Circle Poisson's Integral 317

Appendix A The Existence of Eigenvalues and

8 Some Special Functions of Mathematical Physics 335

Appendix A Legendre Polynomials and Potential Theory 365 Appendix B Bessel Functions and the Vibrating

Appendix C Additional Properties of Bessel Functions 377

5 1 Derivatives and Integrals o f Laplace Transforms 394

53 More about Convolutions The Unit Step and Impulse

56 Homogeneous Linear Systems with Constant Coefficients 427

57 Nonlinear Systems Volterra's Prey-Predator Equations 434

58 Autonomous Systems The Phase Plane and Its Phenomena 440

60 Critical Points and Stability for Linear Systems 455

64 Periodic Solutions The Poincare-Bendixson Theorem 486

65 Introduction Some Typical Problems of the Subject 502

xiV CONTENTS

Appendix B Hamilton's Principle and Its Implications 526

PREFACE TO THE SECOND EDITION

"As correct as a second edition"-so goes the idiom I certainly hope so , and I also hope that anyone who detects an error will do me the kindness

of letting me know, so that repairs can be made As Confucius said , "A man who makes a mistake and doesn't correct it is making two mistakes "

I now understand why second editions of textbooks are always longer than first editions: as with governments and their budgets, there is always strong pressure from lobbyists to put things in, but rarely pressure

to take things out

The main changes in this new edition are as follows: the number of problems in the first part of the book has been more than doubled; there are two new chapters, on Fourier Series and on Partial Differential Equations; sections on higher order linear equations and operator methods have been added to Chapter 3; and further material on convolutions and engineering applications has been added to the chapter

on Laplace Transforms

Altogether, many different one-semester courses can be built on various parts of this book by using the schematic outline of the chapters given on page xxi There is even enough material here for a two­semester course, if the appendices are taken into account

Finally, an entirely new chapter on Numerical Methods (Chapter 14) has been written especially for this edition by Major John S Robertson of the United States Military Academy Major Robertson's expertise in these matters is much greater than my own, and I am sure that many users of this new edition will appreciate his contribution , as

I do

McGraw-Hill and I would like to thank the following reviewers for their many helpful comments and suggestions: D R Arterburn , New

XVi PREFACE TO THE SECOND EDITION

Mexico Tech; Edward Beckenstein , St John's University; Harold Carda, South Dakota School of Mines and Technology; Wenxiong Chen, University of Arizona; Jerald P Dauer, University of Tennessee; Lester B Fuller, Rochester Institute of Technology; Juan Gatica, University of Iowa; Richard H Herman , The Pennsylvania State Univer­sity; Roger H Marty, Cleveland State University; Jean-Pierre Meyer, The Johns Hopkins University; Krzysztof Ostaszewski, University of Louisville ; James L Rovnyak , University of Virginia; Alan Sharples, New Mexico Tech; Bernard Shiffman , The Johns Hopkins University; and Calvin H Wilcox, University of Utah

George F Simmons

PREFACE TO THE FIRST EDITION

To be worthy of serious attention , a new textbook on an old subject should embody a definite and reasonable point of view which is not represented by books already in print Such a point of view inevitably reflects the experience, taste, and biases of the author, and should therefore be clearly stated at the beginning so that those who disagree can seek nourishment elsewhere The structure and contents of this book express my personal opinions in a variety of ways, as follows

The place of dift'erential equations in mathematics Analysis has been the dominant branch of mathematics for 300 years, and differential equations are the heart of analysis This subject is the natural goal of elementary calculus and the most important part of mathematics for understanding the physical sciences Also, in the deeper questions it generates, it is the source of most of the ideas and theories which constitute higher analysis Power series, Fourier series, the gamma function and other special functions, integral equations, existence theorems, the need for rigorous justifications of many analytic processes-all these themes arise in our work in their most natural context And at a later stage they provide the principal motivation behind complex analysis, the theory of Fourier series and more general orthogonal expansions, Lebesgue integration , metric spaces and Hilbert spaces, and a host of other beautiful topics in modern mathematics I would argue, for example, that one of the main ideas of complex analysis is the liberation of power series from the confining environment of the real number system; and this motive is most clearly felt by those who have tried to use real power series to solve differential equations In botany, it is obvious that no one can fully appreciate the blossoms of flowering plants without a reasonable understanding of the roots, stems, and leaves which nourish and support them The same principle is true in mathematics, but is often neglected or forgotten

