Differential Equations with Applications and Historical Notes [1 ed.]

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HEMATICS

m

:ons and Historical Note:

t^P

mm sal

Ti

MAR

DIFFERENTIAL EQUATIONS

INTERNATIONAL SERIES William Ted Martin and

E.

IN

PURE AND APPLIED MATHEMATICS

H. Spanier, Consulting Editors

ahlfors Complex Analysis buck Advanced Calculus busacker and saaty Finite Graphs and Networks cheney Introduction to Approximation Theory Chester

Techniques

in

Partial Differential Equations

coddington and levinson Theory of Ordinary Differential Equations cohn Conformal Mapping on Riemann Surfaces conte and de boar Elementary Numerical Analysis dennemeyer Introduction to Partial Differential Equations and Boundary Value Problems dettman Mathematical Methods in Physics and Engineering Partial Differential Equations

epstein

Elements of Ordinary Differential Equations The Theory of Functions of Real Variables Greenspan Introduction to Partial Differential Equations

golomb and shanks graves

Elementary Theory of Numbers

griffin

hamming

Numerical Methods for Scientists and Engineers

hildebrand Introduction to Numerical Analysis householder Principles of Numerical Analysis householder The Numerical Treatment of a Single Nonlinear Equation kalman, falb, and arbib Topics in Mathematical Systems Theory lass

lass

Elements of Pure and Applied Mathematics Vector and Tensor Analysis

leighton

Introduction to the Theory of Differential Equations

Complex Variables and

lepage

the Laplace

Transform for Engineers

mccarty Topology: An Introduction with Applications to Topological Groups monk Introduction to Set Theory moore Elements of Linear Algebra and Matrix Theory moursund and duris Elementary Theory and Application of Numerical Analysis nef

Linear Algebra

nehari

Conformal Mapping newell Vector Analysis pipes and harvill Applied Mathematics for Engineers and ralston A First Course in Numerical Analysis ritger and ROSE Differential Equations with Applications RITT

Physicists

Fourier Series

Rossi. r

Logic for Mathematicians

RUDIN Principles of Mathematical Analysis saaty and bram Nonlinear Mathematics SAGAN Introduction to the Calculus of Variations SIMMONS

Differential Equations with Applications

si\i\io\s

Introduction to Topology

and Historical Soles and Modern Analysis

SNEDDON Elements of Partial Differential Equations SNEDDON Fourier Transforms STRUBU Nonlinear Differential Equations weinsiock Calculus of Variations weiss /l

Algebraic

MANIAN

Number Theory

Distribution Theory

and Transform Analysis

The sole aim

oj science

is

honor of the human mind, and from this point of view a question about numbers

the

is

as important

as a question about the system of the world.

— C. G.

GEORGE

F.

SIMMONS

J.

Jacobi

Professor of Mathematics, Colorado College

DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES

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PREFACE

To be worthy of serious attention, a new textbook on an embody a definite and reasonable point of view which

old subject should 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 the is

differential equations in mathematics.

dominant branch of mathematics for 300

the heart of analysis. This subject

calculus

is

years,

and

Analysis has been

differential equations

the natural goal of elementary

and the most important part of mathematics

physical sciences. Also, in the deeper questions

it

for understanding the

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,

need for rigorous justifications 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 integral equations, existence theorems, the

of many analytic processes— all these themes arise in our

PREFACE

VIII

one of the main ideas of complex analysis environment of the is the liberation of power series from the confining clearly felt by those who have real number system; and this motive is most

would argue,

for example, that

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.

Fads are as and it is always

common

mathematics as

in

difficult to

in

any other human

activity,

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

of their predominance is that if a 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 satisfying; but

one unfortunate

student doesn't learn a

little

effect

while he

is

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

ductive purpose. This last

has

at least

classic scientific

book

on eggs that did not mention

their repro-

constructed so that each chapter except the

is

one major "payoff '—and often several— in the form of a problem which the methods of that chapter render accessible.

These applications include the brachistochrone problem; the Einstein formula

E = mc 2

;

Newton's law of gravitation; the wave equation for the vibrating string; the harmonic oscillator in quantum mechanics; potential theory;

PREFACE the

IX

wave equation

for the vibrating

membrane;

the prey-predator equations;

nonlinear mechanics;

Hamilton's principle; and Abel's mechanical problem. I

consider the mathematical treatment of these problems to be

the chief glories of

Western

civilization,

and

I

hope the reader

among

will agree.

rigor. On the heights of pure matheargument 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

The problem of mathematical

matics, any

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 not the only

the student grasp the nature

and

to this end,

I

much

and

significance of differential equations;

prefer being occasionally imprecise but under-

standable to being completely accurate but incomprehensible. at all interested in building a logically

I

am

not

impeccable mathematical structure,

and rigorous proofs are welded together 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 11 and Appendices A in Chapters 4 and 5, and B in Chapter 8. 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. in

which

definitions, theorems,

into a formidable barrier

an old Armenian saying, "He who lacks 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

History and biography. a sense of the past

is

There

condemned

is

to live in the

PREFACE

x arts of civilization— it derives

grandeur from the fact of being a

its

human

creation.

dominated by mass culture and bureaucratic 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 In an age increasingly

impersonality,

I

take great pleasure in

The many biographical notes

genius of a few remarkable men. reflect

my

desire to

in this

book

convey something of the achievements and personal

qualities of these astonishing

human

placed in the appendices, but each

is

beings.

Most of

the longer notes are

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, Gauss, Abel, Hamilton, Liouville, Chebyshev, Hermite, Riemann, and Poincare. As T. S. Eliot wrote in :

"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." one of

his essays,

:

History and biography are very complex, and scarcely anything in

my

notes

is

I

am

painfully aware that

actually quite as simple as

it

may

appear.

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, a sufficiently interested student should be able to unravel most of them for himself. At the very least, such efforts may help I

to impart a feeling for the

an aspect of the subject that

immense is

diversity of classical

almost invisible

in the

mathematics—

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 tastes oi"

the instructor

and the backgrounds and purposes of

1.

The Nature

his students.

of

Differential

Equations

2. First

Order

Equations

9. 3.

The Calculus of Variations

Second Order Linear Equations

4. Oscillation

5.

Theory and Boundary Value Problems

Power

Series

7.

Solutions and Special

Systems of First

Order

Equations

10.

Laplace Transforms

Functions

6.

Some

Special

Functions of

Mathematical

8.

Nonlinear Equations

Physics

1

1.

The Existence and Uniqueness of Solutions

and 9 are relatively straightforward, and much of the two of these is often given in calculus courses. Chapter 3 is the cornerstone of the structure. Chapters 4, 5, and 6 deal with the more advanced theory of second order linear equations and with series solutions, and 10 provides a supplementary approach. Chapters 7 and 8 are aimed at second order nonlinear equations, and 8 and 1 1 are the closest in spirit to the mathematical interests of our own times. A word of warning: the material covered in Sections 28 to 35 is rather formidable in places, and the Chapters

1,

material in the

2,

first

instructor giving a short course should perhaps consider reserving these sections for his

most ambitious students.

The in

scientist

it,

and he

does not study nature because delights in

not be worth knowing, living.

Of course

qualities

I

it

because

and

is

it

useful ; he studies

is

it

beautiful. If nature were not beautiful,

nature were not worth knowing,

if

because he delights

it

it

would

would not be worth

life

do not here speak of that beauty that strikes the senses, the beauty of

and appearances; not

nothing to do with science;

I

from

that I undervalue such beauty, Jar

mean

that

if,

but

it

has

profounder beauty which comes from the har-

monious order of the parts, and which a pure intelligence can grasp. Henri Poincare

As a mathematical

discipline travels far

from

empirical source, or

its

still

a second or third generation only indirectly inspired by ideas coming it

is

beset with very grave dangers.

more and more purely Fart pour correlated subjects, which

still

It

becomes more and more purely

Tart. This

need not be bad,

under the influence of men with an exceptionally well-developed

danger that the subject so far from

its

discipline will

along the

will develop

line

source, will separate into a multitude

of

from

mathematical subject

is in

its

is

surrounded by

is

a grave

least resistance, that the stream,

of insignificant branches, and

become a disorganized mass of details and complexities.

at a great distance

it

if the discipline is

But there

taste.

if

"reality,"

aestheticizing,

if the field is

have closer empirical connections, or

more,

from

empirical source, or after

that the

In other words,

much "abstract"

inbreeding, a

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

The individual problem should not be degraded

to the

love for colorful detail.

rank of special

illustration

of lofty

general theories. In fact, general theories emerge from consideration of the specific, and they are meaningless if they do not serve to clarify and order the

substance below. construction, logic

Any one

more particularized

The interplay between generality and individuality

and imagination— this

is

the

profound essence of

,

deduction and

live

mathematics.

or another of these aspects of mathematics can be at the center of a given achieve-

ment. 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

after this flight

of

comes the crucial

thin air

test

where navigation and observation are easy;

of landing

and reaching

specific goals in the

newly

surveyed low plains of individual "reality." In

brief, the flight into abstract generality

must

specific.

start

from and

return to the concrete

and

— Richard Courant

.

