Lectures on Ordinary Differential Equationscaval101/homepage/Differential... · Lectures on...

Lectures

on Ordinary

Differential Equations

WITOLD HUREWICZ

1111111

THE M.1.T. PRE 5 5MASSACHUSETTS INSTITUTE OF TECHNOLOGYCAMBRIDGE, MASSACHUSETTS, AND LONDON, ENGLAND

Copyright © 1958by The Massachusetts Institute of Technology

All rights reserved. This book or any partthereof must not be reproduced in any formwithout the written permission of the publisher.

Second Paperback Edition, March 1970

ISBN 0 262 58001 2 (paperback)

Library of Congress Catalog Card Number: 58-7901Printed in the United States of America

Preface

This book is a reprinting, with minor reVISIons and one correction,of notes originally prepared by John P. Brown from the lectures givenin 1943 by the late Professor Witold Hurewicz at Brown University.Th~y were first published in mimeographed form by Brown Universityin 1943, and were reissued by the Mathematics Department of TheMassachusetts Institute of Technology in 1956. They are now reprinted with the permission of Brown University and of Dr. StefanHurewicz.

An appreciation of Witold Hurewicz by Professor Solomon Lefschetz,which first appeared in the Bulletin of the American MathematicalSociety, is included-m this book, together with a bibliography of hispublished works.

Since this book treats mainly of existence theorems, linear systems,and geometric aspects of nonlinear systems in the plane, a selected listof books on differential equations has been placed at the end of thevolume for those interested in further reading.

The work of the several mathematicians who prepared this newedition was supported by the Office of Naval Research.

NORMAN LEVINSON

Massachusetts Institute of Technology

vii

Witold HurewiczIn Memorianl*

SOLOMON LEFSCHETZ

Last September sixth was a black day for mathematics. For on thatday there disappeared, as a consequence of an accidental fall froma pyramid in Uxmal, Yucatan, Witold Hurewicz, one of the mostcapable and lovable mathematicians to be found anywhere. He hadjust attended the International Symposium on Algebraic Topologywhich took place during August at the National University of Mexicoand had been the starting lecturer and one of the most active participants. He had come to Mexico several weeks before the meetingand had at once fallen in love with the country and its people. As aconsequence he established from the very first a warm relationshipbetween himself and the Mexican mathematicians. His death causedamong all of us there a profound feeling of loss, as if a close relativehad gone, and for days one could speak of nothing else.

Witold Hurewicz was born on June 29, 1904, in Lodz, RussianPoland, received his early education there, and his doctorate in Viennain 1926. He was a Rockefeller Fellow in 1927-1928 in Amsterdam,privaat docent there till 1936 when he came to this country. TheInstitute for Advanced Study, the University of North Carolina, Radiation Laboratory and Massachusetts Institute of Technology (since 1945)followed in succession.

• Reprinted by permission from the Bulletin 0/ the American MathematicalSociety, vol. 63, no. 2, pp. 77-82 (March, 1957).

Ix

x WITOLD HUREWICZ IN MEMORIAM

Mathematically Hurewicz will best be remembered for his importantcontributions to dimension, and above all as the founder of homotopygroup theory. Suffice it to say that the investigation of these groupsdominates present day topology.

Still very young, Hurewicz attacked dimension theory, on which hewrote together with Henry Wallman the book Dimension theory [39].1We come to this book later. The Menger-Urysohn theory, still of recentcreation, was then in full bloom, and Menger was preparing his bookon the subject. One of the principal contributions of Hurewicz was theextension of the proofs of the main theorems to separable metric spaces[2 to 10] which required a different technique from the basically Euclidean one of Menger and Urysohn. Some other noteworthy results obtained by him on dimension are:

(a) A separable metric n-space (= n dimensional space) may betopologically imbedded in a compact metric n-space [7].

(b) Every compact metric n-space Y is the map of a compact metriczero-space X in such a manner that no point of Y has more than n + 1antecedents, where n cannot be lowered, and conversely where this holdsdim Y = n. In particular one may choose for X a linear set containingno interval [6]. '

(c) Perhaps his best dimension result is his proof and extension ofthe imbedding theorem of compact spaces of dimension <n in Euclidean E2n+1 which reads: A compact metric n-space X may be mappedinto En +rn (m = 1, 2, .. '), so that the points which are images of kpoints of X make up a set of dimension <n -=. (k - l)m [26].

This proposition may also be generalized as follows: Any mappingf: X ~ En+m may be arbitrarily approximated by one behaving asstated. Special case: X may be mapped topologically into E2n+1•

Earlier proofs of this last theorem existed. The wholly original proofof the main theorem by Hurewicz rests upon the utilization of the spaceE;+m of mappings of X ~ En +m , as defined by Frechet and the proofthat the mappings of the desired type are dense in E:+m.

A more special but interesting dimensional result is:(d) Hilbert space is not a countable union of finite dimensional

spaces [10].Recall R. L. Moore's noteworthy proposition: a decomposition of

the two-sphere 52 in upper semi-continuous continua which do notdisconnect 52 is topologically an 52. Hurewicz showed [17] that forsa no such result holds and one may thus obtain topologically any compact metric space. This shows that R. L. Moore's results describe avery special property of 52.

1 Square brackets refer to the bibliography at the end.

WITOlD HUREWICZ IN MEMORIAM xi

Another investigation of Hurewicz marked his entrance into algebraic topology. The undersigned had introduced so called LCn spaces:compact metric spaces locally connected in terms of images of p-spheresfor every p < n. One may introduce HLCII spaces with images ofp-spheres replaced by integral p-cycles and contractibility to a point by.-0 in the sense of Vietoris. Hurewicz proved this very unexpectedproperty: N.a.s.c. for X as above to be LCn is HLCn plus local contractibility of closed paths [33]. An analogous condition will appearin connection with homotopy groups.

We come now to the four celebrated 1935 Notes on the homotopygroups, of the Amsterdam Proceedings [29; 30; 34; 35]. The attack isby means of the function spaces Xl'. Let Y be a separable metricspace which is connected and locally contractible in the sense of Borsuk.Let SP denote the p-sphere. Let Xo be a fixed point of Sit -1, n > 1,and Yo a fixed point of Y. Let N be the subset of Y 8"-1 consisting ofthe mappings F such that Fxo = Yo. The group of the paths of N isthe same for all components of M. It is by definition the nth homotopy group 17"" (Y) of Y. For n = 1 it is the group of the paths of Y,and hence generally noncommutative but for n > 1 the groups arealways commutative. Hurewicz proved the following two noteworthypropositions:

). When the first n - (n > 2) homotopy groups of the space Y (sameas before) are zero then the nth 17"" (Y) is isomorphic with H" (Y), thenth integral homology group of Y.

II. N.a.s.c. for a finite connected polyhedron II to be contractible toa point is 17"} (II) = 1 and HI! (II) = 0 for every n > 1.

For many years only a few homotopy groups were computed successfully. In the last five years however great progress has been madeand homotopy groups have at last become computable mainly throughthe efforts of J.-P. Serre, Eilenberg and MacLane, Henri Cartan, andJohn Moore.

Many other noteworthy results are found in the four AmsterdamProceedings Notes but we cannot go into them here. We may mention however the fundamental concept of homotopy type introducedby Hurewicz in the last note: Two spaces X, Yare said to be of thesame homotopy type whenever there exist mappings f: X ~ Y andg: Y ~ X such that gf and fg are deformations in X and Y. This concept gives rise to an equivalence and hence to equivalence classes.This is the best known approximation to homeomorphism, and comparison according to homotopy type is now standard in topology. Identity of homotopy type implies the isomorphism of the homology andhomotopy groups.

xii WITOLD HUREWICZ IN MEMORIAM

At a later date (1941) and in a very short abstract of this Bulletin[40] Hurewicz introduced the concept of exact sequence whose mushroom like expansion in recent topology is well known. The idea restsupon a collection of groups Gn and homeomorphism cPfI such that

and so that the kernel of cPn is cPn+1Gn+2• This was applied by Hurewiczto homology groups and he drew important consequences from thescheme.