XViii PREFACE TO THE FI RST EDITION

Fads are as common in mathematics as in any other human activity, and it is always difficult to separate the enduring from the ephemeral in the achievements of one's own time At present there is a strong current

of abstraction flowing through our graduate schools of mathematics This current has scoured away many of the individual features of the landscape and replaced them with the smooth , rounded boulders of general theories When taken in moderation , these general theories are both useful and satisfying; but one unfortunate effect of their pre­dominance is that if a student doesn't learn a little while he is an undergraduate about such colorful and worthwhile topics as the wave equation, Gauss's hypergeometric function , the gamma function , and the basic problems of the calculus of variations-among many others-then

he is unlikely to do so later The natural place for an informal acquaintance with such ideas is a leisurely introductory course on differential equations Some of our current books on this subject remind

me of a sightseeing bus whose driver is so obsessed with speeding along

to meet a schedule that his passengers have little or no opportunity to enjoy the scenery Let us be late occasionally, and take greater pleasure

in the journey

Applications It is a truism that nothing is permanent except change; and the primary purpose of differential equations is to serve as a tool for the study of change in the physical world A general book on the subject without a reasonable account of its scientific applications would therefore

be as futile and pointless as a treatise on eggs that did not mention their reproductive purpose This book is constructed so that each chapter except the last has at least one major "payoff"-and often several-in the form of a classic scientific problem which the methods of that chapter render accessible These applications include

The brachistochrone problem

The Einstein formulaE = mc2

Newton's law of gravitation

The wave equation for the vibrating string

The harmonic oscillator in quantum mechanics

Potential theory

The wave equation for the vibrating membrane

The prey-predator equations

Nonlinear mechanics

Hamilton's principle

Abel's mechanical problem

I consider the mathematical treatment of these problems to be among the chief glories of Western civilization , and I hope the reader will agree

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PREFACE TO THE FI RST EDITION XiX

The problem of mathematical rigor On the heights of pure mathematics, any argument that purports to be a proof must be capable of withstanding the severest criticisms of skeptical experts This is one of the rules of the game, and if you wish to play you must abide by the rules But this is not the only game in town

There are some parts of mathematics-perhaps number theory and abstract algebra-in which high standards of rigorous proof may be appropriate at all levels But in elementary differential equations a narrow insistence on doctrinaire exactitude tends to squeeze the juice out

of the subject, so that only the dry husk remains My main purpose in this book is to help the student grasp the nature and significance of differential equations; and to this end , I much prefer being occasionally imprecise but understandable to being completely accurate but incom­prehensible I am not at all interested in building a logically impeccable mathematical structure, in which definitions, theorems, and rigorous proofs are welded together into a formidable barrier which the reader is challenged to penetrate

In spite of these disclaimers, I do attempt a fairly rigorous discussion from time to time, notably in Chapter 13 and Appendices A in Chapters 5, 6 and 7, and B in Chapter 1 1 I am not saying that the rest of this book is nonrigorous, but only that it leans toward the activist school

of mathematics, whose primary aim is to develop methods for solving scientific problems-in contrast to the contemplative school , which analyzes and organizes the ideas and tools generated by the activists Some will think that a mathematical argument either is a proof or is not a proof In the context of elementary analysis I disagree , and believe instead that the proper role of a proof is to carry reasonable conviction to one's intended audience It seems to me that mathematical rigor is like clothing: in its style it ought to suit the occasion , and it diminishes comfort and restricts freedom of movement if it is either too loose or too tight

History and biography There is an old Armenian saying, "He who lacks a sense of the past is condemned to live in the narrow darkness of his own generation." Mathematics without history is mathematics stripped of its greatness: for, like the other arts and mathematics is one of the supreme arts of civilization-it derives its grandeur from the fact of being

a human creation

In an age increasingly dominated by mass culture and bureaucratic impersonality, I take great pleasure in knowing that the vital ideas of mathematics were not printed out by a computer or voted through by a committee, but instead were created by the solitary labor and individual genius of a few remarkable men The many biographical notes in this book reflect my desire to convey something of the achievements and personal qualities of these astonishing human beings Most of the longer

notes are placed in the appendices, but each is linked directly to a specific contribution discussed in the text These notes have as their subjects all but a few of the greatest mathematicians of the past three centuries: Fermat, Newton, the Bernoullis, Euler, Lagrange, Laplace, Fourier, Gauss, Abel, Poisson , Dirichlet, Hamilton, Liouville , Chebyshev, Herm­ite, Riemann, Minkowski, and Poincare As T S Eliot wrote in one of his essays, "Someone said: 'The dead writers are remote from us because

we know so much more than they did ' Precisely, and they are that which

we know "