.

CONTENTS

Preface

vii

Suggestions for the Instructor

xi

THE NATURE OF DIFFERENTIAL EQUATIONS

1

1

Introduction

1

2.

General remarks on solutions

3

3.

Families of curves. Orthogonal trajectories

4.

Growth, decay, and chemical reactions

and other

problems

5.

Falling bodies

6.

The brachistochrone. Fermat and

FIRST

rate

ORDER EQUATIONS

the Bernoullis

8

14 19

25

35

7.

Homogeneous equations

8.

Exact equations

38

9.

Integrating factors

42

35

10.

Linear equations

47

1 1

Reduction of order

49

12.

The hanging

52

13.

Simple

chain. Pursuit curves

electric circuits

Appendix

A

.

Numerical methods

58

65 XV

5 1

..

72

SECOND ORDER LINEAR EQUATIONS

72

14.

Introduction

1 5.

The

general solution of the

16.

The

use of a

1 7.

The homogeneous equation with constant coefficients

18.

The method of undetermined coefficients

87

1 9.

The method of variation of parameters

90

20.

Vibrations in mechanical systems

21

Newton's law of gravitation and the motion of the planets

known

76

homogeneous equation

81

solution to find another

83

93 100

Appendix A.

Euler

1

Appendix B.

Newton

1

AND BOUNDARY VALUE PROBLEMS

07 1

OSCILLATION THEORY

115

22.

Qualitative properties of solutions

1

23.

The Sturm comparison theorem

I2l

24.

Eigenvalues, eigenfunctions, and the vibrating string

Appendix A.

A

124 133

Regular Sturm-Liouville problems

POWER SERIES SOLUTIONS SPECIAL FUNCTIONS

1

AND 140 140

review of power series

25.

Introduction.

26.

Series solutions of first order equations

147

27.

Second order linear equations. Ordinary points

151

28.

Regular singular points

159

29.

Regular singular points (continued)

167

30.

Gauss's hypergeometric equation

174

31

The point

180

at infinity

Appendix A.

Two convergence proofs

183

Appendix B.

Hermite polynomials and quantum mechanics

1

Appendix C.

Gauss

196

Appendix D.

Chebyshev polynomials and the minimax property

204

Appendix E.

Riemann 's equation

2

87

1

SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS

219

32.

Legendre polynomials

2 19

33.

Properties of Legendre polynomials

226

34.

Bessel functions.

35.

Properties of Bessel functions

242

Appendix A.

Legendre polynomials and potential theory

249

Appendix

Bessel functions

B.

Appendix C.

The gamma function

and

the vibrating

232

membrane

Additional properties of Bessel functions

255 261

.

SYSTEMS OF FIRST ORDER EQUATIONS

8

265

36.

General remarks on systems

265

37.

Linear systems

268

38.

Homogeneous

linear systems with constant coefficients

276

39.

Nonlinear systems. Volterra's prey-predator equations

284

NONLINEAR EQUATIONS

290

40.

Autonomous

41.

Types of eritical points.

42.

Critical points

43.

Stability

44.

Simple

45.

Nonlinear mechanics. Conservative systems

46.

Periodic solutions.

systems.

and

The phase plane and

its

phenomena

stability for linear

systems

by Liapunov's direct method

critical

points of nonlinear systems

The Poincare-Bendixson theorem

323 332 338

Appendix A.

Poincare

346

Appendix

Proof of Lienard's theorem

349

B.

Some

353

problems of the subject

353

47.

Introduction.

48.

Euler's differential equation for an extremal

356

49.

11

305

316

THE CALCULUS OF VARIATIONS

10

290 296

Stability

typical

Isoperimetric problems

366

Appendix A.

Lagrange

376

Appendix B.

Hamilton 's principle and

its

implications

LAPLACE TRANSFORMS

377

388

50.

Introduction

388

51

A

392

52.

Applications to differential equations

397

53.

Derivatives and integrals of Laplace transforms

402

54.

Convolutions and Abel's mechanical problem

407

Appendix A.

Laplace

413

Appendix

Abel

415

few remarks on the theory

B.

THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 55.

The method of

56.

Picard's theorem

57.

Systems.

successive approximations

The second order

418 418

422 linear equation

433

Answers

436

Index

457

1

THE NATURE OF DIFFERENTIAL EQUATIONS

V

INTRODUCTION

An

equation involving one dependent variable and

its

derivatives with

more independent variables is called a differential equation. general laws of nature— in physics, chemistry, biology, and

respect to one or

Many

of the

astronomy— find

most natural expression in the language of differential abound in mathematics itself, especially in engineering, economics, and many other fields of applied

their

equations. Applications also

geometry, and

in

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

In any natural process, the variables involved

and

their rates of

x.

change are

connected with one another by means of the basic scientific principles that govern the process. When this connection is expressed in mathematical symbols, the result is often a differential equation.

The following example 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 proportionality, so that a

= F/m

or

ma

(i)

CHAPTER

2

THE NATURE OF DIFFERENTIAL EQUATIONS

1.

Suppose, for instance, that a body of mass m falls freely under the influence where g is the of gravity alone. In this case the only force acting on it is mg, acceleration due to gravity. fixed height, then

1

If

y

2

acceleration

its

d y/dt

is

2 ,

to the body from some becomes

down

the distance

is

and

(1)

y = mg m -^j dt

or 2

d y

If

we

alter the situation

air exerts a resisting force pro-

by assuming that

portional to the velocity, then the total force acting on the k(dy/dt\

and

(1)

2

dt (2)

is

mg -

becomes d y —— = mmg9J ~— m^y

Equations

body

and

2

— k—k ,

dv

P)

dt

are the differential equations that express the essential

(3)

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

attributes of the physical processes

dy -ky;

(4)

-b

(5)

dt 2

d y dt

2

=

e~

(6)

IX

d

(1

-

2

dv

y

d^y_ d l+ ~ 2x 2x-^

x 2 )-j±2 dx

dx

2

d y 2

dx

The dependent variable

is

dy

+

x

tt.

dx

+

x

(

p{p

+

- p)y =

variable in each of these equations

either

t

or

x.

The

ordinary differential equation

is

letters k,

one

in

l

these equations

is

ordinary.

0;

(8)

0.

(9)

is y,

and the independent

m, and p represent constants.

which there

variable, so that all the derivatives occurring in

Each of

=

\)y

it

The order of

is

An

only one independent

are ordinary derivatives. a differential equation

g can be considered constant on the surface of the earth in most applications, and mately 32 feet per second per second (or 980 centimeters per second per second).

is

is

approxi-

SECTION

2

GENERAL REMARKS ON SOLUTIONS

3

the order of the highest derivative present. Equations (4)

order equations, and

and

the others are second order. Equations

(8)

(6)

are

and

(9)

first

are

and are called Legendres equation and BesseVs equation, respectively. Each has a vast literature and a history reaching back hundreds of classical,

years.