Still another noteworthy concept dealt with by Hurewicz is that offibre space. In a Note [38] written in collaboration with Steenrodthere was introduced the concept of the covering homotopy, its existence was established in fibre spaces, the power of the method wasmade clear. He returned to it very recently [4-5] to build fibre spaceson a very different basis. In another recent Note [46] written in collaboration with Fadell there was established the first fundamental ad-

- vance beyond the theorem of Leray (1948) about the structure ofspectral sequences of fibre spaces.

Hurewicz made a number of excursions into analysis, principallyreal variables. A contribution of a different nature was his extensionof G. D. Birkhoff's ergodic theorem to spaces without invariant measure[42].

During World War II Hurewicz gave evidence of surprising versatility in distinguished work which he did for the Radiation Laboratory.This led among other things to his writing a chapter in the ServoMechanisms series issued by the Massachusetts Institute of Technology.

The scientific activity of Hurewicz extended far beyond his writtenpapers important as these may be. One way that it manifested itselfis through his direct contact with all younger men about him. He wasready at all times to listen carefully to one's tale and to make all mannerof suggestions, and freely discussed his and anybody else's latest ideas.One of his major sources of influence was exerted through his books.Dimension theory [39] already mentioned is certainly the definitivework on the subject. One does not readily understand how so muchfirst rate information could find place in so few pages. We must alsomention his excellent lectures on differential equations [41] which hasappeared in mimeographed form and has attracted highly favorableattention.

On the human side Witold Hurewicz was an equally exceptionalpersonality. A man of the widest culture, a first rate and" careful linguist, one could truly" apply to him nihil homini a me alienum puto.

WITOLD HUREWICZ IN MEMORIAM xiii

Tales were. also told of his forgetfulness-which made him all the morecharming. Altogether we shall not soon see his equal.

BIBLIOGRAPHY

1. Vber eine Verallgemeinerung des Borelschen Theorems, Math. Zeit. vol. 24(1925) pp. 401-421.

2. Vber schnitte von Punktmengen, Proc. Akad. van Wetenschappen vol. 29(1926) pp. 163-165.

3. Stetige bilder von Punktmengen. I, Ibid. (1926) pp. 1014-1017.4. Grundiss der Mengerschen Dimensionstheorie, Math. Ann. vol. 98 (1927)

pp.64-88.5. Normalbereiche und Dimensionstheorie, Math. Ann. vol. 96 (1927) pp. 736

764.6. Stetige bilder von Punktmengen. II, Proc. Akad. van Wetenschappen vol. 30

(1927) pp. 159-165.7. Verhalten separabler Riiume zu kompakten Riiumen, Ibid. (1927) pp. 425

430.8. Vber Folgen stetiger Funktionen, Fund. Math. vol. 9 (1927) pp. 193-204.9. Relativ perfekte Teile von Punktmengen und M~ngen, Fund. Math. vol. 12

(1928) pp. 78-109.10. Vber unendlich-dimensionale Punktmengen, Proc. Akad. van Wetenschap

pen vol. 31 (1928) pp. 916-922.11. Dimension und Zusammenhangsstufe (with K. Menger), Math. Ann. vol.

100 (1928) pp. 618-633.12. Vber ein topologisches Theorem, Math. Ann. vol. 101 (1929) pp. 210-218.

• 13. Vber der sogenannter Produktsatz der Dimensionstheorie, Math. Ann. vol.102 (1929) pp. 305-312.

14. Zu einer Arbeit von O. Schreier, Abh. Math. Sem. Hansischen Univ. vol. 8(1930) pp. 307-314.

15. Ein Theorem der Dimensionstheorie, Ann. of Math. vol. 31 (1930) pp. 176180.

16. Einbettung separabler Riiume in gleich dimensional kompakte Riiume,Monatshefte fUr Mathematik vol. 37 (1930) pp. 199-208.

17. Vber oberhalb-stetige Zerlegungen von Punktmengen in Kontinua, Fund.Math. vol. 15 (1930) pp. 57-60.

18. Theorie der Analytischell mengen, Fund. Math. vol. 15 (1930) pp. 4-17.19. Dimensionstheorie und Cartesische Riiume, Proc. Akad. van Wetenschappeo

vol. 34 (1931) pp. 399-400.20. Une remarque sur l'hypothese du continuo Fund. Math. vol. 19 (1932) pp.

8-9.21. Vber die henkelfreie Kontinua, Proc. Akad. van Wetenschappen vol. 35

(1932) pp. 1077-1078.22. Stetige abbildungen topologischer Riiume, Proc. International Congress

Zurich vol. 2 (1932) p. 203.23. Ober Dimensionserhorende stetige Abbildungen, I. Reine Angew. Math. vol.

169 (1933) pp. 71-78.24. Ober Schnitte in topologischen Riiumen, Fund. Math. vol. 20 (1933) pp.

151-162.25. Ein Einbettungessatz ;;ber henkelfreie Kontinua (with B. Knaster), Proc.

Akad. van Wetenschappen vol. 36 (1933) pp. 557-560.

xiv WITOLD HUREWICZ IN MEMORIAM

26. aber Abbildungen von endLichdimensionalen Riiumen auf TeilmengenCartesischerriiume, Preuss. Akad. Wiss. Sitzungsber. (1933) pp. 754-768.

27. aber einbettung topologischer Riiume in cantorsche Mannigfaltigkeiten,Prace Matematyczno-Fizyczne vol. 40 (1933) pp. 157-161.

28. Satz iiber stetige Abbildungen, Fund. Math. vol. 23, pp. 54-62.29. Hoher-dimensionale Homotopiegruppen, Proc. Akad. van Wetenschappen

vol. 38 (1935) pp. 112-119.30. Homotopie und Homologiegruppen, Proc. Akad. van Wetenschappen vol.

38 (1935) pp. 521-0528.31. aber Abbildungen topologischer Riiume auf die n-dimensionale Sphiire,

Fund. Math. vol. 24 (1935) pp. 144-150.32. Sur la dimension des produits cartesiens, Ann. of Math. vol. 36 (1935) pp.

194-197.33. Homotopie, Homologie und lokaler Zusammenhang, Fund. Math. vol. 25

(1935) pp. 467-485.34. Klassen und Homologietypen von Abbildungen, Proc. Akad. van Weten

schappen vol. 39 (1936) pp. 117-126.35. Asphiirische Riiume, Ibid. (J936) pp. 215-224.36. Dehnungen, Verktirzungen, Isometrien (with H. Freudenthal), Fund. Math.

vol. 26 (1936) pp. 120-122.37. Ein Einfacher Beweis des Hauptsatzes tiber cantorsche Mannigfaltigkeiten,

Prace Matematyczno-Fizyczne vol. 44 (1937) pp. 289-292.38. Homotopy relations in fibre spaces (with N. E. Steenrod), Proc. Nat, Acad.

Sci. U.S.A. vol. 27 (1941) pp. 60-64.39. Dimension theory (with H. Wallman), Princeton University Press (Prince

ton Mathematical Series No.4), 1941, 165 p.40. On duality theorems, Bull. Amer. Math. Soc. Abstract 47-7-329.41. Ordinary differential equations in the real domain with emphasis on geo

metric methods, 129 mimeographed leaves, Brown University Lectures, 1943.42. Ergodic theorem without invariant measure, Ann. of Math. vol. 45 ~ 1944)

pp. 192-206.43. Continuous connectivity groups in terms of limit groups (with J. Dugundji

and C. H. Dowker), Ann. of Math. (2) vol. 49 (1948) pp. 391-4;06.44. Homotopy and homology, Proceedings of the International Congress of

Mathematicians, Cambridge, J950, vol. 2, American Mathematical Society, 1952,pp. 344-349.