History and biography are very complex, and I am painfully aware that scarcely anything in my notes is actually quite as simple as it may appear I must also apologize for the many excessively brief allusions to mathematical ideas most student readers have not yet encountered But with the aid of a good library, sufficiently interested students should be able to unravel most of them for themselves At the very least, such efforts may help to impart a feeling for the immense diversity of classical mathematics-an aspect of the subject that is almost invisible in the average undergraduate curriculum

George F Simmons

SUGGESTIONS FOR THE

INSTRUCTOR

The following diagram gives the logical dependence of the chapters and suggests a variety of ways this book can be used, depending on the purposes of the course, the tastes of the instructor, and the backgrounds and needs of the students

I The Nature or Difl'ercntial Equations Separable Equations

13 Existence and Uniqueness Theorems

14 Numerical Methods

The scientist does not study nature because it is useful; he studies it because

he delights in it, and he delights in it because it is beautiful If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living Of course I do not here speak of that beauty that strikes the senses, the beauty of qualities and appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmonious order of the parts, and which a pure intelligence can grasp

-Henri Poincare

As a mathematical discipline travels far from its empirical source, or still more, if it is a second or third generation only indirectly inspired by ideas coming from "reality," it is beset with very grave dangers It becomes more and more purely aestheticizing, more and more purely l'art pour I'art This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline

is under the influence of men with an exceptionally well-developed taste But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration

-John von Neumann

Just as deduction should be supplemented by intuition, so the impulse to progressive generalization must be tempered and balanced by respect and love for colorful detail The individual problem should not be degraded to the rank of special illustration of lofty general theories In fact, general theories emerge from consideration of the specific, and they are meaning­ less if they do not serve to clarify and order the more particularized substance below The interplay between generality and individuality, deduction and construction, logic and imagination-this is the profound essence of live mathematics Any one or another of these aspects of mathematics can be at the center of a given achievement In a far-reaching development all of them will be involved Generally speaking, such a development will start from the "concrete" ground, then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy; after this flight comes the crucial test of landing and reaching specific goals in the newly surveyed low plains of individual

"reality." In brief, the flight into abstract generality must start from and return to the concrete and specific

DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES

1 INTRODUCfiON

CHAPTER

1

THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS

An equation involving one dependent variable and its derivatives with respect to one or more independent variables is called a differential equation Many of the general laws of nature-in physics, chemistry, biolog y , and astronomy-find their most natural expression in the

lan guag e of differential equations Applications also abound in mathe­

matics itself, especially in geometry, and in engineering, economics, and many other fields of applied science

It is easy to understand the reason behind this broad utility of differential equations The reader will recall that if y = f(x) is a given

function, then its derivative dy/dx can be interpreted as the rate of change of y with respect to x In any natural process, the variables involved and their rates of change are connected with one another by means of the basic scientific principles that govern the process When this connection is e pressed in mathematical symbols, the result is often a differential equation

The followin g e xample may illuminate these remarks According to Newton's second law of motion, the acceleration a of a body of mass m is proportional to the total force F acting on it, with 1/m as the constant of

As further examples of differential equations, we list the following:

dy

dt = -ky ; d2y

1 g can be considered constant on the surface of the earth in most applications, and is

approximately 32 feet per second per second (or 980 centimeters per second per second)

THE NATU RE OF DIFFE RENTIAL EQUATI ONS SEPA RABLE EQ U ATIONS 3

constants An ordinary differential equation is one in which there is only one independent variable , so that all the derivatives occurring in it are ordinary derivatives Each of these equations is ordinary The order of a differential equation is the order of the highest derivative present Equations (4) and (6) are first order equations, and the others are second order Equations (8) and (9) are classical , and are called Legendre's equation and Bessel's equation, respectively Each has a vast literature and a history reaching back hundreds of years We shall study all of these equations in detail later