We

shall study all of these

A partial differential equation

equations

in detail later.

one involving more than one independent variable, so that the derivatives occurring in it are partial derivatives. For example, if vv = f(x,y,z,t) is a function of time and the three rectangular is

coordinates of a point in space, then the following are partial differential

equations of the second order: d

2

w

+

~d^ a

2

fd 2 w

+

\J^ a

2

2

w + [d^2 fd

d

2

w

Jy I d

2

w

Jf d

2

w

1y

+ + +

d

2

w

5? d

2

=

w\

;

=

~d?) d

2

w\

dw ~di'

=

~d?)

d

2

w

~W

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 develop-

ment of many important mathematical

ideas. In general, partial differential

equations arise in the physics of continuous electric fields, fluid

media— in problems

very different from that of ordinary differential equations, and difficult in

involving

dynamics, diffusion, and wave motion. Their theory is

is

much more

almost every respect. For some time to come, we shall confine our

attention exclusively to ordinary differential equations.

2.

GENERAL REMARKS ON SOLUTIONS

The general ordinary

differential

/

or,

equation of the nth order

dy d 2 y

d"y\

is

...

using the prime notation for derivatives, F(x,y,y',y",...,y in) )

=

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.

CHAPTER

4

THE NATURE OF DIFFERENTIAL EQUATIONS

1.

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 It is

and

derivatives of y(x)

show

to

stituted in the equation, reduce

y

=

e

and these

that y(x)

an identity

to

it

2x

and

y

in x. In this

=

e

when subway we see that

derivatives,

3x

are both solutions of the second order equation

- 5/ +

y"

and,

more

6y

=

0;

(2)

generally, that

y

=

Cl e

2x

+

c2 e

3x (3)

also a solution for every choice of the constants c {

is

and

c2

.

Solutions of

differential equations often arise in the form of functions defined implicitly,

and sometimes explicitly in

it is

difficult

or impossible to express the dependent variable

terms of the independent variable. For instance,

xy is

=

+

log y

c

(4)

a solution of

t = t—Lax

for every value of the constant (4)

and rearranging the

result.

'

(5)

xy

1

c, as we can readily These examples also

verify

by differentiating

illustrate the fact that a

solution of a differential equation usually contains one or constants, equal in In

number

more

arbitrary

to the order of the equation.

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

equation and finding a solution is naturally much more difficult. In due course we shall develop systematic methods for solving equations

differential

like (2) and (5). For the present, however, we on some of the general aspects of solutions.

The

simplest of

all differential

equations

limit ourselves to a few

is

£=/(*), and we solve

it

some

(6)

by writing

y= In

remarks

\f(x)dx

cases the indefinite integral in

(7)

+

c.

(7)

can be worked out by the methods

:

SECTION

GENERAL REMARKS ON SOLUTIONS

2.

of calculus. In other cases this integral. It

is

known,

it

5

may be difficult or impossible to find a formula for

for instance, that

x2

e

dx

and

clx

cannot be expressed in terms of a we recall, however, that

finite

number

of elementary functions.

1

If

/»

f(x)dx is

merely a symbol for a function (any function) with derivative/(x), then we

can almost always give

(7)

a valid meaning by writing

)

The crux limit

x

exists

that

of the matter

(the

its

f(t)dt

The general

is

first

=

f{x).

+

is

is

only a

in the

form

c.

(8)

that this definite integral

the integrand

derivative

to taking n

=

under the integral sign

t

when

is

;

it

is

dummy

a function of the upper variable)

which always

continuous over the range of integration, and

2

order equation

is

the special case of (1) which corresponds

1

**g)-a We

normally expect that an equation

this solution will

(9)

like this will

have a solution, and that

contain one arbitrary constant. However,

has no real-valued solutions at

all,

and

has only the single solution y = (which contains no arbitrary constants). Situations of this kind raise difficult theoretical questions about the existence

and nature of solutions of differential equations. 1

Any

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," J. 2

F. Ritt, "Integration in Finite

This statement

is

We cannot enter here into a

Cambridge University

Terms," Columbia University Press,

one form of the fundamental theorem of calculus.

Press,

New

London, 1916; or

York, 1948.

CHAPTER

6

THE NATURE OF DIFFERENTIAL EQUATIONS

1.

discussion of these questions, but it may clarify matters intuitive description of a few of the basic facts.

For the sake of

simplicity, let us

assume that

(9)

we

if

full

give an

can be solved for dy/dx:

d

r=f(x,y\

(10)

dx

We

also

tangle

R

assume that f(x,y) is a continuous function throughout some recxy plane. The geometric meaning of a solution of (10) can best

in the

be understood as follows (Fig.

If

1).

P =

(x ,y

)

a point in R, then the

is

number f(x

dx/Po determines a direction at

P

Now

.

let

,yo)

P = l

(xi,.Vi)

be a point near

P

in

and use

this direction,

dx

new

to determine a

P

x

in this

new

direction at

direction,

P

i

.

Next,

let

P2 =

(x 2 ,y 2 ) De a point near

and use the number

dxJ P2

P2

we 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 to determine yet another direction at

.

If

we continue

obtain a broken line with points scattered along

the initial point

P

.

each point (x,y) on

this process,

beads;

like

is

required by the differential equation. If we start with a different

then in general

we obtain a

different curve (or solution).

Thus

of (10) form a family of curves, called integral curves. 1

one integral curve of

(10).

This discussion

is

initial point,

the solutions

Furthermore,

appears to be a reasonable guess that through each point in just

if

a solution y = y(x) of equation (10); for at the slope is given by f(x,y)— precisely the condition

This curve it,

it

R

it

there passes

intended only to lend

plausibility to the following precise statement.

THEOREM A. (PICARD'S THEOREM.) If f{x,y) and df/dy arc continuous functions on a closed rectangle R, then through each point (x y ) in the .

interior of

R

there passes a unique integral curve of the equation dy

dx

=

f(x,y). 1

Solutions of a differential equation are sometimes called integrals of the equation because the

problem of finding them

is

more or

less

an extension of the ordinary problem of integration.

SECTION

2.

GENERAL REMARKS ON SOLUTIONS

FIGURE

If

we consider a

fixed value of

that passes through (x ,y

way we

)

is

x

theorem, then the integral curve

in this

fully

1

determined by the choice of y

see that the integral curves of (10) constitute

what

is

.

In this

called a one-

parameter family of curves. The equation of this family can be written

in

the form

y

where

The for

different choices of the

y(x,c),

parameter

(11

c yield different curves in the family.

integral curve that passes

which y

=

v(x

,c).

If

the general solution of (10),

through (x ,y ) corresponds to the value of c we denote this number by c then (11) is called ,

and y

is

=

=

y(x,c

)

called the particular solution that satisfies the initial condition

y

when

x

=

x

(

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 1 1. This proof is quite complicated, probably best postponed until the reader has had considerable experience with the more straightforward parts of the subject. The theorem

and

is

;

CHAPTER

8 itself it

1.

:

THE NATURE OF DIFFERENTIAL EQUATIONS

can be strengthened in various directions by weakening its hypotheses; refer to nth order equations solvable for the

can also be generalized to

would be out of 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 nth order derivative. Detailed descriptions of these results place in the present context,

applications.

PROBLEMS 1

.

Verify that the following functions (explicit or implicit) are solutions of the corresponding differential equations:

d.

= x2 + c 1 y — ex 2 2x + c y = e kx = ce y

e.

y

f.

y

a.

b. c.

g.

y

h.

y

i.

j.

y=

y

= = = = =

v

x2

c 1 sin cl e

2x

\

+

2x

+

Cj sinh

2x

v

+

2x

c2

cosh 2x

-1

2y log y

•

2

;

1.

;

_y

2.

\

;

2

-

= x 2 - ex 2 y - c + c/x m.y = c^ /x n. y + sin = x o. x + y = tan" k.

2x

c 2 cos

c 2 e~

xy x tan x sin

2x;

= 2y; 2x yy' = e = y ky; y" + Ay = y" — Ay = y" — Ay = 0: 2 2 xy' + y = y y/T~- x y 2 2 xy' = y + x + y * /= L y 2 x + y 2 2xyy' = x + y 2 4 2 y + X y = x (y') 2 2 y = y /( xy - x ); - sin y + x)>'' = (y cos 2 2 1 + + y y y' = 0. xy'

j;

!

y

y;

Find the general solution of each of the following differential equations 3x x2 a. y = e x; c. y = xe \

b. xy' 3.

d. y'

1;

=

sin

-1 x.

