[Vol. II] Mathematical Analysis for Engineers [2] 5930002715

Table of contents :
Preface
11
Chapter 13
Number Series 13
34
Definition. Sum of a Series 13
13.2
Operations
on Series
15
13.3
Tests for Convergence of Series 18
13.4
Alternating Series. Leibniz Test 30
13.5
Series of Positive and Negative Terms.
Absolute and
Conditional Convergence 32
Exercises 35
Answers 37
Chapter
14
Functional Series 38
14.1
Convergence Domain and Convergence Hnterval
38
14.2
Uniform Convergence 49
14.3
Weierstrass Test 43
14.4
Properties of Uniformly Convergent Functional
Series 45
Exercises 50
Answers 50
Chapter
15
Power Series 51
15.1
Abel’s Theorem.
Interval
and
Radius
of Convergence
for Power Series 41
15.2
Properties of Power
Series 56
15.3
‘Taylor's Series 59 :
Exercises 70
Answers 7}
Chapter
16
Fourier Series 73
16.1
Trigonometric Series 73
16.2
Fourier Series for a Function with Period 2 76
16.3
Sufficient Conditions for the Fourier Expansion of a
Function 78
16.4
Fourier Expansions
of Odd
and
Even
Functions
82
16.5
Expansion of a Function Defined on the Given Interval
into a Series of Sines and Cosines 86
16.4
Fourier Series for a Function with Arbit
rary Period 88
16.7
Complex Representation of Fourier Series
93
16.8
Fourier Series in General Orthogonal
Systems of Func-
tions @
Exercises
104
Answers 105
hapter
17
First-Order Ordinary
Differential
Equations
106
V7.4
Basic Notions. Examples 106
2
Saiution of the Cauchy
Problem
for
First-Order
Differential Equations 109
7.3
Approximate
Methods
of Integration of the Equation
yi = fix y) 13
17.4
Some Equations Integrable by Quadratu
res il8
175
Riccati Equation 135
IhG
Differential Equations Insolvable for
the Derivative 136
1h?
Ceometrical Aspects
of
First
-Orde
r
Diffe
rential Equa-
Hens. Orthogonal
Traje
ctori
es
142
Exercises 144
Answers 145
Chapter
18
Higher-Order Differential Equations
147
18.1
Cauchy Problem [47
1B.2
Reducing the Order of Higher-Order
Equations 149
18.4
Linear Homogeneous Differential
Equations of
Order a 153
18.4
Linearly Dependent and Linearly Inde
pendent Systems
af Functions 155
18.5
Structure of General Solution of Linea
r Homogencous
Differential Equation 160
Linear
Homogeneous
Differential
Equations
with
Constant Coefficients 164
Equations Reducible to Equations
with Constant
Coefficients 172
Linear Inhomogeneous
Differential Equations
173
Integration of Linear Inhomogeneous
Equation by Var-
iation of Constants
176
18.10
inhomogeneous
Linear
Differential
Equations
with
Constant Coefficients
180
o
(8.41
Integration of Differential Equation
s
Using
Rowet
Series and Generalized Power Series 188
7
16.12
Bessel
Equation.
Bessel
Functions
190
Exercises 201
Answers
208
Chapter 19
Systems of Different al Equations 203
19.1
Essentials. Definition s 203
19.2
Methods of Integra lion of Systems of
Differential
Equations
206
19.3
Systems of Linear Differential Equations 211
19.4
Systems of Linear Differential Equations With
Con-
stant Coefficients 21 p
Exercises
224
Answers
224
Chapter 20
Stability Theory 225)
,
20.1
Preliminaries 225
20.2
Stability in the Sense of Lyapunov. Basic Concepts and
Definitions 227
20.3
Stability of Autonomous Systems. Simplest Types of
Stationary Points 23 A.
20.4
Method of Lyapuno v's Functions 244
20,5
Stability in First (Linear) Approximation 248
Exercises 253
Answers
254
.
Chapter
21
Special Topics of Di {ferential Equations 255
24.1
Asymptotic Behavio ur of Solutions of Differential
Equations as x +
255
21.2
Perturbation Method 257
21.3
Oscillations of Solutions of Differential Equations 261]
Exercises 264
Answers
264
Chapter
22
Multiple Integrals. Double Integral 265
22.1
Problem Leading to the Concept of Double Integral 265
22.2
Main Properties of Double Integral 268
22.3
Double
Integral Reduced to Iterated Integral 270
22.4
Change
of Variables in Double Integral 278
22.5
Surface Area. Surface Integral 286
22.6
Triple Integrals 292
22.7
Taking Triple Integral in Rectangular Coordinates
294
22.8
Taking Triple Integral in
Cylindr
ical
and
Spheric
al
Coordinates
296
22.9
Applications of Double and Triple integrals 302
22.10
Improper Multiple
ntegrals over Unbounded
Domains 307
Exercises 309
Answers 312
8
Contents
Chapter
23
Line Integrals 313
23.1
Line Integrals of the First Kind 313
23.2
Line Integrals of the Second Kind 318
23.3
Green’s Formula 322
23.4
Applications of Line Integrals 327
Exercises 331
Answers 333
Chapter 24
Vector Analysis 334
24.1
Sealar Field. Level Surfaces
and
Curves.
Directiona!
Derivative 334
24.2
Gradient of a Scalar Field 339
24.3
Vector Field. Vector Lines and Their Differential Equa-
tions 344
24.4
Vector Flux Through a Surface and Its Properties 349
24,5
Flux of a Vector Through an Open Surface 354
24.6
Flux of a Vector Through a Closed Surface. Ostrograd-
sky-Gauss Formula 363
24.7
Divergence of a Vector Field 371
24.8
Circulation of a Vector Field. Curl of a Vector. Stokes
Theorem 378
24.9
Independence of the Line Integral of Integration
Path 386
24.10
Potential
Field 391
24.11
Hamiltonian 398
24.12
Differential Operations
of the Second
Order.
Laplace
Operator 402
2413
Curvilinear Coordinates 406
24.14
Basic Vector Operations in Curvilinear Coordinates 408
Exercises: 416
Answers 419
Chapter
25
Integrals Depending on Parameter 420
25.4
Proper Integrals Depending on Parameter 420
25.2.
Improper Integrals Depending on Parameter 425
25.3
Euler Integrals. Gamma
Function. Beta Function 431
Exercises 436
Answers 438
Chapter
26
Functions
of a Complex
Variable 441
-
26.1
Essentials. Derivative. Cauchy-Riemann Equations 44]
26.2
Elementary Functions of a Complex Variable 453
26.3
Integration
with
Respect
to
a
Complex
. Argument.
Cauchy Theorem. Cauchy Integral Formula 461
26.4
Complex Power Series. Taylor Series 476
Contents
26.5
Laurent Series. Isolated Singularities and Their Classifi-
cation 491
26.6
Residues. Basic Theorem on Residues. Application of
Residues to Integrals 503
Exercises
519
Answers 522
Chapter 27
Integral Transforms.
Fourier Transforms 424
;
27.1
Fourier
Integral
524
27.2
Fourier Transform,
Fourier Sine and Cosine
Transforms 528
27.3
Properties of the Fourier Transform
535
27.4
Applications 539
27,5
Multiple Fourier Transforms
543
Exercises 544
:
Answers 545
Chapter 28
Laplace Transform 546 :
28.1
Basic Definitions 546
28.2
Properties of Laplace Transform 551
28.3
Inverse Transform 560
28.4
Applications of Laplace Transform (Operational
Cal-
culus) 565
Exercises 372
Answers 573
Chapter
29
Partial Differential Equations 575
29.4
Essentials. Examples 575
29.2
Linear Partial Differential
Equations.
Properties
of
Their Solutions 577
29.3
Classification
of
Second~ Order
Linear
Differential
Equations in Two Independent
Variables 579
Exercises 583
Answers 584
Chapter 30
Hyperbolic Equations 585
30.1
Essentials 585
30.2
Solution of the Cauchy Problem (fnitial Value Problem)
for an Infinite String 587
30.3
Examination of the D’Alembert Formula 591
30.4
Well-Posedness of a Problem. Hadamard’s Example of
Hi-Posed Problem 594
30.5
Free Vibrations of a String Fixed at Both Ends. Fourier
Method 598 |
30.6
Forced Vibrations of a String Fixed at Both Ends 606
30.7
Forced
Vibrations-of a String
with
Unfixed
Ends 61]
30.8
General Scheme of the Fourier Method 613
30.9
‘Uniqueness of Solution of a Mixed Problem 621
JG.10
Vibrations of a Round Membrane 623
FOUL
Application of Laplace Transforms to Solution, of
Mixed Problems 627
Exercises 60
Answers 632
Chapler
3]
Parabolic Equations 633
3h
Heat Equation 633
SEZ
Cauchy’ Problera for Heat Equation 634
313
Heat
Propagation
in a Finite Rod
640
Fd
Fourier Method
Por Heat Equation 643
Exercises 649
Answers 649
Chapler 32
Elliptic Equations 656
32.1
Definitions. Formulation of Boundary Problems 650
32.2
Fundamental Solution of Laplace Equation 652
32.3
Green's Formulas 653
324
Basic Integral Creen’s Formula 654
42.5
Properties of Harmonic Functions 657 :
52.6
Solution of the Mirichlet Problem for a Circle Using the
Fourier Method 661
32.7
Poisson: Integral 664
Exercises 666
Answers 666
Appendix TI Conformal Mappings 667
index 6o4

Citation preview

Kype BEIciicii

MaTeMaTHuKnh

JUIA WHOKeHepoOB

SD

B gapyx TOMax

M.Kpacnoe, A. Kucenes, T. Maxapenko, E. Ulment

Tom

|

_

MAnaatlhyesmiastifocral f&-Macta"rent Engineers

Volume

In two volumes

|i | Moscow

|

i

‘

Transiated

from

by Alexander

Russian

Repyev,

First. published 1990

Ha

GHanuucKNom

Printed

ISBN

in the

AdblKe Union

5-93-000271-5

of Soviet

Socialist

©

M.

©

I. Maxapenxo, E. Ulneua, £990 English translation, A. Repyev, 1990

ISBN 5-03-0090269-3

—-o-

‘

Republics

Kpacnos,

A.

Kucenée,

Contents

Preface

11

Chapter 13

Chapter

Number Series 13 Definition. Sum of a Series 13

34 13.2

Operations

13.3

Tests for Convergence of Series 18

13.4 13.5

Series of Positive and Negative Terms.

14 14.1 14.2 14.3 14.4

Conditional Convergence 32 Exercises 35 Answers 37 Functional Series 38 Convergence Domain and Convergence Hnterval Uniform Convergence 49 Weierstrass Test 43 Properties of Uniformly Convergent Functional

on Series

15

Alternating Series. Leibniz Test 30 Absolute and

38

Series 45 Exercises 50 Chapter

15

15.1

Answers 50 Power Series 51 Abel’s Theorem.

Interval

and

Radius

of Convergence

for Power Series 41 15.2 15.3

Chapter

16

16.1 16.2 16.3 16.4 16.5

Properties of Power ‘Taylor's Series 59 : Exercises 70 Answers 7}

Series 56

Fourier Series 73

Trigonometric Series 73 Fourier Series for a Function with Period 2 76

Sufficient Conditions for the Fourier Expansion of a Function 78 Fourier Expansions

of Odd

and

Even

Functions

82

Expansion of a Function Defined on the Given Interval into a Series of Sines and Cosines 86

‘yr =

6

Contents

16.4 16.7 16.8

rary Period 88 Fourier Series for a Function with Arbit 93 Complex Representation of Fourier Series Systems of FuncFourier Series in General Orthogonal

tions @ Exercises hapter

17 V7.4 2 7.3

104

Answers 105 First-Order Ordinary

Differential

Basic Notions. Examples 106 Approximate

First-Order

for

Problem

Saiution of the Cauchy Differential Equations 109

106

Equations

of Integration of the Equation

Methods

yi = fix y) 13 17.4 175 IhG 1h?

Chapter

18 18.1 1B.2 18.4 18.4 18.5

res il8 Some Equations Integrable by Quadratu Riccati Equation 135 the Derivative 136 Differential Equations Insolvable for rential EquaDiffe r -Orde First of Ceometrical Aspects 142 es ctori Traje Hens. Orthogonal Exercises 144 Answers 145

147

Higher-Order Differential Equations

Cauchy Problem [47 Equations 149 Reducing the Order of Higher-Order Equations of Linear Homogeneous Differential Order a 153 pendent Systems Linearly Dependent and Linearly Inde af Functions 155 r Homogencous Structure of General Solution of Linea Differential Equation 160 with Equations Differential Homogeneous Linear Constant Coefficients 164 with Constant Equations Reducible to Equations Coefficients 172

Linear Inhomogeneous

173

Differential Equations

Equation by VarIntegration of Linear Inhomogeneous

iation of Constants 18.10 (8.41

16.12

inhomogeneous

176

Constant Coefficients

o

180

s Integration of Differential Equation

Using

Series and Generalized Power Series 188 Bessel

Equation.