A partial differential equation is one involving more than one independent variable, so that the derivatives occurring in it are partial derivatives For example, if w = f(x,y, z,t) is a function of time and the three rectangular coordinates of a point in space, then the following are partial differential equations of the second order:

o2w o2w o2w

- + - + - = 0· ox2 oy2 oz2 '

2 ( o2w o2w o2w ) aw

a ox2 + oy2 + oz2 = at ;

2 ( o2w o2w o2w ) o2w

a ox2 + oy2 + oz2 = ot2 •

These equations are also classical , and are called Laplace's equation, the

heat equation, and the wave equation, respectively Each is profoundly significant in theoretical physics, and their study has stimulated the development of many important mathematical ideas In general , partial differential equations arise in the physics of continuous media-in problems involving electric fields, fluid dynamics, diffusion , and wave motion Their theory is very different from that of ordinary differential equations, and is much more difficult in almost every respect For some time to come, we shall confine our attention exclusively to ordinary differential equations 2

2 The English biologist J B S Haldane (1892-1964) has a good remark about the one-dimensional special case of the heat equation: "In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict new facts of the same kind The catch in this criterion lies in the word 'simplest ' It is really

an aesthetic canon such as we find implicit in our criticism of poetry or painting The layman finds such a law as

much less simple than 'it oozes,' of which it is the mathematical statement The physicist reverses this judgment, and his statement is certainly the more fruitful of the two, so far as prediction is concerned It is, however, a statement about something very unfamiliar to the plain man , namely, the rate of change of a rate of change "

2 GENERAL REMARKS ON SOLUTIONS

The general ordinary differential equation of the nth order is

F X, y,

or, using the prime notation for derivatives,

F (x,y,y ' ,y", , y< n> ) = 0

Any adequate theoretical discussion of this equation would have to be based on a careful study of explicitly assumed properties of the function

F However, undue emphasis on the fine points of theory often tends to obscure what is really going on We will therefore try to avoid being overly fussy about such matters-at least for the present

It is normally a simple task to verify that a given function y = y(x)

is a solution of an equation like (1) All that is necessary is to compute the derivatives of y (x) and to show that y(x) and these derivatives, when substituted in the equation, reduce it to an identity in x In this way we see that

and are both solutions of the second order equation

3 In calculus the notation In x is often used for the so-called natural logarithm, that is, the function log x In more advanced courses , however, this function is almost always denoted

by the symbol log x

THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 5

illustrate the fact that a solution of a differential equation usually contains one or more arbitrary constants, equal in number to the order of the equation

In most cases procedures of this kind are easy to apply to a suspected solution of a given differential equation The problem of starting with a differential equation and finding a solution is naturally much more difficult In due course we shall develop systematic methods for solving equations like (2) and (5) For the present, however, we limit ourselves to a few remarks on some of the general aspects of solutions The simplest of all differential equations is

and I sin x

-X dx

cannot be expressed in terms of a finite number of elementary functions 4

If we recall, however, that

I f(x) dx

is merely a symbol for a function (any function) with derivative f(x),

then we can almost always give (7) a valid meaning by writing it in the form

y = r f(t) dt + c

The crux of the matter is that this definite integral is a function of the upper limit x (the t under the integral sign is only a dummy variable)

4Any reader who is curious about the reasons for this should consult D G Mead,

"Integration," Am Math Monthly, vol 68, pp 152- 156 ( 1961 ) For additional details, see

G H Hardy, The Integration of Functions of a Single Variable, Cambridge University Press, London, 1916; or J F Ritt, Integration in Finite Terms, Columbia University Press, New York , 1948

which always exists when the integrand is continuous over the range of integration, and that its derivative is f(x) 5

The so-called separable equations, or equations with separable variables, are at the same level of simplicity as (6) These are differential equations that can be written in the form

dy

dx = f(x)g(y),

where the right side is a product of two functions each of which depends

on only one of the variables In such a case we can separate the variables

by writing

dy g(y) = f(x) dx,

and then solve the original equation by integrating:

I g t� ) = I r<x) dx + c

These are simple differential equations to deal with in the sense that the problem of solving them can be reduced to the problem of integration, even though the indicated integrations can be difficult or impossible to carry out explicitly

The general first order equation is the special case of (1) which corresponds to taking n = 1 :

We normally expect that an equation like this will have a solution, and that this solution-like (7) and (8)-will contain one arbitrary constant However,