For each of the following differential equations, find the particular solution that satisfies the given initial condition: a.

b. c.

3.

=

= y = y' = y'

xe x y = 3 when x = 1 2 sin x cos x, y = 1 when x log x, y = when x = e. ,

FAMILIES OF CURVES.

=

0;

ORTHOGONAL TRAJECTORIES

We have seen that the general solution of a first order differential equation normally contains one arbitrary constant, called a parameter. When this parameter is assigned various values, we obtain a one-parameter family of

SECTION curves.

3.

FAMILIES OF CURVES

Each of these curves

a particular solution, or integral curve, of

is

the given differential equation,

9

and

of them together constitute

all

its

general

solution.

Conversely, as

we might some

are integral curves of

expect, the curves of first

f(x,y,c)

then

its

differential

any one-parameter family

order differential equation.

=

If

the family

0,

is

(1)

equation can be found by the following

steps. First,

x to get a relation of the form

differentiate (1) implicitly with respect to

(2)

Next, eliminate the parameter c from

(1)

and

(2) to

obtain

(3)

as the desired differential equation.

x is

the equation of the family of

2

+

For example, v

2

=

all circles

c

2

(4)

with centers at the origin (Fig.

FIGURE 2

2).

CHAPTER

10

On

THE NATURE OF DIFFERENTIAL EQUATIONS

1.

differentiation with respect to

x

this

becomes

ax and

since c

is

already absent, there

no need

is

to eliminate

it

and

dy (5)

is

the differential equation of the given family of circles. Similarly,

x2 is

the equation of the family of

(Fig.

3).

When we

+

=

2

y

all circles

2cx

(6)

tangent to the y axis at the origin we obtain

differentiate this with respect to x,

2x

+

2y

x

+

y

— = 2c ax

or

dy

Tx =

c

(7)

-

FIGURE 3

SECTION

3.

FAMILIES OF CURVES

The parameter c is still (6) and (7). This yields

11

present, so

dy

it is

_

neeessary to eliminate

—

2

y

dx

x2 (8)

(6).

interesting application of these procedures,

of finding orthogonal trajectories.

To

through the origin (the dotted

property: each curve in either family

every curve in the other family. in this

way, each

is

is

we consider

the

problem

explain what this problem

observe that the family of circles represented by straight lines

by combining

2 at

as the differential equation of the family

As an

it

(4)

and the family y

lines in Fig. 2)

orthogonal

Whenever two

(i.e.,

we

is,

= mx

of

have the following perpendicular) to

families of curves are related

said to be a family of orthogonal trajectories of the other.

geometry of plane curves, and 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 Orthogonal

trajectories are of interest in the

also in certain parts of applied mathematics.

dy

dx is

= f(x,y)

the differential equation of the family of solid curves in Fig.

slope

(9)

4.

These curves

= — l/f(x,y)

FIGURE 4

CHAPTER

12

THE NATURE OF DIFFERENTIAL EQUATIONS

1.

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

reciprocal of the

dy/dx

first.

= - l/f(x,y)

=f(x,y).

dy

Our method

(10)

of finding the orthogonal trajectories of a given family of curves

therefore as follows:

next, replace dy/dx

first,

we apply

this

find the differential equation of the family;

by —dx/dy to obtain the

orthogonal trajectories; and, If

slope the negative

or

~ is

its

Thus, along any orthogonal trajectory, we have

method

differential equation of the

finally, solve this

to the family of circles

new (4),

differential equation.

we

get

or

dy

y_

=

dx

(id

x

as the differential equation of the orthogonal trajectories.

We

can

now

separate the variables in (11) to obtain

dy

dx

y

x

9

which on direct integration yields

logy

=

+

logx

logc

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 \p is the angle from the radius to the tangent, then tan \jj = rdO/dr (Fig. 5). By the above discussion,

we its

replace this expression in the differential equation of the given family by negative reciprocal, -dr/rd6, 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

orthogonal trajectories

follows from

(8)

that the differential equation of the

is

2xy

dy

dx

x

2

—

v

SECTION

3.

FAMILIES OF CURVES

13

FIGURE 5

Unfortunately, the variables in (12) cannot be separated, so without additional techniques for solving differential equations this direction.

However,

if

we

we can go no

further in

use polar coordinates, the equation of our

family can be written as

2c cos

From

this

we

0.

(13)

find that

dr

— 2c sin

0,

(14)

d6

and

after eliminating c

from

(13)

and

(14)

we

arrive at

rdO

cos 6

dr

sin 9

as the differential equation of the given family. Accordingly,

rd6 _ dr is

sinfl

cos 6

the differential equation of the orthogonal trajectories. In this case the

variables can be separated, yielding

dr r

cos 6 dO sin

;

CHAPTER

14

and

after integration this

)

THE NATURE OF DIFFERENTIAL EQUATIONS

1.

becomes

log

=

r

+

log (sin 6)

log 2c,

so that

=

r

is

2c sinO

(15)

the equation of the orthogonal trajectories.

the equation of the family of the dotted curves in Fig. In Chapter 2

all circles

It

will

be noted that (15)

is

tangent to the x axis at the origin (see

3).

we develop a number

of

more elaborate procedures

for

order equations. Since our present attention is directed more at applications than formal techniques, all the problems given in this chapter solving

first

are solvable by the

method of separation of

variables illustrated above.

PROBLEMS 1.

Sketch each of the following families of curves, find the orthogonal trajectories,

2.

= c\ = ex 2

a.

xy

b.

y

;

Sketch the family y

and focus

Show by 3.

and add them

2

r

d.

y

=

at the origin,

to the sketch:

= =

ce

4c{x

and

+

c(\

cosfl);

x

+

.

c)

of

all

parabolas with axis the x axis

find the differential equation for the family.

that this differential equation

— dx/dy. What

is

unaltered

when dy/dx

is

replaced

conclusion can be drawn from this fact?

Find the curves that a.

c.

satisfy

each of the following geometric conditions:

the part of the tangent cut off by the axes

is

bisected by the point of

tangency; b.

the projection

and the x c.

the projection

and the x d. the

4.

on the x

axis of the part of the

axis has length

on the x

normal between

axis of the part of the tangent

axis has length

\

between (w

v)

1

equals the angle from the radius to the tangent: from the radius to the tangent is constant.

polar angle

\jj

e.

the angle

A

curve rises from the origin in the xy plane into the

\jj

(\.

1

first

quadrant. The

area under the curve from (0,0) to (x,y) is one-third the area of the rectangle with these points as opposite vertices. Find the equation of the curve.

4.

GROWTH, DECAY, AND CHEMICAL REACTIONS

If a molecule has a tendency to decompose spontaneously into smaller molecules at a rate unaffected by the presence of other substances, then it is

SECTION

GROWTH, DECAY, AND CHEMICAL REACTIONS

4.

natural to expect that the

compose

A

number

15

of molecules of this kind that will de-

in a unit of time will be proportional to the total

chemical reaction of this type

is

number

present.

called a first-order reaction.

grams of matter are present initially, and x is the number of grams present at a then the principle stated above yields the following differential

Suppose, for example, that x

decompose later

time

in a first-order reaction. If

r,

equation:

~

=

>

k

kx,

(I)

0.

at

[Since dx/dt

the rate of growth of x,

is

says that the rate of decay of x ables in

(1),

is

after integration

initial

=

we separate

and

(1)

the vari-

-kdt,

= — kt +

c.

condition

x gives c

If

becomes log x

The

rate of decay,

is its

we obtain dx — = x

which

—dx/dt

proportional to x.]

log x

,

so log x

=

when

x

= —kt +

t

log x

,

=

(2)

log (x/x

)

=

—kt, x/x

=

e~

k

\

and x This function

is

=

x e~

kt

(3)

.

therefore the solution of the differential equation

satisfies the initial

condition

(2).

Its

graph

is

given in Fig.

6.

The

(1)

that

positive

measure of the which the reaction proceeds. Very few first-order chemical reactions are known, and by far the most

constant k

is

called the rate constant, for

its

value

is

clearly a

rate at

FIGURE 6

CHAPTER

16 important of these

is

THE NATURE OF DIFFERENTIAL EQUATIONS

1.

radioactive decay.