Exercises 201 Answers

208

Bessel

Functions

with

Equations

Differential

Linear

190

Rowet

7

Contents

Chapter 19

19.1

19.2

Systems of Different al Equations 203 Essentials. Definition s 203 Methods of Integra lion of Systems of Equations

19.3 19.4

Chapter 20 , 20.1 20.2 20.3 20.4 20,5

Differential

206

Systems of Linear Differential Equations 211 Systems of Linear Differential Equations With stant Coefficients 21 p Exercises

224

Answers

224

Con-

Stability Theory 225) Preliminaries 225 Stability in the Sense of Lyapunov. Basic Concepts and Definitions 227 Stability of Autonomous Systems. Simplest Types of Stationary Points 23 A. Method of Lyapuno v's Functions 244 Stability in First (Linear) Approximation 248 Exercises 253 Answers

Chapter

21 24.1

21.2 21.3

254

. Special Topics of Di {ferential Equations 255 Asymptotic Behavio ur of Solutions of Differential 255 Equations as x + Perturbation Method 257

Oscillations of Solutions of Differential Equations 261] Exercises 264 Answers

Chapter

22 22.1 22.2 22.3 22.4

22.5 22.6 22.7 22.8

264

Multiple Integrals. Double Integral 265

Problem Leading to the Concept of Double Integral 265 Main Properties of Double Integral 268 Double

Change

Integral Reduced to Iterated Integral 270 of Variables in Double Integral 278

Surface Area. Surface Integral 286 Triple Integrals 292

294 Taking Triple Integral in Rectangular Coordinates al Spheric and ical Cylindr Taking Triple Integral in Coordinates

22.9 22.10

296

Applications of Double and Triple integrals 302 Improper Multiple Domains 307 Exercises 309 Answers 312

ntegrals over Unbounded

8

Contents

Chapter

23

23.1 23.2

23.3 23.4

Chapter 24 24.1

Line Integrals 313 Line Integrals of the First Kind 313 Line Integrals of the Second Kind 318

Green’s Formula 322 Applications of Line Integrals 327 Exercises 331 Answers 333 Vector Analysis 334 Sealar Field. Level Surfaces Derivative 334

and

Curves.

Directiona!

24.2 24.3

Gradient of a Scalar Field 339 Vector Field. Vector Lines and Their Differential Equa-

24.4 24,5 24.6

Vector Flux Through a Surface and Its Properties 349 Flux of a Vector Through an Open Surface 354 Flux of a Vector Through a Closed Surface. Ostrogradsky-Gauss Formula 363 Divergence of a Vector Field 371 Circulation of a Vector Field. Curl of a Vector. Stokes Theorem 378

tions 344

24.7 24.8

24.9 24.10 24.11 24.12 2413 24.14

i |

Independence of the Line Integral of Integration Path 386 Potential

Field 391

Hamiltonian 398 Differential Operations

of the Second

Order.

Laplace

Operator 402 Curvilinear Coordinates 406 Basic Vector Operations in Curvilinear Coordinates 408 Exercises: 416

Answers 419 Chapter

25 25.4

25.2.

25.3

Chapter

Integrals Depending on Parameter 420 Proper Integrals Depending on Parameter 420 Improper Integrals Depending on Parameter 425

Euler Integrals. Gamma Exercises 436 Answers 438

Function. Beta Function 431

Functions

Variable 441

26 26.1 26.2 26.3

Essentials. Derivative. Cauchy-Riemann Equations 44] Elementary Functions of a Complex Variable 453

26.4

Cauchy Theorem. Cauchy Integral Formula 461 Complex Power Series. Taylor Series 476

Integration

of a Complex

with

Respect

to

a

-

Complex . Argument.

Contents

26.5

Laurent Series. Isolated Singularities and Their Classification 491

26.6

Residues. Basic Theorem on Residues. Application of Residues to Integrals 503 Exercises

Chapter 27 27.1 27.2 27.3 27.4 27,5

Chapter 28

28.1 28.2 28.3 28.4

Chapter

29 29.4 29.2

29.3

519

Answers 522 Integral Transforms. Fourier

Fourier Transforms 424 ;

524

Integral

Fourier Transform,

Fourier Sine and Cosine

Transforms 528 Properties of the Fourier Transform 543

Multiple Fourier Transforms : Exercises 544 Answers 545 Laplace Transform 546 :

Basic Definitions 546

Properties of Laplace Transform 551 Inverse Transform 560 Applications of Laplace Transform (Operational culus) 565 Exercises 372 Answers 573 Partial Differential Equations 575

Essentials. Examples 575 Linear Partial Differential Their Solutions 577 Classification

of

Equations.

30.3 30.4 30.5 30.6 30.7 30.8

Linear

Second~ Order

Equations in Two Independent Exercises 583 Chapter 30 30.1 30.2

535

Applications 539

Properties

Cal-

of

Differential

Variables 579

Answers 584 Hyperbolic Equations 585 Essentials 585 Solution of the Cauchy Problem (fnitial Value Problem) for an Infinite String 587 Examination of the D’Alembert Formula 591

Well-Posedness of a Problem. Hadamard’s Example of

Hi-Posed Problem 594

Free Vibrations of a String Fixed at Both Ends. Fourier

Method 598 | Forced Vibrations of a String Fixed at Both Ends 606 Forced

Vibrations-of a String

with

Unfixed

General Scheme of the Fourier Method 613

Ends 61]

1d

Contents

30.9 JG.10

FOUL

Chapler

3] 3h SEZ 313 Fd

‘Uniqueness of Solution of a Mixed Problem 621 Vibrations of a Round Membrane 623 Application of Laplace Transforms to Solution, of Mixed Problems 627 Exercises 60 Answers 632 Parabolic Equations 633 Heat Equation 633

Cauchy’ Problera for Heat Equation 634 Heat

Propagation

Fourier Method

in a Finite Rod

640

Por Heat Equation 643

Exercises 649 Answers 649 Chapler 32 Elliptic Equations 656 32.1 Definitions. Formulation of Boundary Problems 650 32.2 Fundamental Solution of Laplace Equation 652 32.3 Green's Formulas 653 324 Basic Integral Creen’s Formula 654 42.5 Properties of Harmonic Functions 657 : 52.6 Solution of the Mirichlet Problem for a Circle Using the Fourier Method 661 32.7 Poisson: Integral 664 Exercises 666 Answers 666 Appendix TI Conformal Mappings 667 index 6o4

Preface

This two-volume book was written for students of technical colleges who have had the usual mathematical training. It contains just cnough in-

formation to continue with a wide variety of engineering disciplines. It

covers analytic geometry and linear algebra, differential and integral calculus for functions of one and more variables,

vector analysis, numerical and

functional series (including Fourier series), ordinary differential equations, functions of a complex variable, Laplace and Fourier transforms, and equa-

tions of mathematical physics. This list itself demonstrates that the book

covers the material for both a basic course in higher mathematics and severat special sections that are important for applied problems. Hence, it may be used by a wide range of readers. Besides students in technical colleges and those starting a mathematics course, it may be found useful by engineers and scientists who wish to refresh their knowledge of some aspects ' of mathematics. concisely and without dismaterial l fundamenta the We tried to give on of the basic idcas prensentali the d on concentrate We tracting detail. of linear algebra and analysis to make it detailed and as comprehensible as possible, Mastery of these ideas is a requiremient (o understand the tater

material, The many examples also serve this aim. The examples were written to help students with the mechanics of ‘solving typical problems. More than 600 diagrams are simple illustrations, clear enough to demonstrate the ideas and statements convincingly, and can be fairly casily , reproduced. We were conscious not to burden, the course with scrupulous proofs for theorems which have little practical application. As a rule we chose the proof (marked in the text with special symbols) that was constructive in nature or explained fundamental ideas that had been introduced, showing how they work. This approach made it possible to devise algorithms for solving whole

classes of important

problems.

In addition to the examples, we have included several carefully selected problems and exercises (around 1000) which should be of interest to those pursuing am independent mathematics course. The problems have the form

12

Preface

‘

of moderately sized thearems. They are very simple but are good training

for those learning the fundamental ideas. Chapters

1-6, 26 and Appendix

If were written by E.Shikin, Chapters

7-8, H, 12, 17-21, 27, 28 and 29-32 by M.Krasnov, Chapters 9, 10, 13-16

by A.Kiselev, and Chapters 22-25 and Appendix | by G.Makarenko. There was no general editor, but each of the authors has read the chapters written by the colleagues, and so each chapter benefited from collective advice. The Authors

Chapter Number

13.1.

13

Series

Definition. Sum

of a Series

, Consider an infinite number sequence Gi, Oty vey Gay ve A number a;

:

series is an expression + a

+

of the form

+ dy te

:

(13.5)

A shorthand notation for this is 3) dn. ast

The numbers a@;, a1, ... are called the terms of the series, and the number a, is called the arth or general term of the series. The sum of the finite number 7 of the terms of the series is called the

nth partial sum of the series: A

See a ta

t.. tan =

Dide. kok

Now

consider

the sequence

[S,]

of partial sums

of the series (13.1)

Sy = dj, oz

=

ay

+

On = Oy +

2,

2 be

+ ay

ee

Definition. If the sequence (5, ] has a finite mit

lim 5, = 5, be, [Su Ane oe

converges

to S, then

the limit is called

the sum

of the serics

S) aq

and

ast

the series is said to converge: does

>) aq = 5, If there is no

lim Sy, ie, (Si]

ant

noon

not converge, then the series

>) a, nel

no

sum.

is said

to diverge and

to have

4

{3 Mumber Series

Examples. (1) Show

I

i

i a

3ee 18°

that

the series

l

ee

35

ant — |

|

a

Qa et

converges, “a

We

Using

consider

the ath partial

“ a

I i a a

i

“3

of the series

I

a

3S -

the obvious

sum

dn? — }

relation

_

|

dnt — 1]

(Qe-

Jt

D@n+

8

I AG.

~ i

Qat

)

we represent S, in the form

Passing

to the limit as a — co, we will have lum Sy al = > fen ee

By

the definition

the series comverges

and

its sum

is § = 1/2, or

I

Aat

_ (2) Consider a series known as a geometric progression with.a ratio g

eragtage +.

teg™

4.2

j

“é The-sum

Sa

of m terms is, .

atag+ag a wag’ og

“Sg

Siagh? aml

_—

+o. bag’!

Wray

sagt

poy.

eb

(a = 0).

13.2 Operations on Series

if igi i 11

wel

the series

ote

aoe

diverges.

Le,

lim §, = 0,

hence

lim gq” = 00 and

igi > 1, then

If

At g = —1 we obtain the divergent serlesa -a+a@-~at... Its partial sum is s

(a .

It follows

for odd QO

that

(a #0).

x,

for even A. lim 5,

docs

not

exit.

ato

At

g =

hence

and

| we

will

have

atatad..,

the series

lim va = 0, Ao

:

being iy

and

which

S,

= ra,

series diverges,

ie, the

lim 5S, = Avro

for

Consequently, the series a + aq + ao? ti. 4¢ aq"! +... coaverges lgl < I, its sum

for

13.2

Operations

for

diverges

lgl > Ll.

on Series

Operations on number series may be deduced theorems:

Theorem 13.1. Lf the series ay +a

+.

+ dy +.

'

from

the following

=D) dy ConvErEeS, al

then so does the series obtained fram

it by discarding any finite muumber

of terms in the beginning. Conversely, if a series obtained from the given series by discarding a finite number then the given series converges. “@ In the partial sum Oy

Gp

bt a

tou

bade

+

of terms in the beginning canverges,

daa

cb

we

+

Oh

of the series we denote by ox the sum of the first A(A < a) discarded terms. We

pet

Ba = (dp + dy tw. + a

te (dea

bw

bh a) = oe + Se

16

13 Number

Series t

It

then

lim S,,

exists

there

if

that

follows

there

Hm S,-«

exists

fra od

(k = const) and,

conversely,

Ate

if there exists

lim S,-%,

then there exists

Roto

lim 35,.

/

Remark.

The

resultant

series

dg41

+ @es2

+...

has

the

sum

§ = §— o,, S being the sum of the original series. Theorem

13.2. Let the series a, + a2 +... + Gy +... =

gent and \ #0

be a number.

May +

GQ $a

Then the series

thd

tw

=

3} On be convern=l

DMG n=l

converges and

Shae = San. a=l

tal

~< We write partial sums for the series Sn

= Gy,

+ 2

+...

+ Gas

nm

oF

D)a, and

>> Aan:

a=k

ne

Gn = AG

Clearly, on = \S,. Since, as stated, the series

+ dag

+

ou. + Ady.

5) a, converges,

Le, there exists

as|

lim S,, then from

the last equality there is

woe

lim o,., such foo

that

lim a, = A-9 80

lim AS, = \ lim S,, Le, Ato

Ams

3) Ada

=

>)

On.