THE NATU RE OF DI FFERENTIAL EQUATIONS SEPARABLE EQUATI ONS 7

the existence and nature of solutions of differential equations We cannot enter here into a full discussion of these questions, but it may clarify matters if we give an intuitive description of a few of the basic facts For the sake of simplicity, let us assume that (9) can be solved for

in this new direction , and use the number

to determine yet another direction at P2• If we continue this process, we obtain a broken line with points scattered along it like beads; and if we now imagine that these successive points move closer to one another and become more numerous, then the broken line approaches a smooth curve through the initial point P0• This curve is a solution y = y(x) of equation

(10); for at each point (x,y) on it, the slope is given by f(x,y} and this is precisely the condition required by the differential equation If we start with a different initial point, then in general we obtain a different curve (or solution) Thus the solutions of (10) form a family of curves, called

integral curves 6 Furthermore, it appears to be a reasonable guess that through each point in R there passes just one integral curve of (10) This discussion is intended only to lend plausibility to the following precise statement

Theorem A (Picard's theorem.) If f(x,y) and af/ ay are continuous functions on a closed rectangle R, then through each point (x0,y0) in the interior of R there passes a unique integral curve of the equation dy I dx =

f(x,y)

If we consider a fixed value of x0 in this theorem, then the integral curve that passes through (x0,y0) is fully determined by the choice of y0•

In this way we see that the integral curves of (10) constitute what is called

a one-parameter family of curves The equation of this family can be written in the form

where different choices of the parameter c yield different curves in the family The integral curve that passes through (x0,y0) corresponds to the value of c for which y0 = y(x0,c) If we denote this number by c0, then

(1 1) is called the general solution of (10), and

THE NATU RE OF D I FFERENTIAL EQUATIONS SEPARABLE EQUATIONS 9

The essential feature of the general solution (11) is that the constant c in

it can be chosen so that an integral curve passes through any given point

of the rectangle under consideration

Picard's theorem is proved in Chapter 13 This proof is quite complicated, and is probably best postponed until the reader has had considerable experience with the more straightforward parts of the subject The theorem itself can be strengthened in various directions by weakening its hypotheses; it can also be generalized to refer to nth order equations solvable for the nth order derivative Detailed descriptions of these results would be out of place in the present context, and we content ourselves for the time being with this informal discussion of the main ideas In the rest of this chapter we explore some of the ways in which differential equations arise in scientific applications

(e) y = c1 sin 2x + c2 cos 2x y" + 4y = 0;

(f) y = c1e2x + c2e-2x y" - 4y = 0;

(g) y = c1 sinh 2x + c2 cosh 2x y" - 4y = 0;

(n) y + sin y = x (y cos y - sin y + x)y ' = y ;

(o) x + y = tan- • y 1 + y2 + y2y ' = 0

2 Find the general solution of each of the following differential equations: (a) y ' = e3x - x ; (j) xY + y5 = 0;

(c) y ' = xex2; (1) y ' = 2xy ;

(d) y ' = sin- • x ; (m) y ' sin y = x2;

(e) (1 + x)y ' = x ; (n) y ' sin x = 1 ;

(f) (1 + x2)y ' = x ; (o) y ' + y tan x = 0;

that satisfies the given initial condition:

5 Show that y = ex2 g e _,2 dt is a solution of y ' = 2xy + 1

6 For the differential equation (2) , namely,

y" - 5y , + 6y = 0, carry out the detailed calculations needed to verify the assertions in the text that

(a) y = e2x and y = e3x are both solutions; and

(b) y = c1e2x + c2e3x is a solution for every choice of the constants c1 and c2•

Remark : In studying a book like this, a student should never slide past assertions of this kind-involving such phrases as "we see" or "as we can readily verify"-without personally checking their validity The mere fact that something is in print does not mean it is necessarily true Cultivate skepticism

as a healthy state of mind, as you would physical fitness; accept nothing on the authority of this writer or any other until you have understood it fully for yourself

7 In the spirit of Problem 6, verify that (4) is a solution of the differential equation (5) for every value of the constant c

8 For what values of the constant m will y = e""' be a solution of the differential equation

THE NATU RE OF D I FFERENTIAL EQUATIONS SEPARABLE EQUATIONS 11

Conversely, as we might expect, the curves of any one-parameter family are integral curves of some first order differential equation If the family is