It is

convenient to express the rate of

decay of a radioactive element in terms of its half-life, which is the time required for a given quantity of the element to diminish by a factor of onehalf. If we replace x by x /2 in formula (3), then we get the equation

f=

x e'

kT

for the half-life T, so

kT= If either

k or

T is known

log

2.

from observation or experiment,

this

equation

enables us to find the other.

These ideas are the basis for a scientific tool of fairly recent development which has been of great significance for geology and archaeology. In essence, radioactive elements occurring in nature (with known half-lives) can be used to assign dates to events that took place from a few thousand to a few billion

For example, the common isotope of uranium decays through and an isotope of lead, with a half-life of 4.5 billion years. When rock containing uranium is in a molten state, as in lava flowing from the mouth of a volcano, the lead created by this decay process is dispersed by currents in the lava; but after the rock solidifies, the lead is locked in place and steadily accumulates alongside the parent uranium. A years ago.

several stages into helium

piece of granite can be analyzed to determine the ratio of lead to uranium,

and

this ratio

permits an estimate of the time that has elapsed since the critical

moment when

the granite crystallized. Several

methods of age determination

involving the decay of thorium and the isotopes of uranium into the various

method depends on

isotopes of lead are in current use. Another

potassium into argon, with a

half-life

ferred for dating the oldest rocks,

the decay of

of 1.3 billion years; and yet another, pre-

is

based on the decay of rubidium into

strontium, with a half-life of 50 billion years. These studies are complex and

many kinds; but they can often be checked against one another, and are capable of yielding reliable dates for many events in geological history linked to the formation of igneous rocks. Rocks tens of millions of years old are quite young, ages ranging into hundreds of millions susceptible to errors of

of years are

common, and

3 billion years old.

crust,

and so

for the

the oldest rocks yet discovered are

This of course

age of the earth

is

upwards of

a lower limit for the age of the earth's

itself.

Other investigations, using various

types of astronomical data, age determinations for minerals in meteorites,

and so on, have suggested a probable age years.

For a

for the earth of

about

4.5 billion

1

full

discussion of these matters, as well as

many

other methods and results oi the science

of geochronology, see F. E. Zeuner, "Dating the Past," 4th ed., Methuen, London. 1958.

SECTION

4.

GROWTH, DECAY, AND CHEMICAL REACTIONS

17

The radioactive elements mentioned above decay so slowly that the methods of age determination based on them are not suitable for dating events that took plaee relatively recently. This gap was filled by Willard Libby's discovery in the late 1940s of radiocarbon, a radioactive isotope of

carbon with a half-life of about 5600 years. By 1950 Libby and his associates had developed the technique of radiocarbon dating, which added a second

slow-moving geological clocks described above and made it Age and some of the movements and activities of prehistoric man. The contributions of this technique to late Pleistocene geology and archaeology have been spectacular. In brief outline, the facts and principles involved are these. Radiocarbon is produced in the upper atmosphere by the action of cosmic ray neutrons on nitrogen. This radiocarbon is oxidized to carbon dioxide, which in turn is mixed by the winds with the nonradioactive carbon dioxide already present. Since radiocarbon is constantly being formed and constantly decomposing back into nitrogen, its proportion to ordinary carbon in the atmosphere has long since reached an equilibrium state. All air-breathing plants incorporate this proportion of radiocarbon into their tissues, as do the animals that eat these plants. This proportion remains constant as long as a plant or animal lives; but when it dies it ceases to absorb new radiocarbon, while the supply

hand

to the

possible to date events in the later stages of the Ice

it

has at the time of death continues the steady process of decay. Thus,

piece of old

wood

has half the radioactivity of a living

5600 years ago, and

if it

tree,

has only a fourth this radioactivity,

11,200 years ago. This principle provides a

method

it

it

for dating

lived

if

a

about

lived about any ancient

wood, charcoal, vegetable fiber, flesh, method has been verified by applying it to the heartwood of giant sequoia trees whose growth rings record 3000 to 4000 years of life, and to furniture from Egyptian tombs whose age is also known independently. There are technical difficulties, but the method is now felt to be capable of reasonable accuracy as long as the periods of time involved are not too great (up to about 50,000 years). Radiocarbon dating has been applied to thousands of samples, and laboratories for carrying on this work number in the dozens. Among the more interesting age estimates are these: linen wrappings from the Dead Sea scrolls of the Book of Isaiah, recently found in a cave in Palestine and thought to be first or second century B.C., 1917 + 200 years; charcoal from the Lascaux cave in southern France, site of the remarkable prehistoric paintings, 15,516 + 900 years; charcoal from the prehistoric monument at Stonehenge, in southern England, 3798 + 275 years; charcoal from a tree burned at the time of the volcanic explosion that formed Crater Lake in Oregon, 6453 + 250 years. Campsites of ancient man throughout the Western Hemisphere have been dated by using pieces of charcoal, fiber sandals, fragments of burned bison bone, and the like. The results suggest object of organic origin, for instance,

skin, bone, or horn.

The

reliability of the

.

CHAPTER

18 that

man

Ice Age,

did not arrive in the

some

New World

11,500 years ago,

substantially lower than

it

THE NATURE OF DIFFERENTIAL EQUATIONS

1.

about the period of the

last

the level of the water in the oceans

was

when

now

until

and he could have walked across the

is

1

Bering Straits from Siberia to Alaska. These ideas may seem rather far removed from the subject of differential equations, but actually they rest on the mathematical foundation provided

by equation (1) and its solution as given in formula (3). In the following problems we ask the reader to apply similar techniques to questions arising in chemistry, biology, and physics.

PROBLEMS Suppose that two chemical substances

1

form

in solution react together to

means of the collision and a molecules of the substances, we expect the rate of the of interaction compound be proportional the number of collisions to to formation of the per unit time, which in turn is jointly proportional to the amounts of the compound.

If

the reaction occurs by

substances that are untransformed. in this

manner is

A

chemical reaction that proceeds

called a second-order reaction,

and

this

law of reaction

is

often referred to as the law of mass action. Consider a second-order

grams of the compound contain ax grams of the

reaction in which x

substance and bx grams of the second, where a

first

are

aA grams

second, and 2.

3.

of the

if

x

first

=

substance present

when

=

t

0,

+

b

=

1.

find x as a function of the time

there

t.

Suppose that x bacteria are placed in a nutrient solution at time t = 0. and that x is the population of the colony at a later time t. If food and living space are unlimited, and if, as a consequence, the population at any moment is increasing at a rate proportional to the population at that moment, find x as a function of t. If,

in

Problem

2,

space

is

and food

limited

is

supplied at a constant rate,

then competition for food and space will act in such a the population will stabilize at a constant level

way that ultimately Assume that under

xx grows at a rate jointly proportional and find x as a function of t. .

these conditions the population

and 4.

If

and bB grams of the

initially,

to the difference

Assume

x

1

—

x,

to x

an altitude h above sea level is propormass of the column of air above a horizontal unit area at that altitude, and also that the product of the volume of a given mass of air and the pressure on it remains constant at all altitudes. Up = p at that the air pressure p at

tional to the

sea level, find p as a function of 1

h.

Libby won the 1960 Nobel Prize for chemistry as a consequence of the work described above. own account of the method, with its pitfalls and conclusions, can be found in his book "Radiocarbon Dating," 2d ed., University of Chicago Press. 1955. Sec also G. Baldwin.

His

C

"America's Buried Past," Putnam,

New

York, 1962.

SECTION

5.

AND OTHER RATE PROBLEMS

FALLING BODIES

5.

Assume

19

which a hot body cools is proportional to the it and its surroundings {Newton s law heated to 110°C and placed in air at 10 C. After

that the rate at

difference in temperature between

of cooling

hour

1

it

6.

its

1 ).

A body

is

temperature

to cool to 30

is

60 C.

How much additional

time

is

required for

C?