45. On the concept of fiber space, Proc. Nat. Acad. Sci. U.S.A. vol. 41 (1955)pp. 956-961. .

46. On the spectral sequence of a fiber space (with E. Fadell), Proc. Nat. Acad.Sci. U.S.A. vol. 41 (1955) pp. 961-964; vol. 43 (1957) pp. 241-245.

47. Contributed Chapter 5, Filters and servosystems with pulsed data, pp. 231261, in James, Nichols and Phillips Theory of servomechanisms, MassachusettsInstitute of Technology, Radiation Laboratory Series, vol. 25, New York, McGrawHill, 1947.

48. Stability of mechanical systems (co-author H. Greenberg), N. D. R. C.Report, 1944 (to appear in the Quarterly of Applied Mathematics).

49. Four reports on servomechanisms for the Massachusetts Institute of Technology Radiation Laboratory.

50. Dimension of metric spaces (with C. H. Dowker), Fund. Math. vol. 43(January, 1956) pp. 83-88.

Preface

Witold Hurewicz In Memoriam

Contents

Norman Levinson VB

Solomon Lefschetz ix,Chapter 1 Differential Equations of the First Order

in One Unknown Function

PART A THE CAUCHY-EULER APPROXIMATION METHOD

Section 1234

5

Definitions. Direction FieldsApproximate Solutions of the Differential EquationThe Fundamental InequalityUniqueness and Existence TheoremsAppendix to §4. Extension of the Existence TheoremSolutions Containing Parameters

PART B CONTINUATION OF SOLUTIONS. OTHER METHODS

1257

1012

6 Continuation of Solutions7 Other Methods of Solution

Chapter 2 Systems of Differential Equations

1518

Section 12

Introduction. Vector NotationApproximate Solutions

xv

2325

xvi CONTENTS

3 The Lipschitz Conditions 264 The Fundamental Inequality 275 Existence and Properties of Solutions of the System 286 Systems of Higher Order 32

Chapter 3 Linear Systems of Differential Equations

PART A THE GENERAL LINEAR SYSTEM

Section 12345

Introduction. Matrix NotationLinear Dependence. Fundamental SystemsSolutions Expressed in Matrix FormReduction of Order of a SystemNonhomogeneous Systems

PART B LINEAR EQUATIONS OF HIGHER ORDER

3435384143

6 Fundamental Systems7 The Wronsky Determinant8 Further Properties of the Fundamental System9 Reduction of Order

10 The Nonhomogeneous Case11 Green's Function

PART C LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS

44474850si52

12 Introduction. Complex Solutions 5613 Characteristic Values and Vectors of a Matrix 5714 Vectors Associated with Characteristic Values of a

Matrix 5915 The Solution in the Simplest Case 6016 The Solution in the General Case 6117 Homogeneous Equation of nth Order 6518 Applications 66

Chapter 4 Singularities of an Autonomous System

PART A INTRODUCTION

Section 12

Characteristic CurvesIsolated Singularities

PART B SINGULARITIES OF A LINEAR SYSTEM

7073

3 General4 Nodal Points

7576

CONTENTS xvii

5 Saddle Points 796 - Degenerate Nodal Points 807 Vortex Points 828 Spiral Points 839 Conclusion. Dynamical Interpretation 84

PART C NONLINEAR SYSTEMS

10 Introduction11 Nodal Points12 Saddle Points13 Spiral Points14 Indeterminate Cases

Chapter 5 Solutions of an Autonomous System in the Large

8690959798

Section 1234567

IntroductionGeometrical ConsiderationsBasic LemmasTheorem of Poincare-BendixsonPoincare's IndexOrbital Stability of Limit CyclesIndex of Simple Singularities

References

Index

102104105109111113114

117

119

I

Differential Equations

of the First Orderin One UnknoW"n Function

PART A.THE CAUCHY-EULER APPROXIMATION METHOD

1. Definitions. Direction Fields

By a domain D in the plane we understand a connected open set ofpoints; by a closed domain or region D, such a set plus its boundary points.The most general differential equation of the first order in one unknownfunction is

F(x, y, y') = 0 (1)

where F is a single-valued function in some domain of its arguments. Adifferentiable function y(x) is a solution if for some interval of x, (x, y(x),y'(x)) is in the domain of definition of F and if further F(x, y(x), y'(x)) = O.We shall in general assume that (l) may be written in the normal form:

y' = f(x, y) (2)

where f(x, y) is a continuous function of both its arguments simultaneouslyin some domain D of the x-y plane. It is known that this reduction may

1

1 LECTURES ON ORDINARY DIFFERENTIAL EQUATIONS

be carried out under certain general conditions. However, if the reduction is impossible, e.g., if for some (xo, Yo, Yo') for which F(xo, Yo, Yo') = 0it is also the case that [JF(xo, Yo, Yo')]jJy' = 0, we must for the presentomit (1) from consideration.

A solution or integral of (2) over the interval Xo~ x ~ Xl is a singlevalued function y(x) with a continuous first derivative y'(x) defined on[xo, Xl] such that for Xo~ X < Xl:

(a) (X, y(x» is in D, whence f(x, y(x» is defined(b) y'(x) = f(x, y(x». (3)

Geometrically, we may take (2) as defining a continuous direction fieldover D; i.e., at each point P: (x, y) of D there is defined a line whoseslope isf(x, y); and an integral of (2) is a curve in D, one-valued in X andwith a continuously turning tangent, whose tangent at P coincides withthe direction at P. Equation (2) does not, however, define the mostgeneral direction field possible; for, if R is a bounded region in D,f(x~ y) is continuous in R and hence bounded

If(x, y)1 ~ M, (x, y) in R (4)

(5)dydt = Q(x, y)

where M is a positive constant. If ex is the angle between the directiondefined by (2) and the x-axis, (4) means that ex is restricted to such valuesthat

Itan exl ~ M

The direction may approach the vertical as P: (x, y) approaches theboundary C of D, but it cannot be vertical for any point P of D.

This somewhat arbitrary restriction may be removed by consideringthe system of differential equations

dxdt = P(x, y),

where P and Q are the direction cosines of the direction at (x, y) to thex- and y-axes respectively and hence are continuous and bounded; anda solution of (5) is of the form

x = x(t), y = y(t), to~ t~ tl

which are the parametric equations of a curve L in D. We shall be ableto solve (5) by methods similar to those which we shall develop for (2);hence weshall for the present consider only the theory of the equation (2).

2. Approximate Solutions of the Differential Equation

Since the physicist deals only with approximate quantities, an approximate solution of a differential equation is just as good for his purposes as

DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 3

an exact s~lution, provided the approximation is sufficiently close. Thepresent section will formulate the idea of an approximate solution andprove that the approximation may be made arbitrarily close. Themethodsused will be practical even though rather crude. More important, thedemonstration of the existence of approximate solutions will later leadto the proof of the existence and uniqueness of exact solutions.

Definition 1. Let f(x, y) be continuous, (x, y) in some domain D. LetXl < X < x2 be an interval. Then a function y(x) defined on [Xl' X2] is asolution of y' = f(x, y) up to the error £ if:

(a) y(x) is admissible; i.e., (x, y(x» is in D, Xl < X < x2•

(b) y(x) is continuous; Xl < X < x2•

(c) y(x) has a piecewise continuous derivative on [Xl' X2] which may failto be defined only for a finite number ofpoints, say ~l' ~2'· • ., ~n·

(d) IY'(x) - f(x, y(x))1 ::;: f,· Xl < X < x2' X * ~i' i = 1,· .., n.

Theorem 1. Let (xo, Yo) be a point of D, and let the points ofa rectangleR: Ix - xol < a, Iy - Yol < b lie in D. Let I/(x, y)1 < M, (x, y) in R.Then ifh = min (a, bIM), there can be constructed an approximate solutiony(x) of

y' =f(x, y) (1)

over the interval Ix - xol ::;: h, such that y(xo) = Yo' where the error f maybe an arbitrarily smallpositive number. Observe that h is independent of f.