»

azt

a=l

Theorem

13.3. If the series

and difference, ie, (an

i:

7)

=

then their sum

fi=t

“

33dh

>

+

ba.

n=}

n=l

an

the

>) by, converge, fal

3) (an + bn) and 3) (an — bn), converge, and n=l

oy

>)a, and Ask

Seeaqtat+itae,

S=htht.th

and

ag=

(a, + by) + (a2 + fe} +... + (a, + 5) be partial sums of the series

a

#

tis

Gay i

ot bay.

3 (a, + bn),

ned

Ant

17

on Series

13.2 Operations

respectively. Clearly, gn = Sy + S,. Since by the statement of the thearem

4 #

the series

Sa

and

Sb,

nst

converge, Le, there exist

nal

crit follows from the last equality, lim o,, and that

which

holds

Jim S, and

lim $,,

ne

At

for all #, (hat

there exists

no

is equivalent 3) (ah

+

lim S, +

fim Sh,

vo

lim (S, + S.) = a0

nore

Rage wigs Bags es ent ag

which

lim o, = ate

to

ba)

=

3 On

asi

+

35 On.

n=l

»

n= a

in a like manner,

we can prove the convergence

We

the concept

of

S$ (dn — Bn). AcE

now

introduce

will use later. Delinition.

If we discard

G42

of the remainder

the first rt terms in the convergent

to bd tne

we will obtain the convergent

+ Onan t+

which

is called Ra

=

series

b Omak +

we

serics

= Vm tak

a

2a On tk

the nth remainder of the series anc

3 dn

which

=

:

Gnes + Gyar tn

of a series,

denoted

by

tk

kel

for a fixed x. The original series can >

an

=

On

+

then

be written

Rn.

as

'

A=

If S is the sum

of the series 3) a,, then the remaiider

will be R, =

ant

S—

S, for any n= 1, 2, ...

.

For example,

for the series

series Rp

=

ag”

+

. which is convergent 2—-75

ag*!

yyag"™! asl

++

for Ig! < £.

ag" **-!

its ath

+o

remainder

Sage

kml

will be the

trnt

18

{3 Number Series

13.3

Tests for Convergence

of Series

formulate Cauchy criterion for convergence of a sequence allows to the general criterion for convergence of a number series.

nt condition Theorem 13.4 (Cauchy critecion). A necessary and sufficie os

Py ay LO converge is that for any number € > 0 there

for @ mumber series

Rol

exists a number N=

N there holds

i

1, 2...

3.23

SE

tb Mapl tw

léy # nai Jor af p =O,

N(e) sucii thet for any n>

:

In terms of the partial sums Sao, and S,-1 of (15.2) as iSnea

—_

Sn

~1l

From the Cauchy af @ number Series.

x

3) ay, we can write awl

é.

(he sgcessary test for convergence

follows

criterion ee

Theorem

13.5,

then

converges,

a,

>»

ff series

fmi

lim ay = Q. fees

Pytting po = Oin Theorem 13.4, we will have ja,! < 6, which holds for

«

all a > N(e}.

The

¢ > 0 being arbitrary, we have

number

fim @, = 0. P

nw ot

If

Corollary.

fim a, ig mot equal to zero or does not exist, then

>) an

tae

=

diverges.

“@ Suppose that .

3) am converges, Then by Theorem 13.4, there niust exist nel

°

lim @, equal to zero, Gur assumption has fed us to a contradiction, hence

awe it is wrong. Therefore, the series is divergent. Examples.

(1) The number

series =

~] FOF

i

G+

fa

Neat

Fe

+ Cossphen

,

, cos OS —T a=

diverges, since lima, ge

aoe ap

.

=

lim cos = = cosO = 1 2

aint oe

e

i

0.

I

|

.

|

14.3 Tests for Convergence

|

=

|

(2) The series

DCN!

f-d+¢i-te.=

ne]

.

lim a, =

. lim (-1)"*) does not exist.

ares

note

.

.

a

diverges, since

ion for a series to conRemark. Theorem 13.5 gives a necessary condit lim a, = 0 condition the ie, verge, which is not, however, sufficient, a+ oO ea]

well. may be met for a divergent series aslS] ay -as. series

the number

(3) Consider

Sa Dus

to

it

legtgtetgte=

nel

known

series

mects

necessary

the

since

condition, .

We

series,

Aarmonic

as the

Narmonic

The

im Ato

will prove

1

.

linn ada = 77 oo

— = 0.

for which

that the series is divergent,

a, + an Gn + duet+ dnaa + Se

i

1

ee fe

n+tlo«a¢+2

1 2A

i ne ee

gn

ee

ee

a

ee

nt

Qn

| 2H

ee

1 Zn

ee

a

| re LOR 427

i

ere ne 1

1

i

ee

hn

|

nN

we will

purpose

gives

make use of the Cauchy criterion. Putting p ="

bre OA

be

J

ae Qn

er tg

i 2H

i: 2

It follows that fore < 1/2 The inequality holds for any arbitrarily large n.

y criterion the and p = ” inequality (13.2) is not valid, and by the Cauch . harmonic series is divergent. it possible to eslabComparison tests for series of posilive (erms make by comparing gent} (diver lish whether or not a number series is convergent ent). (diverg gent it with another series that is knowrl to be conver

et

Yaa,

On bo

by t by te

(13.3)

nai

8

be

tw

“,

2 Ds

= a

yt

#4

hb

pO

13.6. Let

=

Theorem

(13.4)

20

13 Nuntber Series

be two series of positive terms,

such that

On & On

(13.5) b

=

for all n. Then, if >) bn converges, n=l

diverges, then

oe

>) a, converges as well: and if >) an ne]

wed

>)by diverges as well. ani

“¢

We

form

the partial sums

Sno a

+ dy t+.

of (13.3) and

+ das

(13.4)

On = by + by +

+

bn.

It follows from (13.5) that 5, < o, for all 7 = 1, 2, (1} Suppose that series (13.4) converges, i.e, there exists

lim o, = o¢ of

ave

its partial sum. Since the terms of these series are positive, then 0 < o, < a, and it follows by (13.5) that 0 < 8, < ¢ fora = {, 2, .... Thus, all the partial sums §, of (13.3) are bounded and increase with n, since a, > 0 for ali n. Consequently, the sequence of partial sums [5,] is convergent, ie., there exists

lim S, = 5, which

implies that

m—-

>) a,

is a convergent

series.

n=i

Now, from the inequality G < S, < ¢, which holds for all natural 1, we obtain as n-* eo the inequality 0 < § < o, ie. the sum § of (13.3) does not exceed the sum o@ of the convergent series (13.4). (2} Suppose that

5) a, diverges. Since all a, > 0, then §, increases with a=]

4, and hence

lim 5, =

+.

From

the inequality

o, > S, (2 = 1, 2, ..)

ast oo

we getSe limo, = +00, ie, 3) On diverges. nel

Remark1. The theorem is valid even when

(13.5) holds not for all a,

but only beginning with some A, ic, for all a 2 k, since when we drop a finite number of terms in the'beginniag, we do not violate the convergence of

the series. ca]

Examples. (1) Examine for convergence the series

> sp AVE aol

“a

We

have

logis w+vn

2"

(3) \2

(n= 0, 1, 2, .).

cere oh

.

13.3 Tesis for Convergence

Since Ss) G ) converges, then by the comparison test the original series .

and

uO

converges as well. oe

1 et

»

the setics

for convergence

(2) Examine

m

;

Mt

#?

nal

“@

From the inequality Ina i

.

2,3,

Vator on

J

. the harmonic

>

follows lnk

l

!

iy. ‘ diverges

a (in this case

diverges

H

nek

-

for convergence

(3) Examine

:

Using the inequality sinx

series converges.

From Theorem 13.6 follows a corollary. Corollary. if there exists a finite nonzero fim

i

=

iQ

Q 0 subject to the condition L — ¢ > 0, there exists a number N such that for alla > NV

ate LE

Qn

bn

) ft)? awd

+o

converges, if the integral

"

\ F(x) dx converges; and diverges, if the integral |

= 2, x45 = 3, 4

te eM

We now consider two stepped figures shown in Fig. 13.1. The acca Q of the curvilinear trapezoid bounded by the straight lines 1,x=

Q=

7, » = 0, and #

| f(x) dx,

the curve » = /(x) will be +t

i

We then take the nth partial sum of the series

Sa = fl) + fQ) + fB) ++ fl.»

Sigates Bort mS

x=

mae ae TE

with xy = 1,2

EEE

take points

ees rages

diverges. “4 We plot f(x) and on the curve.

26

13 Number Series

i

The area of the larger figure will be

~/O

B=f/Q+fO+.+ka=S and

area

the

smaller

of the

figure

will

be

@ = fd) + fl) + /B) t+. + ft - 1) = S,-4. ¥

— FEN] ef2)

oF

f

e

Fig. t3.4 lL is seen

that @
0 for n= 1, 2, .... i

27

13.3 ‘Tests for Convergence

of ps artial sums of the and [herefore it has

Thus, the sequence [S,] and bounded, monotonous

is convergent.

Shen)

that

s « mean

Theorem 7. 9), which

serics is strictly lim Si= § (see

nel

Since f(x) > Oforx B

7 Aix dx diverges.

(2) Suppose now that

i, ihen

1

jim (fe

(se) dx = i

dx =! +00,

|

ane

;

I follows from

M=12 4)

52 |Add i

S1 fu) ciferses. b

i,

lim &, = +4,

that

An ot

and

x Ba, where ais any puynber larger

fo Remark. The theorem also holds

than

~ >

unity. of for conyergence

. . Examples. (1) Examine

n

|

“4 Here f(n) = 1/a°. The integral

we

mad

to,

|(

1

ox is known (Chap. 11) to converge

os lk

converges when p > | for p > 1 and diverge for p < 1. Hence, the series have the harmonic will we 1 and diverges when p < 1. Specialy, al p =

i

i

. series I ++ z + z too. + - Ho

NS. which

has already been shown

H

| j jae

to diverge.

(2) Examine

pr

>

series

for convergence the te

nel

“4 In this case fix) = 1/0? + 1). |The integral :

+= dx

=

:

or


0, the

-%2

it

mn

nok

iq

ye

bx")

o-

™“

+ co
) AC).

If

azl

|

the

FO

TE

1s S(x},

then

it can

be

represented as

S(x) = Sue) + Rol), where

R,(x) is the sum fn+ lx)

+ fre)

|

of the convergent

(on D) series

Fossey

ie, Ralx}

= fas

(x)

+ ta +2€x)

+

a

=

st fx(x)kratl

The quantity &,(x) is called the jath rematuder of the functional series 57 fal), fied

44

14 Functional Series

Take any (arbitrarily small} number ¢ > 0. Then if (14.4) converges exists N = N(e) such that o — on < e, and hence 1S(x) — SiGx)l < ally > N(e) and for all x € 9, Le, series (14.3) converges uniformly on Remark. Series (14.4) is often called the dominant series for the tional series (14.3).

there e for 1). func-

o

Examples, (4} Examine for uniform convergence the series > Sg azl

~4 The

inequality I cos mXle

COS FLX

on

n®

holds

for all a =

I

Si

1, 2, ... and

for all x€({—o,

+00). The

number

series

>} i/n? converges and by the Weierstrass test the original series converges A=i

uniformly and absolutely on the entire axis ~oo 0

such

that

le(x}i < C ¥ xe fa, 8). By the definition of uniform convergence, for any number é > 0 there exists a number NV such that for all x > N and for all x € [a,'6] there hotds

the inequality

ISy(x) ~ SG) < Ee, where S,{x) is a partial sum of the original series. Therefore, we will have

le(x)Sa(0) — g(x) SOX)! = g(x) ESa(x) -- SOI < Ck =e fora > Nand

for any x €

[a, D5], ie,

[a, b}] to g(x) Sx). > Theorem

14.3. [f the series

Se) = Shc) al

2; BCX) fax)

mt

uniformly converges on

t

46

I4 Functional Series

all its terms are continuous, converges uniformly on the interval [a, b] and interval. that on uous then its sum SOx) ig also contin

the interval fa, &]. Since “ G such that for alla > N

Ne)

(14.6)

< 5

SQ) - Sal and

d4.7y

ADL < G,

ISie + Ax) ~ Six

=

.

where S.C)

. are the partial sums of the series 3) f(x). The

.

partial sums

ast

the sums of a finite number Sa(x} are continuous On the interval [a, b] as are r & = 6(e) > 0 for a numbe a be will af functions f(x). Therefore, there [Ax! < 6 we will ying satisf Ax for that such fixed no > Ne) and given « , : : have

ISug(% + BX) ~ Syl) The

of S(x) can

AS

increment

AS = Se

+ Ax)

44.8)

< be written

a$

- S(x) = [S(x + AX} ~ Sa + 429]

4 [Sm(x + Ax) — Sul} + [5.09 - SOI],

whence

LAS] < 18tx # Ax) ~ Sa(x + Ax)! 4+ SaglX + Ax} ~ Spl}!