According to Lamberts law of absorption, the percentage of incident light absorbed by a thin layer of translucent material is proportional to 2 the thickness of the layer. If sunlight falling vertically on ocean water is reduced to one-half its initial intensity at a depth of 10 feet, at what depth is it reduced to one-sixteenth its initial intensity? Solve this problem by merely thinking about it, and also by setting up and solving a suitable differential equation.

FALLING BODIES

5.

In this section

we study

AND OTHER RATE PROBLEMS problem of determining the motion

the dynamical

of a particle along a given path under the action of given forces.

We consider

only two simple cases: a vertical path, in which the particle

falling either

freely

under the influence of gravity alone, or with

is

air resistance

taken into

account; and a circular path, typified by the motion of the bob of a pendulum.

Free

fall.

The problem of a

and we arrived

freely falling

at the differential

dt for this

height.

motion, where y

One

is

body was discussed

in Section

1,

equation

2

~

the distance

0)

9

down

to the

body from some

fixed

integration yields the velocity,

v

=

dy

f =gt

+

Cl

(2)

.

t

Since the constant c t velocity v 09

and

(2)

is

when

clearly the value of v

t

=

0,

it

is

the initial

becomes v

=

—=

gt

+

v

.

(3)

'Newton himself applied this rule to estimate the temperature of a red-hot iron ball. So little was known about the laws of heat transfer at that time that his result was only a crude approximation, but it was certainly better than nothing. 2

Johann Heinrich Lambert (1728-1777) was a Swiss-German astronomer, mathematician, physicist, and man of learning. He was mainly self-educated, and published works on the orbits of comets, the theory of light, and the construction of maps. The Lambert equal-area projection is well known to all cartographers. He is remembered among mathematicians for having given the

first

proof of the

fact that

n

is

irrational.

CHAPTER

20

On

integrating again

we

THE NATURE OF DIFFERENTIAL EQUATIONS

1.

get 1

The constant

c2

is

when

the value of y

+

Vot

=

t

Cl

-

or the

0,

initial

position y

,

so

we

have

finally

=

y as the general solution of

=

so that v

=

y

(3)

=

0t

>

eliminating

£

we have

+

Y gt2

an(*

Vot

body

the

(1). If

u tnen

t?

On

,

= -j9 l +

y

(4)

+

y°

(4)

from

falls

rest starting at

y

=

0,

reduce to

and

y

=

— 1

#r

2 .

the useful equation v

for the velocity attained in

=

y/2gy

(5)

terms of the distance

fallen.

This result can also

be obtained from the principle of conservation of energy, which can be stated in the

form kinetic energy

Since our

body

falls

kinetic energy equals

from its

+

potential energy

1

and

(5)

2

=

a constant.

y = 0, the energy gives

rest starting at

loss in potential

— mv

=

fact that its gain in

mgy,

follows at once.

Retarded

fall.

If

we assume

that air exerts a resisting force proportional

to the velocity of our falling body, then the differential equation of the

motion

is

d 2y

where c

=

k/m [see equation

l-(3)]. If

Tt

On

dy

-g-c-£ dy/dt

=9-cv-

separating variables and integrating,

dv

g

is

-

= cv

we dt

(6)

replaced by

t\

this

becomes 0)

get

.

SECTION

5.

FALLING BODIES

AND OTHER RATE PROBLEMS

21

and \og(g

-

cv)

=

+

t

c

cv

so

—

g

The

initial

condition

v

=

when

=

cv t

= (1

Since c

is

positive, v -^ g/c as

terminal velocity. If we wish,

t

c2 e

.

gives c 2

-

e~

=

g,

so

ct

(8)

).

^> oo. This limiting value of v

we can now

replace v by dy/dr in

another integration to find y as a function of

(8)

is

called the

and perform

t

pendulum. Consider a pendulum consisting of a bob end of a rod of negligible mass and length a. If the bob is pulled to one side through an angle a and released (Fig. 7), then by the principle of conservation of energy we have

The motion of of mass

m

a

at the

mv 2 = mg(a cos Since

s

=

aO and

v

=

ds/dt

=

9

—

a cos

a).

(9)

a (d9/dt), this equation gives

=

ga(cos 6

—

cos

a);

FIGURE 7

(10)

CHAPTER

22

and on solving

for dt

increases (for small

f),

THE NATURE OF DIFFERENTIAL EQUATIONS

1.

and taking into account the

we

d6

a

the period, that

—

cos 6

2g

T is

t

get

dt

If

fact that 6 decreases as

cos a

the time required for one complete oscillation, then

is,

d9

a

—

cos 6

2g

cos a

or

d6 2#

do

—

cos 6

J

cos a

The value of T in this formula depends on a, which is the reason why pendulum clocks vary in their rate of keeping time as the bob swings through a greater or lesser angle. This formula for the period can be expressed more satisfactorily as follows. Since

cos

6=1

—

2 shr

-

2 sin

and cos a

1

2

we have dO 2

9 f

a_

=

sin (a/2)

J

a

-

2

sin (0/2)

dO

2

sin

(12)

2

sin (6/2)

9 Jo

We now that

change the variable from 6 to

increases from

by putting sin (0/2) from to a, and

to n/2 as 6 increases

— cos —6 1

-

d6

=

_

2^Jk

k cos

d(f)

or

2k cos

d(j)

d6 cos (6/2)

2

y/\

-sin 2 (6/2)d.

is

rotated about

acting on a particle of water of mass

x

is its

m at

distance from the axis, and this

mg and

is

The

assumes

centripetal force

the free surface

is

mxou 2 where

the resultant of the

normal reaction

the

axis with

that the surface of the water

the shape of a paraboloid of revolution. (Hint:

gravitational force

its

force

downward

R due

to other

nearby particles of water.) 11.

Consider a bead let

at the highest point of a circle in a vertical plane,

that point be joined to

wire. If the

bead

slides

any lower point on the

down

circle

the wire without friction,

and

by a straight

show

that

it

will

reach the circle in the same time regardless of the position of the lower point. 12.

A

chain 4

Neglect

feet

long starts with

friction,

and

1

foot

hanging over the edge of a

table.

find the time required for the chain to slide off

the table. 1 3.

man

wound around much greater force at the other end. Quantitatively, it is not difficult to see that if Tand T + AT are the tensions in the rope at angles 9 and 9 + A9 in Fig. 12, then a normal force of approximately T A9 is exerted by the rope on the post Experience

a

wooden

tells

us that a

holding one end of a rope

post can restrain with a small force a

in the region

between 9 and 9

+

A9.

It

follows from this that

the coefficient of friction between the rope

and the

post, then

if fi is

AT

is

approximately \iT A9. Use this statement to formulate the differential equation relating Tand 9, and solve this equation to find Tas a function of 14.

A

9,

,u,

load

density

and the

L

is

a. If

force

T

exerted by the man.

supported by a tapered circular column whose material has the radius of the top of the

distance x below the top

if

column

is r

,

find the radius r at a

the areas of the horizontal cross sections are

proportional to the total loads they bear. 15.

The President and

the

Prime Minister order coffee and receive cups of

equal temperature at the same time. The President adds a small amount of cool cream immediately, but does not drink his coffee until 10

The Prime Minister waits 10 minutes, and then adds the same amount of cool cream and begins to drink. Who drinks the hotter minutes coffee?

later.

CHAPTER

34

T

1.

THE NATURE OF DIFFERENTIAL EQUATIONS

+ AT

FIGURE 12

16.

A

destroyer

moment,

is

hunting a submarine

discloses the

in a

dense

fog.

submarine on the surface

immediately descends. The speed of the destroyer submarine, and

it is

known

The

fog

3 miles is

lifts

for a

away, and

twice that of the

that the latter will at once dive

and depart

unknown

What path

at full speed in a straight course of

direction.

should the destroyer follow to be certain of passing directly over the

submarine? Hint: Establish a polar coordinate system with the origin at the point where the submarine was sighted. 17.

Four bugs instant they

sit

all

at the corners of a

square table of side

a.

At the same

begin to walk with the same speed, each moving steadily

toward the bug on its right. If a polar coordinate system is established on the table, with the origin at the center and the polar axis along a diagonal, find the path of the bug that starts on the polar axis and the total distance it walks before all bugs meet at the center.

2 FIRST

7.