Proof. The rectangle

s: Ix - xol ::;: h, Iy - Yol ::;: Mh (2)

is contained in R by the definition of h. See Fig. 1.Let the f of the theorem be given. Since f(x, y) is continuous in S, it is

...------------------"p

s

O~~----:~~--~l(XO+ h,yo)(xo'Yo)

Mh

L..-.-------:--_--.::>IlI-..--h----I Q

Fig. 1


uniformly so; i.e., given E > 0 (which we take to be the E of the theorem)there exists c5 > 0 such that

I/(x, fj) - I(x, Y)I < E

for (x, fj); (x, y) in S; Ix - xl, Ifj - yl < c5.Let Xl" . " X n- l be any set of points such that:

(a) Xo < Xl < x2 < . . . < xn- l < xn = Xo+ h;(b) Xi - X i - 1 < min (c5, c5/M), i = 1,' . " n.

(3)

(4)

We shall construct the approximate solution on the interval Xo< X <Xo + h; a similar process will define it on the interval Xo - h < X< xo'

The approximate solution will be a polygon constructed in the followingfashion: from (xo, Yo) we draw a segment to the right with slope ofI(xo, Yo); this will intersect the line X = Xl in a point (Xl' YI)' From(Xl' YI) we draw a segment to the right with slope I(xlo YI) intersectingX = x2 at Y2; etc. The point (Xl' YI) must lie in the triangle OPQ; fortan (X = M, and I/(xo, Yo)1 < M. Likewise (x2, Y2) lies in OPQ; etc.Hence the process may be continued up to xn = Xo+ h, since the onlyway in which the process could stop would be for/(xk, Yk) to be undefined;in which case we should have IYk - Yol > Mh contrary to construction:Analytically we may define y(x) by the recursion formulas

(5)where

i = 1,' . " n

Obviously by definition y(x) is admissible, continuous, and has a piecewise continuous derivative

i = 1,' . " n

which fails to be defined only at the points Xi' i = 1, . " n - 1.Furthermore, if X i - l < X < Xi

IY'(X) - I(x, y(x»1 = I/(Xi- l , Yi-l) - I(x, Y(x»1 (6)

But by (4), Ix - xi-II < min (c5, c5/M), and by (5)

c5Iy - Yi-ll < Mix - Xi-II < M M = c5

Hence by (3)

I/(Xi - l , Yi-l) - I(x, y(x»1 < E

andIY'(X) - I(x, y(x»1 < E i = 1, 2,' . " n - 1 (7)


Hence y(x) satisfies all the conditions of Def. I, and the constructionrequired by the theorem has been performed. This method of constructingan approximate solution is known as the Cauchy-Euler method.

It is unnecessary to improve the value of h, since, in general, as we shallshow, y(x) is defined in a larger interval than Ix - xol < h.

3. The Fundamental Inequality

With a certain additional restriction upon f(x, y) we shall prove aninequality which will be the basis of our fundamental results.

Deftnition 2. A function f(x, y) defined on a (open or closed) domain Dis said to satisfy Lipschitz conditions with respect to y for the constantk > 0 iffor every x, Yll Y2 such that (x, Yl)' (x, Y2) are in D

(1)

In connection with this definition we shall need two theorems ofanalysis:

(2)(x, y) in D

Lemma 1. Iff(x, y) has a partial derivative for Y, bounded for all (x, y)in D, and D is convex (i.e., the segment joining any two points of D liesentirely in D), then f(x, y) satisfies a Lipschitz condition for y where theconstant k is given by

Idf(x, y) I

k = l.u.b. Jy ,

Proof. By Rolle's theorem there exists a number ~ such that

Jf(x, ~)f(x, Y2) - f(x, Yl) = (Y2 - Yl) ,

dYI.e.,

Idf(x, y) I

!f(x, Y2) - f(x, Yl)1 < l.u.b. Jy IY2 - Yll

since (x, ~) is in D; whence the theorem.

Lemma 2. If D is not convex, let D be imbedded in a larger domain D'.Let 0 be the distance between the boundaries C and C' of D and D' respective~y (i.e., 0 = min (pr), p in C, pi in C'), and let 0 > O. Then iff(x, y) iscontinuous in D' (hence bounded by M, say, in D) and df/dY exists and isbounded by N in D', then f(x, y) satisfies a Lipschitz condition in D withrespect to Y for a constant


Proof. Let PI: (x, Yl) and P2 :· (x, yJ be in D.

(a) If IYl - Y21 > <5, 1fix, Yl) - f(x, yJ 1~ 2M.Yl - Y2 <5

(b) If /Yl - Y21 ~ <5, the segment PIP2 lies wholly in D'.

Hence as in Lemma 1, If(X, Yl) - f(x, yJ 1~ N. Hence the lemma.Yl - Y2

Theorem 2. Let (xo, Yo) be a point of a region R in which fix, y) iscontinuous and satisfies the Lipschitz condition for k. Let y(x), 'o(x) beadmissible functions for Ix - xol ~ h (where h is any constant, not necessarily that of Theorem 1), satisfying

Y' = f(x, y), Ix - xol ~ h

with errors £1 and £2 respectively. Set

Thenp(x) = 'o(x) - y(x), (3)

£Ip(x)I< eklx-xol Ip(xo)I + k (eklx-xol - 1) (4)

This is the fundamental inequality.

Proof. We give the proof only for Xo< x < Xo+ h; a similar processwill give the proof for Xo - h < x <xo. By Def. I, except for a finitenumber of points

I: -Jtx. y) 1< .,. I;:-Jtx·Y)I~ .,. x.<x~ Xo + h

Hence

1:: - :~ 1~ I/(x, y) - f(x, '0)1 + £

~kly - yl + £

by the Lipschitz condition; Le.,

I:~I< klpl + £,(5)

except for a finite number of points at which [dp(x)]/dx fails to be defined.

Case I. Suppose p(x) =1= 0, Xo < x ~ Xo + h. Hence, being continuous, it has the same sign, say without loss of generality p(x) > o.Then a fortiori we can write (5) in the form

:~ kp(x) + £ (5')


This may be written as

e-kx [p'(X) - kp(x)] ~ Ee-kx

or, integrating from Xo to x, where Xo~ x ~ Xo + h

ixe-kx [p'(x) - kp(x)] dx ~ Ei

xe-kx dx

~ ~

(6)

The integrand on the left-hand side may have a finite number of simplediscontinuities but it has a continuous indefinite integral. Hence we maywrite

or

p(x) < ek(x-xo)p(xo) +i [~(x-Xo) - 1] (7)

which is the required inequality.

Case II. If for all x, p(x) = 0, the theorem is obvious.

Case III. If p(x) =1= °where x is some fixed number Xo~ x ~ Xo + h,but p(x) =°for some value of x, Xo~ x < x; since p(x) is continuous,there exists a n'umber Xl' Xo< Xl < x~ Xo + h such that p(xJ = 0,Qut p(x) =1= 0, Xl < X < x. Applying Case I to the interval (Xl' X) wehave

(8)

which is an even stronger inequality than (4).Hence the inequality (4) holds in all cases, since, if p(x) < 0, the same

results follow by considering Ip(x)l.

4. Uniqueness and Existence Theorems

Theorem 3. If f(x, y) is continuous and satisfies a Lipschitz conditionfor y in a domain D, and if (xo, Yo) is in D and y(x) and y(x) are two exactsolutions of y' = f(x, y) in an interval Ix - xol < h such that y(xo) =y(xo) = Yo' then y(x) =y(x), Ix - xol < h; i.e., there is at most oneintegral curve passing through any point of D.

Proof. Applying Theorem 2, we have p(xo) = Yo - Yo = 0, andE = El + E2 = 0; hence p = y - y =0.