X

and since c is already absent, there is no need to eliminate it and

dy

The parameter c is still present,

combining (6) and (7) This yields so it is necessary to eliminate it by

\ \ I

(8)

X

THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 13

As an interesting application of these procedures, we consider the problem of finding orthogonal trajectories To explain what this problem

is, we observe that the family of circles represented by (4) and the family

y = mx of straight lines through the origin (the dotted lines in Fig 2) have the following property: each curve in either family is orthogonal

(i.e , perpendicular) to every curve in the other family Whenever two families of curves are related in this way, each is said to be a family of

orthogonal trajectories of the other Orthogonal trajectories are of interest in the geometry of plane curves, and also in certain parts of applied mathematics For instance, if an electric current is flowing in a plane sheet of conducting material , then the lines of equal potential are the orthogonal trajectories of the lines of current flow

In the example of the circles centered on the origin , it is geometrically obvious that the orthogonal trajectories are the straight lines through the origin, and conversely In order to cope with more complicated situations, however, we need an analytic method for finding orthogonal trajectories Suppose that

dy

is the differential equation of the family of solid curves in Fig 4 These curves are characterized by the fact that at any point (x,y) on any one of them the slope is given by f(x,y) The dotted orthogonal trajectory through the same point, being orthogonal to the first curve , has as its slope the negative reciprocal of the first slope Thus, along any

orthogonal trajectory, we have dy/dx = -1/f(x,y) or

dx

Our method of finding the orthogonal trajectories of a given family of curves is therefore as follows: first, find the differential equation of t he family; next, replace dy/dx by -dx/dy to obtain the differential equation

of the orthogonal trajectories; and finally, solve this new differential equation

If we apply this method to the family of circles (4) with differential equation (5), we get

which on direct integration yields

log y = log x + log c

or

y = ex

as the equation of the orthogonal trajectories

It is often convenient to express the given family of curves in terms

of polar coordinates In this case we use the fact that if 1/J is the angle from the polar radius to the tangent, then tan 1/J = r d(} I dr (Fig 5) By the above discussion , we replace this expression in the differential equation

of the given family by its negative reciprocal , -dr/r d(}, to obtain the differential equation of the orthogonal trajectories As an illustration of the value of this technique, we find the orthogonal trajectories of the family of circles (6) If we use rectangular coordinates, it follows from (8) that the differential equation of the orthogonal trajectories is

THE NATU RE OF DIFFERENTIAL EQUATIONS SEPA RABLE EQUATIONS 15

"

- ,."" ---- ,.""" r ,.""'' d� ­

and after integration this becomes

log r = log (sin fJ) + log 2c,

so that

is the equation of the orthogonal trajectories It will be noted that (15) is the equation of the family of all circles tangent to the x-axis at the origin (see the dotted curves in Fig 3)

In Chapter 2 we develop a number of more elaborate procedures for solving first order equations Since our present attention is directed more at applications than formal techniques, all the problems given in this chapter are solvable by the method of separation of variables illustrated above

2 What are the orthogonal trajectories of the family of curves (a) y = cx4; (b)

y = ex" where n is any positive integer? In each case, sketch both families of curves What is the effect on the orthogonal trajectories of increasing the exponent n ?

3 Show that the method for finding orthogonal trajectories i n polar coordinates can be expressed as follows If dr/dO = F(r, 0) is the differential equation of the given family of curves, then dr/dO = - r2/ F(r, 0) is the differential equation of the orthogonal trajectories Apply this method to the family of circles r = 2c sin 0

4 Use polar coordinates to find the orthogonal trajectories of the family of parabolas r = c/(1 - cos 0), c > 0 Sketch both families of curves

5 Sketch the family y2 = 4c(x + c) of all parabolas with axis the x-axis and focus at the origin, and find the differential equation of the family Show that this differential equation is unaltered when dy /dx is replaced by -dx/dy

What conclusion can be drawn from this fact?

6 Find the curves that satisfy each of the following geometric conditions: (a) The part of the tangent cut off by the axes is bisected by the point of tangency

(b) The projection on the x-axis of the part of the normal between (x,y) and the x-axis has length 1

(c) The projection on the x-axis of the part of the tangent between (x,y) and the x-axis has length 1

(d) The part of the tangent between (x,y) and the x-axis i s bisected by the y-axis