+

— Say(x)!-

1S}

his means that

~ bo

fel ait

lAs| 0 there is N(e) > 0 such that for all 1 > N(e) and all x¢ ia, b} we will have

IS(x) - Sul But

< 5 eo

-a

then

| \ (i) dt —

| S(t) a|

| aa Me

ast

= 2 Udx) — fo)

ant

= tx)

ne]

wx E

- Sho)

= $(x) ~ S(29).

n=l

But since the function (x) is continuous as the sum of a uniformly convergent series of continuous functions, then by differentiating

[ o(f}dt = S(x) — Sx) we will get

| or

at]

=S'(1},

°

Boe} = KO~ nxt

As!

be,

o(x) = S’(x)

50

14 Funcuianal Series

Pexercises

the convergence

Find

intervals

on

x

Loma

=

a

2

Loe

“Toyl

é.

ZY -

ih

(ht xp

é. » | in a

li. »

.

>|

x H fan

x

S|

9.

Ing eee,

naj

fs}

Az}

Ase

H=l

nal

ral

He

;

| In” x.

4

.

2

4,

3. > /e

i>,

series below:

for the functional

i

Sift si

nad

Using the Weierstrass tesi, prove that the following converge uniformly in the specified intervals:

functional series

oo

ae]

/

Ҥ

cong a>,~fnintn

+ (4 ~ ¥7}

erg DEKE

~

-

“|

+ 5), -igx jn(

a

Answers i-bEc

Bettecaxecen b

§G Ix,l. According to the treatment above it must converge at x = x2, since

Ix21
hah, then it will diverge in the infinite intervals (—, ~ Ix!) and (layzt, +00) as well. The series| diverges

lop

4 ~[ee}

i

$44 + fd

Fhe series

converges

absolutely

e 0

' ;

$5 a

ep

|

.

The series diverges

4 i]

“~

x

Pig. 15.1

It follows from the above that two points exist on the x-axis (symmetri-

cally about O), which demarcate the interval of convergence from that of divergence (Fig. 15.1). Theorem 15.2. There is a unique number R > 0 Jor every power series ot Cnx"

which

converges

n=O

not only at the point x = 0, R

being & such

that

the series couverges absolutely when |xi < R and diverges when tx| > R. “a Let & be the set of all points x at which the series converges. The set & is bounded. The theorem states that there are points on the x-axis at which the series diverges. We take one such point, say x,. Abel’s theorem

states that for any x €é’ we have Ixi < (x,}. However, in a set bounded above there is a unique upper boundary sup ix}. Suppose sup lxf = R. XEG

Since by definition R> 0.

the series converges

not only at x = 0, we find that

We now take any x for which ix! < R. By definition of the upper boundary, we can find x» €é’such that Ix! < Ixol < R, whence as follows

15.1 Abel's Theorem

from Abel’s

theorem

53

the series must converge

absolutely for the chosen

x. [any x is taken for which Ll > R, then v¢e" Consequently, the series we ‘ must diverge at this x. > a hus, the region of (absolute) convergence of a power series 3) cnx" a ned is the interval (~R, R) centered on the origin. Definition. The interval (-R, R), where R > 0, at every point xé

(-,

R) of which the series converges and at points such that

Ixl > R

the series diverges, is called the convergence interval for the power series LJ

>

Cax® The number R ts called the radnis of convergence of the series.

=}

At the ends

of the

x = R the power Remark.

interval

(-&, R),

ie,

serics either converges

The

power

series

at the

points

x =

—R

and

has

the

or diverges.

>) Cx(¥~ x0)"

where

xy #0

n=f asd

same radius of convergence as

(xo ~ Ry xn + R). When

>) cnx", but its convergence interval is

neo

a finite Limit :

lenag!

iim er

= “L,

where 0
56o!") di = 1

ai

=

xt

a= it

for any x in the interval (—R, R). ~t Any point x in the convergence interval (—R, R) can be included in the interval [—a@, a], where 0 < a < R. On this interval the series will converge uniformly and, since its terms are continuous, it can be integrated term by term by Theorem 14.4, e.g., from 0 to x, where 0 < Ix} < R. Then x

mm

oo

x

| (Qe) d= 2 fen 0

n=O

2

atl

ar =

thea G

ork n+]

Q

xe (~R, R).

n=O

We now find the radius of convergence RX‘ of the series obtained, under the condition that there exists a finite

lim lc,l/lc,4,1

= R. The

radius

FE + >

of convergence

A’

will then

be

Cr R’

=

[iim .-

Ay

=

aon | Cree

iim

nt2

noo

V+

“ Feat

1 bengal

atl

= tim

2 tim

norco ft

I

n=

lel oR

eR,

Mpa il

Thus, integration does not change the radius of convergence of the power series, Differentiation of power series. Differentiation of power series obeys ihe following theorem.

Theorem

15.6. The power series SQ) =

3) tax” can he differentiated and

term by term at any point x in its convergence interval (-- R,'R), R > 0, and

S'(j = ( 3 cnx") anf

=

5 Fax", w= FE

58

«@

LS Power Series

of the series

fet ® be the radius of convergence

(15.6)

o) eux" G

rie

be the radius

R’

and

of convergence

of the series

(15.7)

Sonex, Aas

let there be a finite or infinite limit

and

len |

lim aay fit

Then,

, &

lene

t we will obtain Do ii lncal R= Tin + Dena il

fim ot a eR R. e lita (1 + ari) gace (nail +] ne

Thus,

&’

BR. Denote

=

by eC)

the sum

of (5.7). The

series converges

an the interval ‘ R, RB). Series (15.6) and (15.7) converge uniformly on a], where O< a< R. Ail the terms of (15.7) will then any interval [-@, be conlinwous; they are derivatives of the corresponding terms of (15.6). ef}, By Theorem 14.5, we will then have o(@) = S’ (x) in the interval [- a, & R. < a since and hence in the interval (- 2, R) as well, Corollary. A power series 3) ¢ax” may be termwise differentiated any A= O

rlumber of times at any point x in its convergence interval (—R, R), Le. its sura Six) has derivatives of all orders at each point xé (—R, A), viz, SMa)

=

» afa-

Ooo.

(a

Dex" *

k+

...)-

1,2,

(k=

nek

The

convergence

radius of ‘his series is equal to the convergence

radius

2

of the original series S(x) = By applying derivailves

the

theorem

3) cnx" to

the

S' Od) = cy + Qeax + Segx* e+

series

Hep

containing

the

first-order

eax,

be ae |

We get and then to the series containing the second-order derivatives, ete.,

the formuta for S&x) for any x.

is said to be expanded into a power series

Definition. A function f(x)

.

i.

on the interval (-- R, R) it the series converges

+) cx"

on

the interval

|

me @

and

29

Taylor’s Series

15.3.

oo

a

cere

_

15.3 Thylar’s Series

its sum

ie.

is f(x),

Dp en",

fos =

(15.8)

xl -R, R)

Aa=O

assumed

the interval being

into a point.

not to idegeri¢rate

nt expanWe shall first prove that the function cannol have two differe

sions into power series of the form (15.8). ed into a power series Theorem 15.7. if a@ function fi) can be expand expansion is unique, 2, (15.8) on the interval (- &, R), Rv 0, then this defined hy tis sunt. the coefficients of the series (15.8) are uniquely 0, in the interval (- &, Ry, Bo -@ dct the function (Go be expauklable into

4 power

serics

can be done

By differentiating this series # times, which (~R, R) due to Theorem 15.6, we get

1x 2x de...

PPOs

in the interval

4 (am Nae

0. KH

42x39

(15.9}

GA be

fO) = coh ON + ONE

Dee t Den iy +.

When x = 0 we obtain

0) = 1K 2X3...

KH

Dace

or PPO)

= ate,

v=

0,1,

2.

whence

7) Cr

(15.10)

ae fee (0)

nl

given that /“(0) = /(0), OL = 1.

in (15.8) are The coefficients c, (tt = 0, 1, 2) ...) of the paweer series

thus uniguely determined by (15.10). Remark.

Uf f(x) oo

is expanded

fo) = Si ealx — x0)", n=

in powers

of the difference x ~ Xo, Le,

XE (0 — R, x0 +R), R> O,

64

15 Power Series

According to Theorem 15.9, sin x can be expanded into a Taylor series in x in the interval (—0o, +9) that converges to it. Since

sin

=

f(0)

7

=

>

(-I)r

for

a= 0, 2, 4,

for

x» =

1, 3, 5,

we will have .

SINK =X

3

Get

x

8

ap

SS

yortl

we. + (-1) Gran

*

vee

nt

(-1)"

Grae D:

(15.15)

n=

‘The radius of convergence is R = +00, (3) f(x) = cos x. In a like manner, we obtain _ x2 x4 cosx=

1-3

= Dy .

ayn

+47:

+ (~1)" Seay +

ee, R= +0, xO

(15.16)

-

asf

(4) f@) = (1 + x)", where x > —1 and o is any real number. The function obeys

(lL + x)f’ @) = af)

(15.17)

and the condition /(0) = i. We will look for a power series, such that its

sum S(x) would meet (15.13) and the condition $(0) = 1. Let

SQ) = 1 bein bax? + gxt to.

ak +

(15.18)

Hence

S’ (x) = 1 + 2eax + Jeux? 4 0. t+ meax™ he ll, Substituting (15.18) and (15.19) into (15.14) gives

(E+ xfer + 2eax + Besx7 +... + neax™ + 20.)

= al + crx + cnx? + 37 + 02, + ax +.) oT

cy + (er + 2e.)x+ Qe + 3e3)x7 +... + [rte + Gt+ Dene ilx™

=

+ OX + ocx?

+...

4... FOG

+o...

(15,19)

65 15.3 Taylor's Series

cosa,

C+

22

= act,

GF

«0, Henk

Qe, + Fey = 02,

Hence

an

= 1+ art

oes

= On,

nl

2)... fa-at)

pees

into (15.19) gives

these coefficients

Sg

Den+1

ga a= Ma=2),

c _ a(a — Ifa ~ wer Ca =

Substituting

either side gives

of xon

powers

the coefficients at the same

Equating

ofa 7 ~ 1) 2, ofa - iIa= 2) ys (a-atlyy

2 De > 2 +... + He

ag.

We find the tadius of convergence of (15.20) for the case where a is not a natural number. We have afa- Il)...

tebe

+ ie A) ean

it

R= lim poy = lim ay fim tS now

le

(a@-a+ tf

Rn

tim Al

nooo

{n+

1!

rai

poet = at

ti

Series (15.20) thus converges for Ix! wet

.

¢

r

{ cos mx cos mxdx

+ by \ COS MIX Sin HX ax)

-*

.

*

The trigonometric system being orthogonal, all the integrals on the righthand side are zero, save for one, which corresponds to n = i. Therefore, ¥

n

{ FO) cos nix dx = an \ cos? ax dx = dint,

~

7

hence

f, 2,

Gn

| f(x) cos nxdx

Gin =

...).

—F

or

Likewise, multiplying both sides of (16.2) by sin mix and integrating [rom tom, we obtain , r

rT

| (9 sin mex dx = bun { sin? sux dv = Dm,

—-

-F

hence

bia = L [ye

sinaaxdx

i, 2,

(at=

...).

T Y

Now let f(x) be an arbitrary function with period 24 and integrable

on the interval {—2, a]. We do not know beforehand whether or not it can be represented as the sum of a certain convergent trigonometric series. But using (16.3) we can calculate a, and &,. Definition. The trigonometric series a

ao

x +

:

>

an cos AX + Oy, sin 1x),

n=l

whose coefficients a9, da, and by are defined through f(x) by the formulas r

an = y | 709 cosnxdx

(n=0, 1, 2, ...) '

bn

ai

~

=-

| fl) sinnnxdx i

(1= 1,2, 2),

78

16 Fourier Series §

is called the Fourier series of f(x), and @,, ba, defined by these formulas are called the Fourier coefficients of f(x). Each function Oo integrable on [-—-2#, w] corresponds to its Fourier RCPICS ws

fi~

* + >; (a, Cos my + By sin Ax),

(16.5)

nowt

ie, & itigonometric series, whose coefficients are given by (16.3). If, however, we only require that /(x) be integrable on [~, a], then, generally speaking, we camnot replace the symbol of correspondence in (16.5} by the symbol of equality. Frequently, a function f() needs to be expanded into a trigonometric series defined only on the interval [~ «x, z] and consequently it is not periodic. & Fourier series can be written for such a function because the coelficients of the Fourier integral in (16.3) are calculated for [—a, w]. Hf however, the funetion 7(x) is extended neriadically along the x-axis, ie., over the interval (— oo, +0), then we get a function F(x) that has period 2y and coincides with /G@) on the interval [-—#w, #], le, FG) = fo for allxin [-a, a]. The function /() is called the periodic extension of #(x}. Foo does not have a single value at ws ta, +3, +5m, ... or at the points of discontinuity of f(x} in [--#, al.

The Fourier series for Fc) will be identical to that for f(x}. If the Foarier series for f(x} converges to F(x), then the sum of the series, being a periodic function, will yield a periodic extension of /Q) on [-— a, a} over the whole of the x-axis. Thus when considering the Fourier series for /(x) defined on [—-, a, we are also considering the Fourier series for F(x). It is sufficient, therefore, that the tests for convergence of a Fourter series be formulated only for periodic functions.