(2)

(1)

For, if

(b)

y-:::::=o

(

y = (iX)3/2 X ~ 0

Y = 0 x< 0

Of course, yt does not satisfy the Lipschitz condition at y = o.Yl = band Y2 = - b

We may state this in the form that under the above assumptions, twointegral curves cannot meet or intersect at any point of D.

Observe that without the additional requirement of a Lipschitz condition uniqueness need not follow. For consider the differential equation

dy = ytdx

f(x, y) = yt is continuous at (0, 0). But there are two solutions passingthrough (0, 0), namely

(a)

(3)If(Yl) - f(Y2) I= _1

Yl - Y2 b2/3

which is unbounded for b arbitrarily small.

Theorem 4. If f(x, y) is continuous and satisfies the Lipschitz conditio';for Y in a domain D, then for (xo, Yo) in D there exists an exact solution ofY' = f(x, y)for Ix - xol < h, where h is defined as in Theorem 1, such that

y(xo) = Yo

Proof. Given a monotone positive sequence {En} approaching 0 as alimit, by Theorem 1 there exists a sequence {Yn(x)} of functions satisfying

Yn'(x) = f(x, Yn(X)) up to En> Yn(XO) = Yo

over Ix - xol < h; i.e.

IYn'(x) - f(x, Yn(x)) I< En' IX - xol < h (4)

except for the finite set of points xi(n l , i = 1,· .., mn.

Part I. The sequence {Yn(x)} converges uniformly over Ix - xol < hto a (continuous) function y(x).

For let nand p be positive integers; and apply Theorem 2 to Yn(x),

Yn+ix)

IYn(x) - Yn+ix)I< En +kEn+p [ek(x-xo) - 1]

< 2;n [ekh _ 1]

(5)

whence the assertion.


Part II. The sequence {i~ f(t, Yn(t» dt} approaches l!(t, yet»~ dt

uniformly for Ix - xol < h.Let R be the rectangle Ix - xol < h, Iy - Yol < Mh, by hypothesis in

D. Since f(x, y)is continuous in the closed domain R, given E > 0, thereexists 0 > 0 such that

(x, YI)' (x, Y2) in R,

Likewise for this 0 > 0 there exists N such that

IYn(x) - y(x)1 < 0, n>N, (7)

in virtue of Part 1. Hence if n > N

Vex, Yn(x» - f(x, y(x»1 < E, n> N, Ix - xol < h (8)

Therefore the sequence {/(x, Yn(x»} converges uniformly to f(x, y(x».Hence, by a well-known theorem we may reverse the order of integrationand pass to the limit; which completes the proof of Part II.

Part 1/1. y(x) is differentiable, y(xo) = Yo, and

y'(x) = f(x, y(x», Ix - xol < h

Integrating each side of (4) from Xo to Xl we have

(9)

But since Yn(t) is continuous

IYn(x) - Yo - LX f(t, Yn(t» dt I< Enh.co

(10)

Approaching the limit by Part II, we have

y(x) - Yo - LXf(t, yet»~ dt = 0Xo

(11)

(12)ekh

- 1i.e., N= k

from which the assertion follows by an elementary theorem on functionsdefined by definite integrals.

Theorem 5. Let y(x) be an exact solution of Y' = f(x, y(x» under theassumptions of Theorem 4, and iJ(x) an approximate solution up to the errorE, for Ix - xol < h, such that iJ(xo) = Yo' Then there exists a constant Nindependent of E such that liJ(x) - y(x)I< EN, Ix - xol < h.

Proof. By Theorem 2

liJ - yl < i(ekh- I);


Theorem 5 justifies our description of the function of Theorem 1 as an"approximate solution,'; for its actual value differs from that of the exactsolution by a multiple of £; whereas in our definition we only required itto satisfy the differential equation to within an error £. To recapitulateour results: Iff(x, y) is continuous in D and satisfies a Lipschitz conditionwith constant k, and (xo, Yo) is in D, then there exist quantities h ~nd Nsuch that

(a) There is a unique exact solution y(x) of y'(x) = f(x, y(x» passingthrough (xo, Yo) and defined for Ix - xol :::;; h.

(b) There may be constructed by the Cauchy-Euler method approximate solutions y(x) such that, if £ > 0

1Y'(x) - fix, y(x»1 :::;; £, Ix - xol :::;; h, x =1= ~l' ~a,· .., ~nIy(x) - y(x)1 :::;; N£, Ix - xol :::;; h (13)

•y(xo) = y(xo) = Yo

Appendix to §4. Extension of the Existence Theorem

We have seen that without the Lipschitz condition on f(x, y) thesolution need not be unique. The condition is, however, superfluous forthe proof of existence of a solution. This may be proved directly; wegive a proof based upon Theorem 4 by the use of Weierstrass' polynomialapproximation theorem.

Theorem 6. /ff(x, y) is continuous in D, there is a solution ofy' = fix, y)such that y(xo) = Yo where (xo, Yo) lies in D, defined for Ix - xol :::;; h =min (a, bIM).

Proof. We use the following two theorems of analysis:

Lemma 1 (Weierstrass). /f fix, y) is continuous in the region R, given£ > 0, there exists a polynomial P(x, y) such that

IP(x, y) - f(x, y)1 :::;; £, (x, y) in Rand

P(x, y) :::;;f(x, y), (x, y) in R

Definition 3. A set S of functions fix) is equicontinuous over aninterval [a, b] ifgiven £ > 0 there exists 6 > 0 such that if Xl' xa in [a, b],IXI - xal :::;; 6, then If(xJ - f(xa>I:::;; £,for allfin S.

Lemma 2. An infinite set S of functions uniformly bounded and equicontinuous over a closed interval [a, b] contains a sequence converginguniformly over [a, b].

DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 1t

By virtue, of Lemma 1, let Pix, y) be a sequence of polynomials suchthat

IPix, y) - I(x, Y)/ ~ £n' Ix - xol ~ h, Iy - Yol ~ Mh

where hand M are defined as usual, and the sequence {£n} approaches O.Then {Pix, y)} is uniformly bounded; and it may be assumed that Mis the common bound of I and Pn. Since Pix, y) satisfies Lipschitzconditions in Y, by Theorem 4 there exists a sequence of functions {Yn(x)}defined over Ix - xol ~ h, satisfying Yn'(x) = Pn{x, Yn(x)}, and such thatYn(xO) = Yo' By Theorem 4

IYn(x) - Yn(Xo)I~ Mh, Ix - xol ~ h

i.e., {y,,(x)} is uniformly bounded over Ix - xol < h.Furthermore, {y,,(x)} is equicontinuous over Ix - xol < h for

Y,,(x,J - Yn(xI ) = Yn'(~)(X2 - Xl)' Xl < ~ < X2

for all Xl' x2 in [a, b]. But

IYn'(~)1 = IPn(~' Yn(~»1 ~ MHence

'IYixJ - yixl ) \ ~ IX2 - xilM

Given £ > 0, if <5 = £/M, then if IX2 - XII < <5; Xl' x2 in Ix - xol ~ h,then IYn(x,J - yixl ) I~ £ all n. Hence by Lemma 2, {yix)} has asubsequence converging uniformly over Ix - xol ~ h to a (continuous)function y(x). It will do no harm if we denote the subsequence by thesame symbol {Yn(x)}.

As in Theorem 4, Part II, we can prove that

~~CXlL:I(t, Yn(t» dt ~i:I(t, yet»~ dt

uniformly for Ix - xo\ < h. By definition

Yn(x) - Yo = f~Pn(t, Yn(t» dt

Hence we write

IYn(x) - Yo - i:/(t, Yn(t» dt I<L:IPit, yit» - I(t, Yn(t»1 dt

But ifn > N

IX - xol ~ h, Iy - Yol < Mh

Hence for n > N

IYn(x) - Yo - LXI(t, Yit» Idt < £nhXo


Letting n ~ 00 the above is valid with Yn(x) replaced by y(x) and so

Iy(x) - Yo - L~f(t, yet»~ dtl = 0, Ix - xol < h

since En tends to 0. As in Theorem 4, this last formula proves that y(x)actually is the required solution.