ORDER EQUATIONS

HOMOGENEOUS EQUATIONS

Generally speaking, equations.

it

is

very difficult to solve

dy

jdx cannot be solved its

of

solution in first

first

order differential

Even the apparently simple equation

a = f(x,y) \

in general, in the sense that

all cases.

On

no formulas

exist for

obtaining

the other hand, there are certain standard types

order equations for which routine methods of solution are available.

In this chapter

we

shall briefly discuss a

applications. Since our

main purpose

is

few of the types that have

to acquire technical facility,

many

we

shall

completely disregard questions of continuity, differentiability, the possible

The

vanishing of divisors, and so on.

matical nature will be dealt with

relevant problems of a purely mathelater,

when some

of the necessary

background has been developed.

The

simplest of the standard types

is

that in

which the variables are

separable:

dy

dx

=

g(x) h(y)

35

>

:

CHAPTER

36

To

we have only

solve this,

to write

in the

it

2.

FIRST

form

h(y)

ORDER EQUATIONS dy

=

g(x)

dx and

integrate:

h(y)

dy

g(x)

+

dx

c.

We have seen many examples of this procedure in the preceding chapter. At the next level of complexity is the homogeneous equation. A function /(x,y)

is

called

homogeneous of degree n

=

f(tx,ty) for all suitably restricted x, y,

are

homogeneous of degrees

and

2, 1,

is

said to be

homogeneous

if

n t

f(x,y)

Thus x 2 + xy, yjx 2 + y 2 and and 0. The differential equation t.

,

+

M(x,y) dx

same

if

N(x,y) dy

M and N are

sin (x/y)

=

homogeneous functions

degree. This equation can then be written in the

of the

form

where f(x,y) = —M(x,y)/N(x,y) is clearly homogeneous of degree 0. The procedure for solving (1) rests on the fact that it can be changed into an equation with separable variables by means of the substitution z = y/x. To see this,

we note

that the relation f(tx,ty)

permits us to set

t

=

1/x

=

t°f(x,y)

=f(x,y)

and obtain

f(x,y)=f(l,y/x)=f(l

Then

since y

=

= (1)

z

+

x— dx

(2)

becomes dz

Z

and the

z).

zx and

dx equation

9

...

x

+ x-=/(U),

variables can be separated

dx

dz /(l, z)

We now complete the

-

z

x

solution by integrating and replacing z by y/x.

.

SECTION

HOMOGENEOUS EQUATIONS

7.

EXAMPLE

We

+ y)dx -

Solve (x

1.

(x

37

-

=

y)dy

0.

begin by writing the equation in the form suggested by the above

discussion:

dy

_

dx Since the function on the right that

it

+ —

x

x

clearly

is

y y

homogeneous of degree

can be expressed as a function of z

=

y/x. This

is

easily

0,

we know

accomplished

by dividing numerator and denominator by x:

We

dy

1

dx

1

next introduce equation

y/x

1

y/x

1

+ —

z z

and separate the

(2) (1

On

+ —

— z)dz 2 1 + z

variables,

which gives

dx X

integration this yields -

1

tan

and when

z

— log

-

z

replaced by y/x,

is

tan"

+

z

2 )

=

+

log x

c;

we obtain

—x =

l

(1

log

Jx +

2

2

y

+

c

as the desired solution.

PROBLEMS 1

Verify that the following equations are 2

—

2 2y )dx

a.

(x

b.

x 2 y'

-

3xy

c.

x 2 y'

=

3(x

d.

x sin

y

2

2.

xy'

=

y

+

—=

+

xy dy

2y

=

2

2

_1

— + xy; x

h x;

x

.

Use rectangular coordinates family of

all circles

homogeneous, and solve them:

0;

y

.

y sin

2xe~ y/x

=

0;

y )tan

dy

x dx e.

-

+

to find the orthogonal trajectories of the

tangent to the y axis at the origin.

•

CHAPTER

38

Show

3.

y'

=

+

by

+

= f{ax +

by

+

that the substitution z

ax

into an equation with separable variables,

FIRST

2.

ORDER EQUATIONS

changes

c

c)

and apply

this

method

to

solve the following equations: y'

=

b. y'

=

If

ae

a.

4.

a.

(x sin =/=

+ 2

2

y)

bd,

\

—

(x

+

y

show

1).

x

=

dy

z

—

= F

5.

y

= w —

in

such a

way

k reduce

(ax'+ by

+

+f

ey

/

homogeneous equation,

to a

to

h,

\dx

dx

b. If

and k can be chosen

that constants h

that the substitutions

ae

=

one

bd, discover a substitution that reduces the

in

equation in

(a)

which the variables are separable.

Solve the following equations:

dy

x

dx

x

a.

dy

_

x

b.

dx

x

+ —

y

+ +

y

y

y

+ —

4

+ —

4

6

6

EXACT EQUATIONS

8.

If we start with a family of curves f(x,y) can be written in the form df = or

=

c,

— dx + — dy = ox cy

then

its

differential equation

0.

For example, the family x 2 y 3 = c has 2xy 3 dx + 3x 2 y 2 dy = as its differential equation. Suppose we turn this situation around, and begin with the differential equation

M(x,y)dx If

+

N(x,y)dy

=

0.

(1)

there happens to exist a function f(x,y) such that

4-

=

dx

then

(1)

can be written

in the

M

and

(2)

form

I* + g* -

or

df-

0,

:

SECTION

and

its

EXACT EQUATIONS

8.

general solution

39

is

f(x,y) In this case the expression

and

(1) is

M dx

=

c.

+ N dy

said to be an exact differential,

is

called an exact differential equation.

It is sometimes possible to determine exactness and mere inspection. Thus the left sides of

x

— dx 1

+ xdy =

y dx

and

~dy

l

y are recognizable as the differentials of xy

=

general solutions of these equations are xy

find the function / by

=

y

and x/y, respectively, so the c and x/y = c. In all but the

simplest cases, however, this technique of "solution by insight" impractical.

What

the function/

We

Suppose that equations

(2).

needed is a test for exactness and a method develop this test and method as follows. is

(1) is

exact, so that there exists a function

We know

partial derivatives

is

clearly

for finding

/

satisfying

from elementary calculus that the mixed second

of/ are equal

(3)

dx dy This yields

dM _ dN dy so

(4) is

a necessary condition for the exactness of (1).

also sufficient satisfies

(4)

dx

by showing that

equations

(2).

We

(4)

We shall prove that

it is

enables us to construct a function/ that

begin by integrating the

of equations

first

(2)

with

respect to x:

f==1\Mdx

+

(5)

g(y).

The "constant

of integration" occurring here is an arbitrary function of y since must disappear under differentiation with respect to x. This reduces our problem to that of finding a function g(y) with the property that/ as given it

The reader should be aware that equation (3) is true whenever both sides exist and are continuous, and that these conditions are satisfied by almost all functions that are likely to arise in practice. 7) is

that

all

Our

blanket hypothesis throughout this chapter (see the

the functions

we

discuss are sufficiently continuous

the validity of the operations

we perform on them.

and

first

paragraph

in

Section

differentiable to guarantee

CHAPTER

40 by

(5) satisfies

to y

the second of equations

and equating the

N, we

result to

M dx

(2).

FIRST

2.

ORDER EQUATIONS

On differentiating (5) with

respect

get

+

g'(y)

=

N,

so

=

g'(y)

N-

Mdx. dy

This yields

g(y)

N-

=

provided the integrand here

is

—

\Mdx) dy,

a function only of

derivative of the integrand with respect to

question

x

0;

is

y.

(6)

This

will

be true

if

the

and

since the derivative in

2

r

is

dN Mdx =

dx\

d

dxdy

dx

dy J

Mdx J

2

dN

d

dx

dydx

dN

dM

dx

"dy

Mdx m

9

an appeal to our assumption (4) completes the argument. In summary, we have proved the following statement: equation (1) is exact if and only if dM/dy = dN/dx; and in this case, its general solution is f(x,y) = c, where f is given by (5) and (6). Two points deserve emphasis: it is the equation f(x,y) = c, and not merely the function f which is the general solution of (1); and it is the method embodied in (5) and (6), not the formulas themselves, which should be learned.

EXAMPLE

1.