16.3

Sufficient

CondHions

for the Fourier Expansion We will now

of a Function

find the sufficient test for convergence of a Fourier

series. Definition. A function f(x) is called piecewise monotone on the interval

(a, &], U the interval can be broken

up by a finite number

of points

@Ox, Seo... < xy. < & into tervals (e, 1), Og, 2), -.., Oe-1, in each af which f(x) is monoione, Le, it is cither nondecreasing or nonin-

creasing (see Fig. 16.1).

79

16.4 Sufficient Conditions for Foutier Expansion

piccewise monotone im the Examples. (1) The function f(t) = * is be broken up into two intervals interval {— oo, +0) since the interval can

(- 09, O)-and (0, +),

creases. Fy

it inin the former it decreases anc in the latter

‘

one The function f(x) = cos X is piecewise Monot

on the interval

into two intervals (- a, 0} and [-7, a], since the interval can be divided ses from ~ Plo & i, in the sccond (0, #), in the First of which cos x iricrea it decreases from +1 to -1.

i 1 t I

~

Sih

nee

:

4

Bo

4 “a

xO}

t

7

tf

i

o

t

i

i

!'

":

7a rene

oe

liv. 162

Fig. i611

hounded on fa, 4] (Le, Hf the function f(x) is piecewise monotone and tinuities in Unis interval, m < f(x)

f(x) (Pig. 16.2). period Im is piecewise then its Fourier series af the series

(a cos nx + By sin AX)

as]

obeys: (SQ)

=f)

f-ra

.

2

,

cos

lal

i?

with

o0

16 Fourier Series

wi

a i

ho

where

al

mt mgeny u

(COS Mor

nx

¢

| .

)

co

for

a=

J, 3,5, ...,

0

for

a=

2, 4,6,

....

Substituting these values of the Fourier coefficients into the series gives

“cos ££ pat. for

Af (

cos 24%

fy.

[4

at

fy

cog 3% st

te,

-fexg lh» Note one important property of periodic functions. Hf f0o has a period T and is integrable, then for any number a we will

have

.

ast | fG) dx = #

Poros | fod dx. G

ie, the integral over an interval of length T has the same value regardless

of the position of the interval on the number axis. Indeed, se

é

r

.

\ JGodx = { fioddx + st

a

a+T

[ [0d dx.

r

91

16.6 Series for a Function with Arbitrary Period

7,dx = di, This gives

In the second integral we change the variablex = 1+ a

a

a

ant

\ -f(jdx = | fet Pdt= \ {dt = \ fix) de, a 4 a Fr

. we

hence

r @ as. r a+T | SO) dx = | Sax) dx + | Jwdx = \ fijde + | faydx u

a

Gi

@

8

T

=

\ fixdx. %

the areas hatched Geometrically, this property implies that if f(x) 2 0 Fig. 16.10 are equal.

in

Fig. 1610

Specifically, x

for f(x) with period) T= 2" at a=

\ fd dx =

“~F

—w

we will get

iy

{ fix dx.

6

Examples. (1) The function /() = sin’? x is a periodic function with 7 = dex. Therefore, without even taking the integrals we can state that for any a we will have atin

{ a

7

2x

sin’ xax =

\ sin’ xd =

o

:

j sin’ xdx = 0,

-*

since the function is odd. The property implics, in particular, that the Fourier coefficients of a periodic function f(x) with period 2/ can be worked out by

92

16 Fourier Series

a+if

dn = ;

fix) cos "2 dx (a = 0,1, 2 ..),

(16.6)

fo) sin 7 dx (n= 1,2, ...,

(16.7)

a

a+27

Dn =F where

a is an

sin a turn

arbitrary

real

number,

since

the

functions

have period 2/, and products of functions with

be functions

with

cos =

and

period 2/ will in

period 2/, »

. : (2) Expand the function /(x) =

{

a-~x i

for for

O 0 js called the step size of

the

mesh.

Since

by

definition

Aer bay YO)as

the

derivative

dy/dx

is

the

limit

of

0, then substituting this ratio for the derivative

we, instead of (17.14), will obtain the difference equation (Euler differenc e

scheme)

Foe

Nhs 1—

¥e

HL

Ie) (=D,

(17.16)

or

Yer. = Ye + Af (es, ve)

(kK = 0, 1, 2, ...).

(17.17)

By iteration we find yp = y (xy), remembering that by (17.15) vo = ¥ (x) is a known quantity. As a result, instead of the solution ¥ = y(x) we find the function Je = y (Xx) Of the discrete argument x% (mesh function), which yields an approximate solution of the problem (17.14)-(17.15). Geometrically, the desired integral curve y = y (x) that passes through point Mo (Xo, Yo) is

replaced by an Euler broken line Mog Af; M)... with vertices at points Mr (xx, Ye) (Fig. 17.5). The Euler method is a single-step method, which, to comput e a point (4h 21, Jeo a), requires a knowledge of the previous point (xz, ¥) only. To

estimate the error of the method in one step of the mesh we expand the exact solution y = y (x) in a neighbourhood of the mesh points x = xy by

the Taylor formula

¥(%e41) = Ye +h) = yOu) +’ OA + OU) = Y(t) + Af (cK, Ye} + OCA).

“

(17.18)

117

.

17.3 Approximate Meibods of Integration

Comparison of (17.17) and (17.18) indicates that they coincide up to the terms of the first order in A, and the error of (17.17) is O(A7), We say then is of the first order.

method

that the Euler

at

o

¥

ela

My, Yo

| My ( Xtal

a

—

Y=Yq* OF Xy sth)

ty

Ey=Xgeh

x

Fig. 17.5 the Euler method,

Using

Example.

dy a

solve the Cauchy

problem

_

x,

p(O}=2

on the interval 4

[0,0.5] with step A = 0.1, In this case, f(y y} = ¥ — x, 40 = 0, Yo = 2. Using (17.17), we obtain

Jeai = Ye t Mf Xe Ye) we find consecutively v=

Pe t

yo =v and

so on. We tabulate

fev. yo) = 2+

t Afi

yy)

1

= 224+

- O) = 2.2;

0102

-O)=

2.4]

°

. ihe

results:

&

ak

re

0 i 2 3 4 5

0 OF 0.2 0.3 04 0.5

2.0000 2.2000 2.4160 2.6310 2.8641 9S

-

flv

hflxa, Fed

ye}

. 2.0000 8.2100 0.2219 0.2331 0.2464

2.0000 2.4000 2.2100 2.33510 2.464]

.

/

Exact yee

: :

-

: .

solution txt

2.0000 24,2052 2.4214 2.6499 2.8918 3.1487

f

if we consider

the Caucay

problem

Hea y-x, yO! & > 0, we will obtain v= E+ 4, on any interval [0, a] with any step ens that the Euler broken line “‘straight y= b+ 2A, ys = i+ 3h, etc, so soluexact the y = x + 1, ie., with out” and coincides with the straight line lem. tion of the Cauchy prob method is fairly simple but inacQQ) Runge-Kutta method. The Buler me,

licating the difference sche curate, Accuracy can be improved by comp

by the Runge-Kutta method. 4)-(17.15). We will again tabulate We return to the Cauchy problem (17.1 the solution y = »y OG) of UT14 at the approximate values Va, Fa, » 0 is the step ‘size of the mesh. ities ¥i+1 are computed by the in the Runge-Kutta method the quant

eg.

following scheme:

vier = FE+ 4. (ky + 2k, + 2ka + Ka), where

ky = ftxis Pd: ka =f («

+ 4, Yi + Ue),

Ra

+

I =

i

€

“

Ak Mi

*

SY)

3

kee = FOG + By ye + Fey).

L714 Seme

Equations

A, differential equation

Integrable

by Quadratures

if is said to be integrable by quadratures

obtained as a result of a finite its general solution (generat integral) can be integrations known sequence of elementary operations with vely relati are ions equat Such of those functions. able integr not is y* x? = yp" ple the equation the first some kinds of differential equations of ratures. Separable

equations.

AQhdy=fpojdx

Tquations

functions and few in number. For examby quadratures. Consider order integrable by quad-

re

of the type

:

* (17.19)

17.4 Equations

HY

.

Integrable by Quadratures

tere equations with separated variables, are called separated equations, OF s. ment argu ctive ions of respe fi: 0),f2 @) are known continuous. funct the equation. Then if we substitute

Suppose that y (x) is a solution of y, and if we integrate it we will y (x) inte (17.19), we will obtain an‘identit

“find the finite (nat differential)

equation

(17.20)

(A Oidy = \f (pdx t Co,

pf (17.19) (C is an arbitrary constant). which is satisfied by all the solutions is a solution of the differeetial Conversely, each solution of (17/20) turns fujnction y (x, when substituted, equation (17.19). Indeed, if some y O) thal shows ity ident this rentiating (17.20) into an identity, then diffe ion. equat al renti diffe this of ral integ also satisfies (17.19), ie., is the general ated equation, Hf we write if in separ ¢ is 0 = ydy + xdx nce, insta For both parts, we will find the general the form ydy = ~xdx and integrate :

integral of the equation: vty An

equation

sic

of the form

721)

AarerOrdr = fea) 4

can be factored into components where the coefficients at the di Fferenttials called a separable differential equathat depend only on x and only on » is 4 0) # 9, reciuce it to a separated tion, since we can, by division by @lO) 7 equation:

J

ge Mg AO 0) Aw"

“4

Pyxdx= OF Example. Integrate the equation (1 4 yy {dividing both sides of the equation by Gt

x") yep. x") # 6 pives

equality, we will ‘get If then we integrate both sides of the resultant

ind +7) = in(l +37) +,

I +

“4

Ty

XS

2

2.

a loss of solutions that Notice that division by ¢1 (v)f2 (x) may lead to " turn gy (¥)fz (x) into zero. dx/x. = dy/y gives dx y = For example, separating the variables in.x dy

(here C can asIntegration yields In by = In |x| + In IC], whence y = Cx

divided by y sume both positive and negative values, but C # 0). Having the general we have lost the solution y = 0, which cai be included into 2 0. = C value the solution y = Cx if we allow C to take on

then we should If we assume that x and y may both enjoy equal rights,

120

First-Order Ordinary Differential Equations

supplement the equation dy/dy = y/x, which makes no sense at x = 0, by

the equation dxAdy = x/y, which has the obvious solution x = 0. In the Renesascase, along with ihe differential oa we should

equation

= f(y)

(17,22)

also consider

= fi

¥);

(17.22"}

where fi (x ¥) = I/f(% y) in so doing, we should (17.22) makes no sense, and (17.22') is meaningful.

use

(17.22'},

where

By a change of variables we can reduce some differential equations to spare uations Consider the equation of the form 4

-- = flax + by + ©),

(17.23)

where J{z}is a continuous function, a, b, and care constants. A substitution = ax + by + c yields the separable equation dz

Go

=

+

Ome

=

at

bf{z},

hence dz

= dN.

at bf) : . Integration gives

~~

dz | ati

=x+C.

find the general integral of (17.23). Examples. ({} Integrate the equation ~¢ We pul z =x + ¥, then dz

“de

dy ={+—2

dy/dx = (x + »)*.

dz “t=1+427,

o,f

“dx

: Changing z for ax + by + c, we

hence

dx

ae

jee

dx.

Integrating gives tan”! z =x + Corz= tan(x+ C). Substituting x + y for z, we obtain the general solution y = tan(y4+ C)-—x »

(2) It is common knowledge that the rate of radioactive decay is proportional to the amount x of the radioactive substance that has not yet decayed. Find the variation of x with time ¢, if at f = = fo there was x = xX» of sub-

stance. “@ The process is described by ‘the differential ‘equation dx

ao

—

Om

.

.

:

:

Sone

’

toe toe at

ee a

~

Ae

4)

i2}

17.4 Equations integrable by Quadratures

Here k > Ois the decay constant, which is assumed to be known; the minus sign is to indicate that x decreases with f. Separating the variables in (+) and integrating yicids

wt

Inj] = -At + InfC,

x= Cem

From the initial condition xc = x=

= Xo we find C = xgeke, therefore (##}

xpe hE O%),

Any process (not only radioactive decay), in which the rate is propor tional to the amount described by equation The equation

3

dx —-=kx,

of substance (+).

that

has

not

yet been

involved,

+a (+94)

k>o

which only differs by the sign on the right form (*), describes a multiplication process, c.g., the multiplication of neutrons in chain reactions or the multiplication of bacteria on the assumption that the rate of their multiplication is proportional to the available number of bacteria.

Equation (+++) subject to the condition Xr +f = Xo has the solution x() = xpe*"") which, unlike the solution of («*#), grows with 4 The equations (+) and (##*} can be merged to yield

ay

dt

-sky

k= const.