5. Solutions Containing Parameters

We consider next changes in the solution of a differential equationcaused by either a small change in the initial conditions or a small variationin the function f(x, y). Our general results will be that such small changeswill bring about only small changes in the solution. This again will beof interest to the physicist, since it will assure him that small errors in thestatement of a problem cannot change the answer too greatly.

We have seen already that the solution y(x) of

y' = f(x, y), Y(Xo) = Yo

contains the initial value Yo as parameter. We determine first a minimumrange of values of x and Yo for which we can be assured that y(x, Yo) willexist and be unique.

Theorem 7. Iff(x, y) is continuous and satisfies a Lipschitz condition ony in Ix - xol < a, Iy - Yol < b, (xo, Yo) some fixed point, there exists aunique solution y(x, Yo) of

y' = f(x, y),in the region

Ix - xol < h', Iyo - Yol < bj2, Iy - Yol < Mh' (1)

where h' = min [a, (bj2M)], M being the bound off(x, y) in Ix - xol < a,

Iy - Yol < b.

Proof. The proof follows immediately upon applying Theorem 4 tothe point (xo, Yo), Iyo - Yol < bj2. Observe that the h' of this theorem isnot less than half of the h for which the original existence theorem wasproved.

Theorem 8. If in some domain D, f(x, y) is continuous and satisfies aLipschitz condition on y for some k, and if a solution y(x, Yo) of

y' = f(x, y), Y(Xo) = Yo

exists for some rectangle R, Ix - xol < h, Iyo - Yol < I, then y(x, Yo) iscontinuous in x and Yo simultaneously in R.


Proof Consider the solutions y(x, yoll», y(x, yo(2», yo(l), and yo(2)satisfying IYoli) - Yol < I, i = 1,2. Applying Theorem 2 to these twofunctions over Ix - xol < h, we obtain

IY(x, yoll» - y(x, yo(2»1 < IYo(1) - Yo(2)lek\ Ix - xol < h (2)

This means that in R, y(x, Yo) is continuous in Yo uniformly in x. Sincey(x, Yo) is continuous in x for any value of Yo, it follows by a generaltheorem of real variables that in R, y(x, Yo) is continuous in x and Yosimultaneously.

Geometrically (2) means that two integral curves close enough togetherfor one value of x stay close together for Ix - xol < h.

With a little stronger restriction upon f(x, y) we now prove a furtherproperty of y(x, Yo), namely:

Theorem 9. Under the assumptions of Theorem 8 and if in addition(df/dY) (x, y) exists in D and is continuous in x and y simultaneously, then[dy(X, YO)]/dYo exists for (x, Yo) in R and is continuous in x and Yosimultaneously.

Proof Let fio be an arbitrary value of Yo which is to remain fixed untilthe end of the argument, satisfying lfio - Yo/ < I. Let Yo be a variablein the same interval. For convenience we write

y(x, fio) = fi(x), y(x, Yo) = y(x)

We shall prove that the function

y(x) - fi(x)p(x, Yo) = _, Yo =1= fio

Yo - Yo

(3)

(4)

approaches a limit as Yo - fio, which limit must be [dy(X, Yo)]/dYo atYo = fio; and that this limit has the required properties.

In the first place we write

ddx [y(x) - fi(x)] = f(x, y(x» - f(x, fi(x»

or by the mean-value theorem

= [y(x) - fi(X)] [:y f(x, fi(x» + !5{y(x), fi(x)} ] (5)

where as IY(x) - fi(x) I- 0, !5{y(x), fi(x)} - O. Since by Theorem 8

lim y(x) = fi(x) uniformly for x, Ix - xol < hYo- Yo

it follows that

uniformly in x.

lim !5{y(x), fi(x)} = 0Yo-Yo

(6)

t4 LECTURES ON ORDINARY DIFFERENTIAL EQUATIONS

By (4) and (5) we may write

op(x, y) [0 ]ox 0 = p(x, Yo) dy f(x, y(x» + ~{y(x), y(x)} (7)

Since p =1= 0 for Yo =1= Yo by the general uniqueness theorem, we maydivide (7) by P and integrate explicitly with respect to x

p(x, Yo) = exp{i~[f"(x, y(x» + ~{y(x), y(x)}] dX} (8)

where the constant of integration is determined by the fact that p(xo, Yo) =1. Equation (8) is valid for all Yo' But lim p(x, Yo) exists; for by (6)and (8) 110""'0

dy~x, Yo) = lim p(x, Yo) = exp{iZf,,(X, y(x» dX} (9)Yo 1I0....iJo Zo

The right-hand side of (9) is continuous in x and Yo simultaneously; for(oflCJy) (x, y) was continuous by hypothesis, and y(x) = y(x, Yo) is continuous by Theorem 8, Yo now being considered the variable point. Thiscompletes the proof.

By similar (but considerably more complicated) reasoning we can pro\reunder the same assumptions that if f(x, y) in addition has a continuousfirst derivative with respect to some parameter p in a certain interval, thesolution y(x) is also continuous and differentiable in p. We omit theproof since this will appear as a special case of the general theory ofChapter 2.

We finally prove that the solution is only slightly changed if f(x, y) isonly slightly changed; precisely:

Theorem 10. If in some domain D,

(a) F(x, y) andf(x, y) are continuous,

(b)f(x, y) satisfies a Lipschitz condition on y for some k,

(c) IF(x, y) - f(x, y)1 ::;: €, (x, y) in D, (10)

(d) for some point (xo, Yo) in D, y(x) and y(x) are admissible in Ix - xol :::;:h and satisfy

y'(x) = f(x, y), y'(x) = F(x, y), y(xo) = y(xo) = Yothen

IY(x) - y(x) \ < ~ (ekh- 1), Ix- xol::::;;;h (11)

Proof. By (10), y(x) is an approximate solution of y' = f(x, y) witherror €; applying Theorem 5 we immediately obtain (11).

DIFFERENTIAL EQUATIONS OF THE FIRST ORDER ts

For example, in the neighborhood of the origin we can replace y' =sin (xy), which cannot be integrated explicitly, by y' = xy; which can; andfor sufficiently small values of x and y the error will be arbitrarily small.

PART B.CONTINUATION OF SOLUTIONS. OTHER METHODS

6. Continuation of Solutions

In Part A we proved essentially the following: If in a domain D thefunction f(x, y) is continuous and satisfies a Lipschitz condition in yuniformly, for every point (xo, Yo) of D there is a rectangle Ro such thatthe integral curve y(x) of y' = fix, y) passing through (xo, Yo) can beextended at least to the right and left sides of Ro. Since Ro lies in D, byapplying Theorem 4 to the point at which the integral' curve goes out ofRo, we can extend the region in which the curve is defined. We nowprove that if D is bounded, the integral curve passing through any pointof D may be extended up to the boundary of D.

We remark first that since D is bounded, and the integral curve cannotpass out of D, there exist positive numbers I and m, such that the integralcurve passing through (xo, Yo) can be defined in the open interval

Xo-I < x < Xo + m (1)

but not outside it. Our theorem will demonstrate that as x --+ Xo+ mthe integral curve approaches the boundary of D; a similar proof holdsfor approach on the left.