Test the equation e y dx

and solve it if it is Here we have

+

(xe y

+

2y)dy

exact.

M

=

ey

and

N=

xe y

+

so

dM ^y

=

dN e>

and dx

=

e>

2y,

=

for exactness,

SECTION

EXACT EQUATIONS

8.

41

Thus condition (4) is satisfied, and the equation exists a function /(x,y)

— = dx Integrating the

first

is

exact. This tells us that there

such that ey

—- =

and

+

xe y

2y.

dy

of these equations with respect to x gives

/=

ey

+

dx

g[y)

=

xe y

+

g(y),

so

df

=

j-

+

xe*

dy

g'(y).

+

Since this partial derivative must also equal xe y

=

so d(y)

2

y

and/ = xe y + y 2

.

All that remains

+

xe y is

2

y

=

2y,

we have

g'(y)

=

2y,

to note that

is

c

the desired solution of the given differential equation.

PROBLEMS Determine which of the following equations are exact, and solve the ones that are.

(

2.

(sin

3.

4. 5.

6.

7.

—

+

x

1.

dy

+ ydx =

0.

+ 1) dx + cos x sec 2 y dy = 0. 3 3 iy - x )dx + (x + y )dy = 0. 2 (2y - 4x + 5) dx = (4 - 2y + Axy) dy. (y + y cos xy) dx + (x + x cos xy) dy = 0. 2 cos x cos y dx + 2 sin x sin y cos y dy = 0. y dy = (e* + cos x cos y) dx. (sin x sin y — xe x tan y

)

sin

8.

y 9.

)

(1

—X dx

H

10.

(2xy

11.

dx

+

y)dx 3

=-

l

sin

(1

-

x)rfy

y 1

-

~-^dx z x 2y

=

0.

2

+ ycosx)dx +

=

0.

y

y

y

+

—XX—dy = (3x y

2

4-

sinx)dy

x

+ 1

-

j-^dy.

x L yz

=

0.

CHAPTER

42

FIRST

2.

ORDER EQUATIONS

INTEGRATING FACTORS

9.

that exact differential

The reader has probably noticed

equations are

depends on a precise balance in the form of the equation and is easily destroyed by minor changes in this form. This being the case, he may have doubts that exact equations are worth discussing at all. In the present section we shall try to answer these doubts. rare, for exactness

comparatively

The equation

is

ever,

if

which in this

{x y

dM/dy =

be nonexact, for

easily seen to

we multiply through by

1

the factor 1/x

(1)

and dN/dx 2

=

the equation

,

2xy — 1. Howbecomes

exact. To what extent can other nonexact equations be made exact way? In other words, if

is

+

M(x,y)dx is

- x)dy =

2

+

y dx

=

N(x,y)dy

(2)

not exact, under what conditions can a function n(x,y) be found with the

property that

+ N dy) =

fi(M dx is

exact?

for

(2).

Any

Thus

function 1/x

2 is

\i

that acts in this

always has an integrating factor

Assume then

if it

by

c

"fdx

follows from

called

(1).

We

an integrating factor shall prove that (2)

has a general solution.

(2)

=

c,

differentiating:

dx It

is

for

that (2) has a general solution

f{x,y)

and eliminate

way

an integrating factor

and

(3)

= +*fdy ay

0.

(3)

that

M

dy _

dx

df/dx

~ "77 " ~df/dy

so

If

we denote

the

common

df/dx

df/dy

M

N

ratio in (4)

by

n(x,y), then

(4)

SECTION

INTEGRATING FACTORS

9.

=

r

43

and

fiM

On

multiplying

(2)

by

=

r

OX

fiN.

(Y

becomes

it

//,

/.iM

+

dx

=

//N dy

or

'

Px

whieh it

is

dy

argument shows that

exact. This

has at least one integrating factor

tegrating factors; for

if

F{f)

so fiF(f)

Our

also

is

has a general solution, then has infinitely

it

many

in-

any function of/ then

is

+ Ndy) =

liF(j)(M dx

if (2)

Actually

//.

an integrating factor

F(j) df

for

=

d

F(f)df

(2).

discussion so far has not considered the practical problem of finding

is quite difficult. There are a few cases, which formal procedures are available. To see how these procedures arise, we consider the condition that \i be an integrating factor

integrating factors. In general this

however,

in

for (2):

_

d(fiM)

d(nN)

dx

dy If

we

write this out,

we obtain

dM

dfi

dy

dy

dN

AJ

d(t

or

m-t 1

It

/

A,,

P\,,\ ,

dM P)A/f

AM dN

dy

dx

(5) '

appears that we have "reduced" the problem of solving the ordinary equation (2) to the much more difficult problem of solving the

differential

partial differential equation

general solution of

And from

this

(5)

d\ijdx

=

dfi/dx

On

the other hand,

we have no need

for the

since any particular solution will serve our purpose.

point of view,

instance, that (2) has

Then

(5).

(5) is

more

fruitful

than

it

looks. Suppose, for

an integrating factor which is a function of x alone. and dp/dy = 0, so (5) can be written in the form jj,

— 1

d\i

[i

dx

=

dM/dy - dN/dx

N

(6)

CHAPTER

44 Since the

left

side of this

=

N (6)

ORDER EQUATIONS

a function only of x, the right side

is

dM/dy - dN/dx

then

FIRST

2.

is

also. If

we put

g(x),

becomes dix

1 -

ax

\i

.

,

=

-j-

g(x)

or d{\ogfi)

so

=

log//

g(x)dx

and

=

H This reasoning (6) is

is

e$

g{x)dx

obviously reversible:

if

a function only of x, say g(x), then

only on x and

equation

satisfies

(5),

(7)

.

and

the expression on the right side of

(7) yields is

a function

\i

that depends

therefore an integrating factor for

(2).

EXAMPLE

1

In the case of equation

.

dM/dy - dN/dx _

-

1

N which

is

-

—

x

x y

ll

is

(2xy 2

a function only of

an integrating factor

= for

we have

1)

_ -2(xy - 1) x(xy — 1)

_

2

x

Accordingly,

x.

e

(1)

h i2/x)dx =

(1),

as

£~ 21o g*

— x -2

we have already

seen.

Similar reasoning gives the following related procedure, which cable whenever

(2)

is

has an integrating factor depending only on y:

appliif

the

expression

dM/dy - dN/dx

n —M is

^'

a function of y alone, say h(y), then li

=

e^

h(y)dy

(9)

)

SECTION

is

INTEGRATING FACTORS

9.

which

also a function only of v

integrating factor for

There

45

satisfies

equation

(5),

and

is

consequently an

(2).

another useful technique for converting simple nonexact equations into exact ones. To illustrate it, we again consider equation (1), is

rearranged as follows:

- {xdy -

x 2 ydy

The quantity

ydx)

=

0.

(10)

parentheses should remind the reader of the differential

in

formula

y\ = xdy-yte which suggests dividing into y dy

—

d(y/x)

—

0,

(10)

so

through by x 2 This transforms the equation .

general solution

its

1

In effect,

we have found an

combination

xdy — ydx

following are

is

evidently

y

2

integrating factor for

and using

some other

(11)

t

(1)

by noticing

in

the

it

(11) to exploit this observation.

The

formulas that are often useful

differential

in

similar circumstances:

ydx - xdy

d(j\ = =

d(xy)

d(x

2

+

/

y .

tan'

d

2

1

)

x dy

=

—x\

+

y dx;

2{xdx

=

ydx y

y J

V

(12)

;

x

+

(13)

ydy);

(14)

— xdy

7 2

2

_

-\-

*

....

(15)

;

y

-

dL g ^-) = ydX *y Xdy

(16)

-

y

V

We

see from these formulas that the very simple differential equation — x dy = has 1/x 2 l/y 2 l/(x 2 + y 2 \ and l/xy as integrating factors, dx y and thus can be solved in this manner in a variety of ways. ,

EXAMPLE

2.

,

Find the shape of a curved mirror such that

source at the origin will be reflected in a

The mirror

will

beam

light

have the shape of a surface of revolution generated by the law of

APE (Fig. 13) about the x axis. It follows from reflection that a = p. By the geometry of the situation, = a +