{++**)

This equation is the simplest model of the dynamics of populations (the multitudes of individuals of one species of plant or animal organisms). Let y(¢} be the number of the members of the population at a time ¢. If we suppose that the rate of the variation of the population is proportional to the size of the population, then we arrive at equation (**+**). We then put k =m — nr, where #7 is the coefficient of the relative birth rate, and nis the coefficient of the relative death rate, then & > 0 for m> aA and

k 0 the Bernoulli

Remark.

;

(*) we will then: derive

has the obvious

soluuion

.

Q. we can make use of the subatituTo integrate the Bernoulli equation is any nontrivial solution of the cquation ¥ (x) = u(x) vO), where v (xy) ion of

ys

tion

= 4,

v' GQ) + pry

is

u(x)

and

defined

as

the

solut

Uva (xp.

dome du “ae = TOY

ulli equation Example. Find the solution of the Berno — yptany=

yi

“i We y Gey =

-- y? cos Xx.

form in the equation of the yx) the solution seck gives tion equa nal origi the Ox) v x). Substituting y = uv into

-uvianx =

~H v? cos XxX,

u'v+ (vi - ytanxyu =

—y? v? cos x.

viv duv' or

nonzero solution of the equation We choose v (x) such that it will be lsorne

py’ — vtanx = 0. We integrate it: dy

wena

Qe

sin x

x,

=

:

[32

- First-Order Ordinary Differential Equations

Since we are interested in any particular solution, we put C = 1, Le, take v= I/cosx. Then for u(x) we will get the equation t

os

ue,

litegration gives u(x) = I/G@ + C).

The general solution y(x} of the original equation is given by

yO=uQ)

vy) = EE:

Eexaet differential equations. The equation Aft,

dx + NGG

dy = 0

(17.36)

is said to be an exact differential equation if its left-hand side is the total differential of a certain and y, ie, Ms

yidx + NOG

function

w(x, y) of two independent

dy = du = ou

dx + +

variables x

ay.

Here u(y FY) = C will be the general integral of (17.36). We assume that the function M4 (xy, ¥) and N Gs y) have continuous partial derivatives in y and x, respectively, in a simply connected region D in the xy-plane. Theorem 17.4. The necessary and sufficient condition for the left-hand

side of (17.36) to be the exact (total) differential of a function u(x, y) of two

independent

variables x and ¥ is

aM _ aN ay “i

(17.37)

ax

Necessity. Suppose that the left-hand side of (17.36) is the exact differen-

Hal of uty y), Le, ate Mix yjdx + Nix yidy = du = “ay

du dx + 3y dy.

Then M = du/dx, N = du/dy. We differentiate Mf with respect to y, and NV with

respect to x

aM

ul

ay

Since

dy ax’

the mixed

OM ago

_

ON ax

axay’

derivatives are equal,

aN Ox

This proves the necessity of (17.37).

17.4 Bquations

133

by Quadratures

Integrable

Sufficiency. Suppose that the condition (17.37) is also sufficient and that dv = At (x yy dv + NOY) dy, or construct

find w(v ¥) such

due

ay

-

ot

aa

au eo

iy

AM PY (x, co ¥),

=

|

N(x,( y)

a.

7. (17.38)

.

‘First of all we find « (x, y) satisfying the first of (17.38). with respect to x (assuming y to be constant) gives

(17.39)

dx + oO),

\M (x

“=

Integrating this

where p(y) is an arbitrary Function of y. We select » (y) so that the partial derivative of u given by (17.39) with respect to y would be N(« ¥). It is always possible to find such a function ¢(y) subject to (17.37). Indeed, from (17.39),

ce Z\Ma nae +9 Q). Equating

the right-hand

yg w=

NOY -

side of this to N(y, y) gives (17.40)

¥) dx.

Z\ue

The left-hand side of this is independent of x. We will now see that, provided (17.38) is satisfied, its right-hand side does not includex either. With this in mind, we will show that the partial derivative with ‘respect to x of

the right-hand side of (17.40) is indentically zero. We thus have

_an_ Oxal

but

4, |mar|

a

fa

Ox

ax. 3

|orae|

_ait aN _

2|

-{was],

a mes yodx = M(x, ¥), fA ax

hence

-

aN

"axo[v—2.[mas| = an

aM

By =

Now, integrating (17.40) with respect to y, we will get . eo

{I

- Z| marl

av +e

where C is the constant of integration.

Soe

Substituting this into (17.39), we

-

Equations

Fisst-Order Ordinary Differersial

{ad

arrive af gic desired

+

| ara

=

wiv)

function:

dy+C

fy \war

\ ly -

vdy whose exact differential, as is easily verified, is Ad Gy 3) dy + N&y equaing integrat This procedure of constructing « (y, ¥) is a method of tian (17.36), whose left-hand side is an exact differential. Example. Check that

(x)

e~tex — (2y + xe “¥ydy =O is an exact differential equation, and integrate it. —(2y + xe -7) “§ In this case Af = e@-!, Ne

aM

we

Sy and

ny

aN

_

gy

aM

GS

Renee

eT

aN,

_

ai

so {*) ig an exact differential equation. We now want to find wu (see (17.39):

w= (MQ, y)dx + eO) = le ~¥dx + @ GF} or

(re),

xe we + oO),

w=

Finding du/dy from (ox) and equating du/dp to N(x, y a we obtain —xevy

be’

(ype

Gj =

—2¥,

and

@

Thus,

eby= Substituting

-iy-

vec.

~p+C,

C= const.

this into (#*) gives

xer- js? = C,

ie, the general integral of the original equation.

find

to

possible

sometimes

is

ETF,

hence

uaxey—yPtC) lt

= 2pm

a

Be

function

p(x, y)

such

that

aif dx + pNdy will be an exact differential, although Mdx + Ndy may not

be one.

it can

Such

be

a function

shown

that

«(x y) is called an integrating factor.

for

the

first-order

equation

M(x y)dx +

with Aft, y) and N&, ¥) subject to certain conditions there Nox y)= 0 condition always exists an integrating factor, but to deduce it from the

en.

a Suey

tial equation, which

in the general case means to integrate a partial differenag 4 rule is a more difficult task.

135.

__.

17.5 Riceati_ Equation

the linear differential equation Problem. Find the integrating factor for

dy _ te * pQ)y = 9@)..

ffint: Seck the factor in the foym pp

a?

17.5

(y).

Riccati Equation The equation

a7.4t)

Ba g(x) + play + 00",

functions, is called the Riceali equa where q(), p(X), and r(x) are known it is integrated by separating the tion. Vf p, g, and r are constants, then variables

dyfn

KA,

| Gt py + ry

and when ¢ (4) = 0, it is the HerWhen r(x) = 0, equation (17.41) is linear, (17.41) is not integrable by noulli equation. In the general case, equation quadratures.

Riceali equation. We will now discuss some properties of the of the Riceali equation, solution Theorem 17.5. Given one particular

ity general solution can be found by quadratures. y = sy (xy) of (74D, ~¢ Suppose we know the particular solution

HW # FONT 09.

wid = a) + PW

then

(17.42)

a new desired function, we oblua, Puttingy = y1 CO + 200, where z (Xx) is

by (17.42),

c)z= rO9z.

ad: — (pO) + Wy oo

Thi

equation,

is-the Bernoulli

Example

is iritegrated

which

Integrate the Riccati equation

|

et bey yn? 4 depe er

if we know its particular solutionyi = ~@

Putting y = e* +z,

dz _ ya

“ie = zg’,

we will have

hence

for z(x)

1

z= re

The solution of the original equation will ,be

yO) ae

bai

Cmx

by quadratures.

£46

First-Order

A

Ordinary

Differential

special case of (17.41) is the special Riccati equation

(>. 0),

v

dy B+ ay? 2 = bx" ix

where

Equations

a, 6, and

«

are

(17.43)

constants.

At a = 0 we have dy/dx = b ~ ay? and the equation is integrated by separation of variables.

At a=

—2 we get dy/dx + ay* = b/x’. Setting y = 1/z, where z is a

new unknown

function, we get

hence

+=

~ 1), and a zero of arder at ~ 2 or higher of p(x) GL at > 2), thert there extsts at feast ove nontrivial solution of (8.82) in the form of the sum of the generalized power series + weep woVth bu baal - wt p(x) = dole ~ de)? + ale where o is a real manber,

18.12 Bessel Equation.

no? necessarily an integer

Bessel Functions

An equation of the form

pt exp +O?

(18.84)

wy = 0,

has where pis a real number, is called the Bessel equation. This equation equathe in derivative highest the at coelficient (the 0 = a singularity atx for tion vanishes at x = 0). Comparing (18.82) and (18.84) indicates that

»*, since x = 0 is the Bessel equation po(x) = x, pix) =X a(x) = a zero of the second order (#1 = 2) of the function po{x), is a zera of the p.(x) Gf first order of the function p(x), and is no zero of the function of solution a exists y # 0). Therefore, by virtue of Theorem {8.17, there ({8.84) in the form of the sum of the generalized power serics

yO) = x’ Soax*,

(18.85)

a #0,

kad

where ¢ is the characteristic exponcal We rewrite (8.85) in the form

lo be determined,

‘s Soeex**?

yeh =

kad

and

find the derivatives

wos

Skt

a) axe tent,

kel

yt a Ves ak +o ~ Yaex a

ked>

kbar d

'

Substituting these into (18.84) gives

AR et ok to = Yatton? ,

fad

$x

ke

(kh + oh apt tt

OH

yy 3" xk? =

Qo

0,

491

18.12 Bessel Equation, Bessel Functions

if we then equate to zero the cocfficients at whl

wt Ko,

we will

)

(886)

get the system of equations

a

x

[o* ~ v*]ay = 0,

xi

[Ca +

ver?

[pe ean

atk

[(o

1)’ —- y?] a

(k= 23,

eda ta =O, vy] a

-

ky

+

= 0,

+d

=O,

) it follows that gi — y= Since a = 0, then from the first of (18.86 qa

Now

0, oF

kp.

the second

from a

of (18.86) we will have

= 0.

We will first take the case of ¢ = » > 0. We rewrile , the Ath (k> 1) in (18.86) in the form

equation

vo

k+

(ot

+k

- vy dg +. de 2 = 0.

to determine ay in terms OF a - 2 Trom this we derive the recurrence formuta mee

FR

Ci

(gtk that a;

Considering

tik -2 em

tvfotk-¥) = 0, we obtain

fan. 4 2 0. On the other through the previous one =

Gam

from

this that a;

= U and,

coclficieat

hand, each even by the formula

can

in general,

be expressed

et — ¥)'

“(oe dat + vo +

i

|

Or, Since o = », =

ffm

If we apply through

be)

“50

this formula

several times we will be able to express dam

a i

pe

=

EX ix (+ a

ay

a=

.

“FRI TDF “wax =

ce)

pe eg

ZC + Dt

+ 2)

192

18 Higher-Order

Differential Equations

Or, in general, ’

Tim Ts

We

now

a

( -H . 2” nlp

+ DO

+ 2). _(

+ my

substitute the values of the coefficients

yu(x) = as (: ' y=

x

into (18.85):

xn

ey pom e ye OS =).

.

~ t)

(18.87)

mo.

.

It can easily be verified that the series on the right of (18.87) converges in any case on the positive x-axis and defines there the function yi(¥), Le,

a particular solution of the Bessel equation. Consider now the case when ¢ = ~». If » is no posilive integer, then we can write the second particular solution that is deduced from (18.87) by the change of » by ~» (in (18.84) » appears evenly):

vax)= dox™ (6 deo

ett y"

nlp

+ Kovac x

mel

em):

(18.87°)

(if » is a positive integer, then the solution (18.87’) is no longer valid, since beginning with a in (18.87') will be for all values ofx dent. Really, their

certain number one of the factors in the denominator zero.) The series on the right of (18.87’) also converges > 0. The solutions ¥4(x) and j2(x) are linearly indepenratio x?

POD

nw PCP ED

vile)

x

(n+

41)

is not constant. For our further discusston we will need some of the properties of Fuler’s Y-function. The latter is defined as follows:

P(p) =

\ x’ le-*tdx,

Rep> 0.

a

Integrating T'-function

by

parts

we

obtain

the basic

functional

equation

Pip + 1) = perp}. Since

T() = 1,

then

T(2)=1-P0)=

for the

(18.88) 1,

general

1G) = 27@Q) = 2!,

and ,

Tint

ant

(n=0, 1,2, ..).

in

193

18.12 Besset Equation. Hessel Functions

It can be shown that °'(i/2) = Va. Using the functional equation (18.88) we can derive the gamma-function for negative values of the argument. if we represent (18.88) in the form P(p) = P(p + I/p, we notice that for small p we have I'(y) = l/p. Similarly, if 2 is a positive integer, then for p close to —m we have oe

(-y"

l(p) = aT nl

:

1

ara pb mm

;

It can be shown that P(p) # 0 for all p, therefore the function {/T'(p) will be continuous for all p if we put

VP(-m)

=0

(n=9,

1, 2, .. ):

“

Let us return to the solution of the Bessel equation, (18.84). Here ao is still arbitrary. If » # —n, where n > 0 is an integer, then putting ih

=

1

27

rrr

na

+ 1)

gives yy

4)” Sam 2" =tal (p

(

= (~J)

+

JIT

(r+

2). fy

+ ante

1

22" *°T On + D(H +o 4 ly

+1)

|

Substituting this into (18.86) gives

m-6

Series (18.89) defines JCx) == 2a ;

Tar

)

Dat

x

oe.

a

yt

_

.

yo

wi)

+ OC

+ at

ate

‘

53)

(18.89)

the function J

o iy Cn

1

+ Or»

x

+ =

2m +e

yG)

’

(18.90}

which is a solution of the Bessel equation and is called the Bessel function of the first kind of order », ‘The series _

Jr)

=

= di

I

yy

)

Con + OP(-2 + a+

corresponds to the case of o =

x

ant

1)(3)

- + (# is a noninteger) and defines the se-

cond solution of (18.84), which is linearly independent of /,(x). tt

75

194

‘

|

18 Higher-Order Differential Equations

We thus conclude

that if # is a noninteger (v #0,

+1,

+2,

..), then

the functions A(x) and J_,(x} form a fundamental set of solutions of the Bessel equation

y=

(18184) and

Ce)

For integral

+ Ci?

its general solution

will then

have the form

0).

yw, we have the lincar dependence (8.91)

Jauin(xy = (1) F(x).