Theorem 11. Let D be an arbitrary bounded domain, and let f(x, y) becontinuous in D and satisfy a Lipschitz condition in y locally in D,· i.e.,for every point ~fD there must be a neighborhood N in which the conditionis satisfied uniformly. [For example, it is sufficient that (J/O'g)f(x, y)exist and be continuous in D.] Let the solution y(x) ofy' = f(x, y) passingthrough (xo, Yo) be defined on the right only for

Xo< x < Xo + m (1')

Then ifp(x) is the distance of the point P, (x, y(x» from the boundary C ofD

lim p(x) = 0 (2)a:..-:to+ m

Proof. Let € > 0 and let S be the region in D consisting of all points ofD having a distance greater than € from C. Suppose the solution(x, y(x» has points in S as x --+ Xo + m. Then there exists a monotone


increasing sequence {Xi}' j = 1, 2,' . " such that Xi ~ Xo+ m and thepoints Pi' (Xi' y(Xi)) are all in S. Since S is bounded, the points Pi haveat least one cluster point P, (x, fi), which is in S or on its boundary andhence is inside of D. Since Xi ~ Xo+ m, x = Xo+ m so that Xi < xfor allj.

Since P is in S, P is the center of a rectangle R which is in D and withedges X = x ± a, y = fi ± b for some positive a and b. Let M be abound on III in R. Then by decreasing a if necessary we can assumethat a < b/(4M).

Let RI be the rectangle with edges X = x - a, X = x, and y= fi ± lb.All points Pi must be in RI for j large enough. In particular, then, forsome n, Pn , (xm y(xn)) is in RI • Hence from Theorem 4 it follows that thesolution of y' = f(x, y) through Pn must exist to the right of X n and stayin R. Indeed for the case here, h of Theorem 4 is given by

h = min (x + a - Xn , lb/M)

and since x+ a - Xn ~ 2a and lb/M > 2a, it follows that h = x+ a - xn•

Hence the solution y(x) exists for Xn < X ~ X + a. Since x = Xo+ m,this contradicts the fact that the solution y(x) is in D only for Xo - I <X < Xo + m. Hence as X~ Xo + m, the solution can have no point~ inS. Since € is arbitrary, this proves the theorem.

y

y=o

Fig. 2

y =(2/3%)3/2

This does not necessarily mean that the integral curves will actuallyapproach some particular point of C; they might oscillate over a finiteinterval in the neighborhood of the boundary. Also, a number ofintegral curves may approach the same point of the boundary; e.g., ifD is any bounded domain such that in D, X > 0, and f(x, y) = yl, thenany integral curve lying between y = (Ix)! and y = 0 will approach theorigin on the left (Fig. 2).


If D is unbounded, Theorem 11 does not hold as stated; however, wemay replace it by a corollary:

Corollary. If D is unbounded, under the other conditions of Theorem 1I,as x-+- Xo + m either

(a) y(x) becomes unbounded, or(b) y(x) actually approaches the boundary of D.

For if y(x) does not become unbounded, we may replace D by a suitablebounded domain, and Theorem 11 is applicable.

Under cenain conditions we can go farther and state that, even thoughD is unbounded, the integral curves will approach a real boundary. Forexample:

Theorem 12. Let f(x, y) be continuous in D; Xl < X < x2, - 00 <y < 00, and satisfy a Lipschitz condition on y uniformly in D. Then theintegral curve y(x) passing through any point (xo, Yo) of D is defined over thewhole open interval Xl < X < x2•

Proof. Suppose there is a point X, X o < X < x2, beyond which y(x)cannot be continued. Since the Lipschitz condition is satisfied uniformlyfor some k, we may apply Theorem 2 to the approximate solution'o(x) =Yo over the interval Xo< X < X - €, for any € > O. The errorof 'o(x) is

d-max d~ - f(x, '0) =max If(x, Yo)I, (3)

This maximum is finite, since f(x, Yo) is continuous over the closed intervalXo< X < X < x2 ; let it be M. Then by Theorem 2

Iy(x) - Yol < ~ [ek(f-Xo) - 1], (4)

Hence y(x) is bounded in the neighborhood of X = x. Therefore bythe Corollary to Theorem II, (x, y(x» must approach the boundary ofD as x -+- x. This is manifestly impossible, and therefore y(x) must bedefined for Xo< x <x2• Similarly, y(x) must be defined for Xl < X < xo•Hence the theorem.

For example, a solution of the equation

:; = P(x) cos Y + Q(x) sin y

[where P(x) and Q(x) are polynomials] passing through any given pointmust be defined for all values of x.


Now that we.have shown the existence and uniqueness of solutions invery general domains, we prove that the Cauchy-Euler method may beused to approximate an exact solution arbitrarily closely between any twovalues ofx for which the solution is defined.

Theorem 13. Under the hypotheses of Theorem 11, let y(x) be an exactsolution of y' = f(x, y) in some domain D, passing through (xo, Yo) anddefined for x = Xl. Then the value of y(XI ) may be determined by theCauchy-Euler method to any degree of accuracy by a sufficiently fine subdivision of the interval Xo< X < Xl.

Proof. The boundary C of D and the curve y(x), xo :::;;: x:::;;: Xl' havea nonzero distance d, being both closed, disjoint, bounded sets. Let 1] beany number, 0 < 1] < d. The strip S, Xo < x:::;;: Xl' Iy - y(x)1 :::;;: 1], liesentirely in D. . Hence there is some value k of the Lipschitz constant forf(x, y), valid throughout S. We prove that if by the Cauchy-Eulermethod we begin to construct an approximate solution y(x) at (xo, Yo),satisfying the differential equation with an error €

(5).then y(x) will be defined over the whole interval xo :::;;: x:::;;: Xl' andIy(xl) - y(xl)1 < 1].

We know by Theorem 2 that y(x) will certainly be defined in someinterval to the right of xo, say Xo< X :::;;: i. Then by Theorem 5 we cancompute the actual error; it is

ek1f - ZoI - 1Iy(i) - y(i)1 < 1] ek1x,. -Zol _ 1 (6)

Hence if i < Xl' Iy(i) - y(i)1 < 1]; and therefore y(x) still is in thestrip S. As long as y(x) stays in S, we can continue the approximatingprocess, since in S, f(x, y) satisfies the conditions of Theorem 2. Hencey(x) is defined up to X = Xl' and from (6) Iy(xl) - y(xl)1 :::;;: 1]; whencethe theorem.

7. Other Methods of Solution

Another method of proving the local existence of a solution of y' =f(x, y) is Picard's method of successive approximations. The idea behindthis method is as follows. Suppose y = 4>(x) is an approximate solutionof

dydx = f(x, y), y(xo) = Yo (1)

(2)

(3)


with the initial condition ~(xo) = Yo' Let us replace the unknownfunction y in the right-hand side of (1) by the function ~(x). This yieldsthe differential equation

dydx = f(x, ~(x»,

which can be solved by a quadrature. It seems very plausible that thesolution of (2) will better approximate the exact solution of (1) than will~x). Picard's existence proof consists of repeating this process to forman infinite sequence of approximate solutions which are· then shown toconverge to an exact solution. The process also produces approximatesolutions of any required degree of accuracy. Observe the analogybetween this method and Newton's method of approximating solutionsof algebraic equations.

It will be more convenient to transform (1) into an integral equation;we state formally:

Lemma 1. Let f(x, y) be continuous oJ<er a domain D, and let y(x) be anycontinuous function defined over some interval Xl < X < x2, and admissible[i.e., such that (x, y(x» is in D]. Let Xobe any point such that Xl < X o < x2•

Then a necessary and sufficient condition that y(x) be a solution of thedifferential equation (1) is that it be a solution of the integral equation

y(x) = Yo +f~ f(t, y(t» dt,

The proof is trivial; we have already used the lemma implicitly.We shall demonstrate the existence of the solution of (3) by Picard's

method under the same assumptions as in Theorem 4. This theorem isclQsely related to the so-called fixed point theorems of topology, whichstate that given a continuous transformation of an abstract space S intoitself, under certain conditions, there must exist points which remainunchanged under the transformation. Now we may take as the "points"of S a set of continuous functions ~ over Xl < X ~ x2• If we considerthe transformation L(~)

L(~) = Yo +i~ f(t, ~(t» dt (4)

it can be shown that Sand L satisfy the conditions ofa certain very generalfixed-point theorem, and hence that there exists at least one "point" ~

for which L(~) = ~, i.e., for which (3) holds.We prove our theorem, however, by more elementary and constructive

means.