Indeed, we have, \ foalX)

=

s

(~

"Fn

x

+ bE Dr Gn

—_

raizi

The m= fhe

n +

iG

2m

4

}

first a terms of the series vanish, since 1/PQn- a+ = 0 al O01, 0, 2-41, and LT(n-a+ i) = 1 Introducing the notation k +m, we find Jun(x)

=

3 io} ny" tn

i

fm

Pk. A+

Ae

(3)

DP (K 4 + 1) ken

a

eed

we

om

een

i pg

«

iy Tk +1 ire tnt

on

(3 )

te

n

CMa.

u vo

vp

Y2d(X)

EOS

a fac,

ome E5520

era meee

8.854

SSeS

‘are

Fig. 15.4 We write the series for the Hessel function of the first kind of order zero (a= OF and order one G1 = Lb:

(2)

yer

x =

x

,

4 (4)

4

_

1 fx}

ar (=)

1

+ aha

Lf)

6

my

20.

ay (3) Fon

G) : ay(3)

d= | Ao)

2

{xP

(3)

ees

The functions Je(x} and JC) (@ig. 18.4) often occur in applications. For Loe he them detailed tables are available.

18.12 Bessel Equation.

Bessel Furctions

195 :

’

d

=

ax

a . — yl Of hCG) = x4, 10.

In exactly

the same

d

pony ACY)

mix ax

manner =

we

veep

XS

E 18.92)

: :

find

Lt

-

(18.93)

x).

5|

Expanding the derivatives in the left-hand sides of (18.92) and (18.93),

we obtain

Ix) + = Ix) = J,.1(X),

(18.94)

Idx) ~ EJ) = ~ Jovi).

(18.95)

By adding and subtracting (18.94) and recurrence formulas:

(18. 95}, we obtain

,

two important

Ma) = F110) = Jon I!

(8.96)

rat) + Jeol) = *y JAX).

(18.97)

Formula (18.96) indicates that derivatives of Bessel functions are ¢xpressed through Bessel functions. It follows from (18.97) that, knowing

J,02 and J,i1(x), we can find J,41(*). Specifically, all Bessel functions

of whole numbers are expressed through Jo(x) and /;(x). Relation (18.91) comes in handy here. At v = | we find from (18.97), eg. |

°

fal) = 2 Jax) ~ Jax). Besse! functions for half-integer a. Consider a special class of Bessel

-

functions with half-integral odd a: This class occurs in applications and is noted for the fact that in the case under discussion Bessel functions can be expressed through elementary functions. So at » = 1/2 we readily find

,

Sisa(x) Likewise,

Ui

a

hae 2 FU des ape

Recurrence formulas. Using the'formula (18.90) we make sure, by direct check, that

"AME Mae nae

——

at yp =

J oaitx) =

2SIX.

~—~ HX

~1/2

[2

cos Xx,

.

196

[8 Higher-Order Differential Equations

The

above

furmulas

A(x) _= Using

can

[2 ee

be rewritten

cos(x

the recurrence

~'e_ 4

formula

as

mr) i)

yo y

(18.97) we find,

haw) = 4A) = Lang) = and

so

yl 5

(18.98)

for example,

{2 (em

that

~ cos x),

on,

Zeros of Bessel functions. In many applications it would be instructive to have an idea of the distribution of zeros of Bessel functions. The zeros

of Jiva(x} and J_1/2(x) coincide with the zeros of sinx and cos x, respectively. It can readily be shown representation (cf. (18,98))

JAx) =

2

that

for large x we have

cos( x — > _ x) + O(x~ 7),

x7

the asymptotic

+00,

(18.99)

that holds for any integer or noninteger ». (Here f(x).= O(p(x)) means

that the ratio f(x)/(x) remains bounded as x + 0.) Formula (18.99) shows

the behaviour of the Bessel function with increasing argument. This oscillating function becomes zero an infinite number of times and the amplitude of the oscillation tends to zero as x -> +0,

The distribution of zeros of the Bessel function with positive integral n, Le,

the roots of the equation At)

=0

(n=,

I, 2, ...)

is established by the following theorem. Theorem 18.18. The function J.{x) (1 = 0, 1, 2, .-) las no complex

zeros, but has ant infinite number of real zeros, arranged symmetrically about the point x = 0, which is one of them if n = 1, 2, .... All the zeros of the

Junction are simple, multiplicity n,

Orthogonality

except for x = 0, which at n = 1, 2, ... is @ zero of

and

norm

of Bessel

property of orthogonality,

functions, We

first consider the

It can readily be verified’that the equation xp" + oxy’ + (Mx - ? yy = 0, (18.100) where \ is some nonzero numerical parameter, is satisfied by the Bessel function J.0\x). We rewrite (18. 100) in the form z

yt ay! + (» - ay =0

.

tod

(18.101)

and denote y1 = J.0x), 32 = J,02Xx), where \;, \2 are some values of }.

18.12 Bessel Equation, Bessel Functions

We

will then

have the identities

yt i

+

I yt x t

x2

ptt tyes ze

1

oes

ee x

y=

197

0 “

.

20 xeNy te

i

Multiplying the first one by y2(x), the second by yi(x) and subtracting one

from the other, we will get

yitye — yt + ‘ Grin — yd) + OF — My yun = 0. this identity by x, we notice. that it can be written as

Multiplying

5

aod kOe = yD) = (ME ~ Ni) Integrating

yr.

this with respect to x from 6 to 1, we will have

xiv

yO

}

- = (AE — A) | xyi(d yal) de. o

or

AAOt) Soha) — 22) SO) x SAX) J.Qax) ax.

= (M — AP)

(18.102)

e

(1) Let A: fg. Then from (18.102) it follows that if A1, A2 are zeros of J.C), then the left-hand side of (18.102) and, hence, the right-hand side as well, are zero. Then i

[Ox) Jax) dx = 0. t

This implies that, by definition, /,(A.x) and J,Q2x) are orthogonal weight g(x) = x on the interval (0, I]. The Bessel function J,(.1) has the countable set of zeros

with

O —1 it has an infinite number of positive roots, but it has no complex roots, save for the case of (~ A + ¥) < G, where there are two purely imaginary roots.

If we write the left-hand side of (18.102) in the form ae

aa

Aa)

7

Ande

ayn

= JO V02) Fercnus ~

ee,

we will see that Bessel functions are orthogonal in zeros of the linear combination

xJ/(x) — AJdix) = 0 of the Bessel L

i

function and

its derivative

.

| whe, Oy xidy = 0,

if,

a where The

ky (kK = 1, 2, ...) are the roots

of (8.105).

quantity

IAQ)

"gl 4/2 = ({ x #0s)de) 0

‘

is called the narm of the Bessel function J,(\x). (ising the equality (18.1043, we can show that

F.Qux) H? = in particular, for #Qx,

Iho cince

[700

+ ( ~ +)

Boo:

-

(18.106)

where d is a zero of the Bessel function we have

= 5 BSA) = 5 POO,

HOA) Ss ACA).

Ser

eo hah.

[99

19.12 Bessel Equation. Bessel Functions

Neumann (Weber) functions. Any nontrivial solution of the Bessel integer, equation (18.84) is called a cylindrical function. When » is not an of the solutions of set tal fundamen a form J-,(x) the functions J.(x) and linear the have we integer an is a jie, = p When Bessel equation (18.84).

“dependence

Jon) = (-1)ale). onal to i, ‘To suppelement the solution J,{x) by one that ig not prapaiti function we proceed as follows. When » is not an integer, we form the .

N,(t) = (

J,.(x}co

~ Je

(x)e0s ty ee

)

(18.107)

@)

Sin wy

it is a linear combination of solutions of the linear homogeneous equation (18.84), and so it is itself a solution of the equation. y ost

ye ig (ai

°

2

mB

a

ion

Fig, 18.5 In the limit as

»-* a, we, by UElospital’s

n Ody

Nyt} =

rule, will have.

as-

Na 7 ar eae (-1)"9

yen

:

One distinction of N(x) (Bessel function of the second kind) is presence of a singularity at the origin of coordinates (Fig. 18.5).

N(x) ~2n,

y = LBL. xwoO+

Natx)

~

the

nee e

(2) x

0.

(n21,2,.) ay

The solution Ny(x) of the Bessel equation (18.84) at » = m constitutes

200

18 Higher-Order Differential Equations

togelher with J,(x) a fundamental set of solut ions of the equation

ay" + xy’ +O

— ny = 0,

The function N,(x) is also called the Neum ann

(or

At sufficiently large x J,(x)

2 ~~

N(x) ~

are

2.

cas

Pr

(

3

sin (« ~ >

Weber) function.

5 ~

i)

~ 3)

Tatle 18.2 Forms of particular solutions of inhomogeneous Hnear equations with constan t cocfficients for various right-hand sides Right-hand side* of equations

i. P(x)

Roots

of characteristic cyuations

Fortns

Number @ is no root of characteris.

B08

Number

x PG

lic equation equation

2. e Pate)

3. P(x} cos Bx + Osx) sin fx

0 is a root of characteristic of multiplicity ¢ >

of particular solution

1

Number a is no root of characteris. tic equation

mfx}

Number @ is a root of characteristic equation of multiplicity r 21

xe

Numbers £/8 are no roots of characteristic equation

P00) cos fix + Ge(x) sin Bx,

FL}

A = max (mn, s] Numbers

4/8

are roots

of charac-

X'(PAG) cos px

teristic equation of multiplicity r 4, e“LP,.() cos Bx + Os(x)} sin Bx]

Numbers @

i§ are no roots of |

characteristic equation

+ Ok(x) sin Ax} _

Numbers o + if are roots of charac

leristic equation erst

of multiplicity r

nerve

* The first three kinds of right-hand sides are special casts of the fourth,

& (8.0) cos Bx + Oey

sin Ax),

& = max

[m, 5]

xe (F(x)

cos Bx

+ Geto) sin Bx}

Exercises

201

This suggests

that at large distances

trom

the origin of coordinates

the

cylindrical functions of the first and second kinds are related as (hose of sine and cosine, Owing to the factor 1/vx the functions decay as x grows. These functions are convenient to represent standing cylindrical waves, 7 In analogy with exponential functions (Euler formulas} we can construct a linear combination of J.(v) and N,(x) to obtain functions associat-

ed with running waves. This brings us to the Bessel functions of the third kind or Hankel functions given by

HOY (x) = Fc) + IN, (x), FEBx) = Oe) — INC). Exercises

Find the general solution of the equations:

1. (l tx)" yore

+ dxy =x,

QyM tanh

sy”.

Bypp = yp 2

Alyy

+

Find the solution of the Cauchy problem: 5.y" + [8sinycos’y = 0, y(0) = 0, (0) = 3.

6." = 18)", y(I) = 1, oC) = 3. Thy”

= 4(yt — 1, (0) = V2, 9 (0) = V2.

Integrate the equations. Where required, find particular solutions. By"

~4yo

Wey”

t+ 4y nO

Oy"

—3y' + 2y = 0, p=

-2r'

0,

-3y

y=0, yDerO=ey"M=H0 16. pO = 0, Uy

ty’

Ly”

by”

bay’

+yssin

LI

— y=

x. 2y"

+ Sy = 0,

- y' = 0, By”

4 yP-yHd

Find the general solutions for Wy" type lL iy? ~ 2p ty exe

Qy’ + ys2e.

sO.

PV O)=

y+ de®

-

y =O. 20 yp" -

+ p= cosx— 2sinx,

23. y" + 4y = Ixsinx. 24. y" + 4y = 2sin?x. Find the form of particular solutions for Boy" — yl = Jt xe + sin. oy" ty

Wy” ~ yp" = ltxe+2xcosx. 29.9" — ym 1 + xe" + e* cos x.

Woy”

a leust

xe® + xsinx,

+ y" =x + xe + xsinx,

Integrate by variation of constants

30." + y = MVeasx. SL y” — yp’ =e sine’, Integrate the following Euler equations:

3Lxty” — xy’ — Sy = 0. 33.7yp” tay

ty =O.

Answers

:

xox Lys a5 74+ Boye Cele,

CrtanT hx + Cy.

dyt = Oe + CGY + Ch.

2.¥ = Cy coshx + Cox? + Cex t+ Ca. S.y=tan'3x,

6

y= 4—a3x"

18 Higher-Order Differential Equations

202

Fo yoe via eo 8. pe Ce + Core 9 y= Cye7% + Coe™, Ci *singx, Eye eM ~ eo Eye Ch+ het + Gel Cre’ + Cae Cre

~ YE yen

4+ Cycosx

cas

VE5

xe

+ Ca sin x.

me Cee

9

15. p= Cet?

ve

45in

ane

thy

= Cycopa + Cysiny + 118 p=

ie

des

cas x.

Mv

V2.3

= Cye+

= Creagw

Cove

+ Cosine

!

a

+ ace’.

ley

Cre

= Cp

w+

3

Cet? cos 5 sin wa.

22.x +e ett gin V2.7 y +

+ Cork

+ Cave’ tet

Pho pe

fox (cos

cos

10. p & Che” “cos 2x + tye ldy =

Cyet

2 Wy x4

a

= Oye Cre"?

Cyt

4

Coat

+ Cet .

sin “8.

a

23. y = Creos2x + Cy sindx +

I

x Sia

; cosx.

Wey

= Crcasax

¢ Cisin ax + i _ $sin 2x.

Bye’ 4 (4zx’ + By + Dijoosx + (43x7 + Bix + Dyjsin x.

25. yp = Ax

+ x(Aix

+

26.¥ = x(A4x + BD +

fAyat + Bix + Dae’ + x[(Aga + Byhoos xy + G4ax + By)sinax]. 2h y = Axt 4 x(Ayet+ Bye’ + C4ax + Ayeosx + (4gx + ijsin x, BR. ym (Ax t Bpst + (Aye t Bye" +

{daw + Babeose + (ripe + By}sing.

29 py = A+

(4ex + Bije™ + e(4acos x + Asin x).

30.7

= Creos.y + Cysin x + cos x Infcos x| + x stax.

Shy

e Ox

a a

Hy

= Cpcastinx)

Bip

+ Cusmiin x}, x > 0.

ew Cy

+ Cre

— sine®.

19

Chapter

Equations

of Differential

Systems

(19.1 Essentials. Definitions

even the simplest problem A system of differential equaticns models the law

acting on a particle, find of particle dynamics: given the forces ¥ = YU), 2 = z(t}, which express of motion, ie, find the functions x = x(), ng particle on time. The system movi a the dependence of the coordinates of the form that results in the general case has

d*x ween

ae

SS

ay —_ dt’

=

az

de

vege

( MGT we

roe

i,

e

iyi

(« MP

( Ie i

,

+

?

dx

dy

a

4)

dt

~~

~—s

~~

dx

dy Tu a

dz “) 2)

dy

ad

—

dx oT

at

“sy

‘) we

ye

19. (19-4)

P?

pe

particle, / is time, fg. A Here x, y, z are the coordinates of a (ravelling are known

functions of respective arguments.

canonic system. Turning to A system of the type (19.1) is known'as a own m of m differential equations with m unkn

the general case of the syste treat as canonic the system. functions (0, suff, ..-+ xXm(f) of t, we will of the form ka . / ’ », xu ++ 4) k Amy » Xm; 9, xm cess Mis iy xf d = fill, £ (19.2) i= 1, 2, ..., #7 tives. A system of first-order which is solvable for the highest-order deriva 3 desired functions equations solvable for the derivatives of the

ee fiy te oo

is called a normal system lf xf xf.

CANA)

(19.3) :

ary functions, xfeen ) in (19.2) are taken to be new auxili

replaced by an equivalent then the general canonic system (19:2) can be *

204

19 Systerns of Differential Equations

normal system consisting of N= fore, it is sufficient to consider

ky + ko +... only normal

+ Ke equations. There-

systems.

For example, one equation d’x/dt? = —x is a special case of the canonic system. Setting dx/dt = y, we will have dy/dt = —x from the original equation. As

a result, we will have the normal

system

of equations

ax a

dy

ae equivalent to the original equation. Definition. Any system of 4 functions x,

=

x;(f),

2

=

xa),

eres

Ay

=

xa,

(19.4)

differentiable on the interval a < ¢ < b, such that it turns the equations of

(19.3) into identities in ¢ on the interval (a, b) is called a solution of the normal system (19,3) for ¢ defined on the interval (a, 5). ‘The Cauchy problem for the system (19.3) is formulated as follows: find

the solution (19.4) of the system such that at ¢ =

it obeys the initial con-

ditions

”

Xifrete= 2%, Theorem

19.1

(on

2] eww = KS, ..., Me fray = x2. existence

and

uniqueness

of the

(19.5) solution

of the

Cauchy problem), Let (19.3) be a normal system of differential equations.

Suppose that the functions fi (t, Xi, x2, ..., Xn), i= 1,2, ..., 4, are defined in a certain (n + 1)-dimensional domain D of the variables t, Xp, Xa, oes Xn. Uf there exists a neighbourhood Q of a@ point Mo (to, X71, x4, ..., x2) where fj are continuous in the multitude af arguments and have bounded partial derivatives in x, X2, .-., Xn, then there will be an interval fo ~ lip < it < ty + Ag where there exists a unique solution ef the normal system (19.3) satisfying the initial conditions x, | veto = x2, X2 | tal *

XS, .e0a Mn fees = Xe.

Definition, A system of ” functions

HHO Cy oy GC)

FLAW M

019.6)

of fand n arbitrary constants C), Cp, ..., Cy, is called the general solution

of the normal systern (19.3) in a certain domain Q where there exists a

unigue solution of the Cauchy problem, if

Do

oO

(i) for any permissible values of Cy, C2, ..., Cy the system (19.6) turns

equations (19.3) into identities; cs (2) in 2 the functions (19.6) solve any Cauchy

Foe problemi-0o

. 20. -

205

19.1 Essentials. Definitions

Solutions that are deduced

from

the general one

Cor concrete values

of Ci, Cay wees Cn are called particular solutions. We will turn for definiteness lo the narmal syslem of two equations a

as

=

1 (f,

Xts 22);

19.7 om

09-7)

= fo (i, X1, 2).

We will treat the system of values of f, x1, x2 as the Cartesian coordinates ates _ of a point in a three-dimensional space with the system of coordin at which (19.7), Otx,x2. The solution x) = xi(0, x2 = x2{ of the system through passing line a f = fg assumes the values x9, x2, defines in the space

the point Mo(fo, x?, x2). This line is known as the integral curve of the

folnormal system (19.7). The Cauchy problem for (19.7) can be given the find to required itis %2 x1, f, of jowing geometrical treatment: in the space the integral curve passing through a given point Molto, x, x§) (Fig. 19.1). Theorem 19.1 establishes the existence and uniquenes of such a curve.

Rt

2

ol

aa

Fig. 19.2

Fig. 19.1

The normal system (19.7) and its solution can also be treated as follows: as a parameter, and the solution we will view the independent variable

X= x4(f), X2 = 20(f) of the system as parametric equations of the curve in the x1x2-plane. This plane of the variables x1, x2 is called the phase plane. In the phase plane the solution x) = 4),

% = soft) of (19.7), which

at

t= fy takes on the initial values x?, x2, is shown by the curve AB, passing through Mo(x?, x) (Fig. 19.2). This curve is termed the path ((rajectory)

of the system (phase path). The path of (19.7) is the projection of the in-

tegral curve on the phase plane. From an integral curve the phase path can be determined

uniquely,

but not vice versa,

206

19 Systems of Differential Equations

19.2

Methods of Integration of Systems

of Differential Equations

One of the integration methods is integration by elimination. A special case of a canonic system is one equation of order 7 solvable for the highest-order derivative

0 oe 6, KK

oe KY),

Introducing the new variables x; = x‘(2), x = x"(D, ..., Xne 4 & XO" OCD, we replace this equation by the following normal

SS

system

x,

of n equations

(19.8)

wt PCE,

X,

Mtg

seep

Xn~ ths

ic, one equation of ath oréer is equivalent to the normal system (19.8). The reverse is also true, Le, generally speaking, a normal system of a equations of the first order is equivalent to one equation of order nm. It is on this that the elimination method of integration of a system of differen-

tial equations is based. it is done as follows. Let there be a normal system of differential equations

BB A fe, xis ay oes toh

f ax: cx

—ar = Salt

Wa,

ey

oc ry nds

voce teucnucsunsvesurees

(19.9)

We differentiate, for example, the first equation in (19.9) with respect tof

déxsagt _ eran oh 4, oh ae Ge at ae dt Substituting a,

.--;

@j/df

on

afi ed de ax dt

°

the right-hand side

y cen Gi mm Bn ax, dt’. by

the expressions A(%,

X41,

Xa) yields

af’xy am ade

ai

at

Gee aUi Re Ls ah. Ul, Xt, Xty vey : Xiah aay Bag A Bx, fod, Xr, % **)

407

of Integration

19.2 Methods neem

of the form

ie, an expression

.

.

.

d?x;

“a0 * P(t, 4, Xa...

,

.

:

Xn).

(19.30)

respect to ¢. Using (19.9), we get This equation is again differentiated with

OF

OF,

Px,

efi te

oe) eo

a"

We

aFy

tet

xa

or

dx

- es

Xi, Xa,

; = FSU,

we get

If this process is continued, dix

We Si

assuine that the determinant Fy,

sere

Fy

(the Jacobian

ot

Fy

i)

_.

“Phe, tay esXd

OF

fx, Diy AFayee. oe) Fa)

feeb

PM,

Kay

ony

of functions

afiOX aps

OX

OX,

| AK?

for the values of x2, 47,

7

af

rT

is nonzero

of the system

wah, diy 8x)

1)

Fa,

DU.

Xs

Xnbe

++

Xp, Ka,

= Fy,

ar"

cere

RBp

May

Fall,

=

“ae

Xn).

OF yw

OF nt

ax

OXn

.-.. %n in question,

viz., 9.1

Ob (

Xn)

in (19.9) and the The system of equations containing the first equation equations d*x,

ae

d"~'x;

me Pay

eee

he

a By

ity

Xa,

Xt,

oss

Xn)

terms of 4 M1, will be solvable for x2, X3, .--. Xn, Which are expressed in

208

19 Systems of Differential Equations

dy, /di, ..., d"7 'x,/dt"~'. d"x

net

dt”

Pah,

Tintering these expressions into

Xp, Nay

eee

Xn)

yields one equation of order.#

a"x

dx;

a" hx

Tad = #(1 a bogs

cat):

(19.12)

It follows from the way it was derived that if x1(), x2(0, ..-, xa(0 are solutions of (19.9), then x:(4) will be a solution of (19.12). Conversely, let xi(f) be a solution to (19.12). If we differentiate this solution with respect to

dx.

d"~ tx;

i, we get “a

aT

. We get these values as known

functions of f¢

in the systern of equations dx;

—— 7

fit, Jt,

Xt,Xu, Xa,Xz

6-0

Mh)

d*x,

og at?

= Palit, at, M1,X1, Xa,Xa

---, Mad) oe

d"~'y arc

= Faith

XMiy

Xty

vass

Xn).

_

By our assumption, this system can be solved for v2, ¥3, ..., Xn, Le, Xa, X3, ..., ¥, can be found as functions of ¢. It can be shown

that the system

xy = x(t), x2 = x2,

of functions

«26, Xa = Xnft)

constructed in this way is a solution Example. Integrate the system dx

dy

a

dt

=a FF

a

to (19.9).

19.13

*

(19.13)

d’x

~i Differentiating the first of these, we will get ai

dy

ae

whence,

;

using

2y

the second

equation,

we will have as

+x=

0, ie, a linear differential

equation of the second order with constant coefficients and one unknown function. Its general solution has the form

x(t) = Cicosf + Cysing. From

the first equation we find WO =

—-Cysini+

Crcosé

It is easy to verify that x(/) and y(/ at all C, and C, satisfy the given system.

19.2 Methods

of Integration

Functions xf) and ¢(@) can

ve Asin(gQ-ba),

;

be represemed

209

in the form

re cleosth ta

7

49.44)

And so the integral curves of (19.13) are belical fines with lead A = 29 and

general axis x = y = 0, which is also an integral curve (Fig.

19.3).

Eliminating in (19.14) the parameter f, we obtain the equation eeyp= A’, since the phase paths of the system are circles with the centre at the origin of coordinates, ie, projections of the helical lines on the

"y

xy-plane.

Fig. 19.3

At A = 0 the phase path consists of one point x = 0, y = 0, called the stationary (ot rest} paint of the system, & Remark. Ho may happen that the functions Xa, Na, ..., 4%, Cannot be : . cy ei" 'y found

in derms

of

f,