Lemma 2. Let R: /x,- xol ~ h, Iy - Yol ~ Mh be the rectangle ofTheorem 2 in which f(x, y) is continuous and bounded by M. Let ePo(x) bean arbitrary function with continuous first derivative and admissible in R[e.g., set ePo(x) == Yo]' Define ePl(X) by

ePl(X) = Yo +ix

f(t, ePo(t» dt, Ix - xol < h (5)Xo

Then ePl(X) is admissible and ePl(XO) = Yo'

Proof. Obviously

lePl(x) - Yol <L:I/(t, ePo{t»1 dt ~ max /I(t, ePo{t»IIx - xol

~ Mh, Ix - xol ~ h (6)Hence ePl(X) is admissible.

By repeated applications of the lemma we may define an infinite sequenceof admissible functions by the recursion formula

eP?l+l(X) = Yo +ix

f(t, ePit) dt, n = 0, 1, 2,' . "Xo

We now state our main result:

Ix - xol ~h'(7)

(8)Ix - xol ~h

Theorem 14. If in addition to the assumption of Lemma 2, f(x, y)satisfies a Lipschitz condition on Y in R for some constant k, then thesequence {ePn(x)} converges uniformly and absolutely to afunction satisfyingthe integral equation (3) and hence the differential equation (1).

Proof. Let D be the error with which ePo(x) satisfies (1) in Ix - xol ~ h

IdePo(X) Idx - f(x, ePo(x» ~ D,

Integrating (8) from Xo to x, we obtain

lePo(x) - ePl(X)I = lePo(x) - Yo -ix f(t, ePo{t» dtl < Dlx - xol (9)Xo

We wish to provekn-lD Ix - xoln

lePn(x) - ePn-l(X)I< , ' Ix - xol ~ h (10)n.

We give a proof by induction for x >xo. Equation (10) is certainly truefor n = 1. Assume (10) for n = m. By the definition of ePn(x)

lePm+l(X) - ePm(X)I"<ixI/(t, ePm{t» - f(t, ePm_l(t»/.dtXo


and by the Lipschitz condition

< ki~ Ic/>m(t) - c/>m-l(t)\ dt (11)

Since we have assumed (10) true for n = m, we have

(12)

whence

(13)

(14)Ix - xo\ < h

This being (10) written for n = m + 1, the proof by induction is complete.Equation (10) may now be written

kn-1Dhn

Ic/>n(x) - c/>n-l(X) \ < "n.

This proves that the sequence {c/>n(x)}, or equivalently the series

ex>

c/>o(x) + I (c/>n(X) - c/>n-l(X»n=1

(15)

cOI}verges uniformly to a continuous function c/>(x). But in (7) we maypass to the limit and interchange the limit process and integration process,as we have done several times before. Hence

c/>(X) = !~~ c/>n(x) = Yo + :~~ L: f(t, c/>it» dt

= Yo +L~ f(t, c/>(t» dt (16)

This completes the proof.Observe that the existence of a solution has been proved only for the

interval Ix - xol < h. Thus Picard's method (unlike the Cauchy-Eulermethod) is suitable for finding approximate solutions only in a smallinterval about xo' But in such an interval it is very convenient. Theerror after the nth approximation may be easily computed; it is

ex> D ex> (kh)"Ic/>(x) - c/>n(X)I< .,~n Ic/>"+l(x) - c/>.,(x)\ < k "=~+1---;l

< D (kh)n I (khY = D (ekh _ 1) (kh)n (17)- k n! .,=1 v! k n!


Finally we remark that the Cauchy-Euler method may be generalizedby using more refined types of approximation to the exact solution.For example, the method of Adams and StOrmer uses parabolic arcswhose extra constants are so chosen as to fit more closely the part of thecurve already constructed. Runge's method uses line segments whoseslope is determined by the value of f(x, y)at several points, using aninterpolation formula. These methods supplant our crude ones in thesolution of any practical problem.

2

Systems ofDifferential Equations

1. Introduction. Vector Notation

We now consider a system 01 differential equations of the form:•

dXldt = Il(xl, X2,' • " X,,; t)

dX2dt = !t(Xl , X2,' • " x,,; t)

(I)

dx"dt = lixl' X 2,' • " x,,; t)

where It, 12" • " I" are single-valued functions continuous in a certaindomain of their arguments, and Xl' X2,' • " X" are unknown functions ofthe real variable t. A solution of (I) is a set of functions xl(t), x2(t),' • "x,,(t) satisfying (I) for some interval tl < t < t2•

To treat the system (1) it is convenient to introduce vector notation.An n-dimensional vector X is defined as an ordered set of n real~umbers,

Xl' x2,' • " X", called the components of X, and X is designated by

X = (xb X 2,' • " x,,)23


We shall always use capital letters for vectors and small letters for realvariables (scalars). We assume that the reader is familiar with theelements of vector theory. By IXI we denote the norm of the vector X

Ixi = (Xl2 + X 22 + ... + X n2)1

If X is a function of a real variable t

X(t) = {xl{t),· .., xn{t)}

it is continuous in t if and only if each of its components is continuous int. If each of its components is differentiable, X(t) has the derivative

dX(t) = {dXI{t) dx2(t) . . . dXn{t)}dt dt ' dt' 'dt

If X is a function of the components YI' Y2'· . ., Yn of another vector Y,it is continuous in Y if and only if it is continuous in the n variables YI'Y2'· ••, Yn simultaneously.

Let us therefore in terms of the system (1) define the vectors

(2)

where by assumption F is continuous in X and t simultaneously. Thenthe system (1) may be written as the single vector equation

dXdt = F(X, t) (3)

The vector equation (3) is obviously analogous to the scalar equation ofChapter 1

dxdt = I(x, t) (4)

and is in fact equivalent to it when n = 1. We shall discover that all thetheorems of Chapter 1 may be generalized so as to hold for the vectorequation (3).

We shall also at times consider the set of n numbers (Xl' x2,· ••, xn)

as defining a point in n-dimensional Euclidean space. Then of courseevery such point defines a vector and vice versa. We shall find it convenient to use the two ideas of vector and point interchangeably; e.g., weshall define the range in which the vector X may vary by saying that thepoint (xlJ x2,· • ., xn) (or even "the point X") may vary through a certainregion R of n-dimensional space. We shall use vector notation whereveralgebraic processes are involved. This double terminology will lead tono confusion.

In §§ 2-5 of this chapter the theory will closely parallel that of Chapter1, so we shall give detailed proofs only where there is a real difference from

SYSTEMS OF DIFFERENTIAL EQUATIONS 25

the simpler case. In these sections the theorems are numbered so as tocorrespond with Chapter 1.

2. Approximate Solutions

Let Xo = (xIO' x20,' • " xnO) and to be fixed. Let F(X, t) be continuous in the (n + I)-dimensional region R

IX - Xol < b, It - tol < a (1)

Since F(X, t) is continuous, IFI is bounded by some number M in R.Then as before we define an approximate solution of

dXdi = F(X, t) (2)

with error € as a continuous admissible function X(t) with piecewisecontinuous derivative, such that

and satisfyingIX'(t) - F(X(t), t)1 < €

(3)

(4)

for all points of some interval at which the left-hand side is defined. Asbefore, we have immediately:

theorem 1. Given € > 0, there can be constructed an approximatesolution of (2) with error € over the interval

It - tol < h = min (a, ~) (5)

For exactly as before, there exists <5 > 0 such that

IF(XI, tI) - F(X2, tJ\ < €

for (Xl' tI)' (X2, t2) in R, IXl - X2 1 < <5, Itl - t21 < <5. Choose numbersti , i = 1, 2,' . " m - 1 such that

to < tl < t2 < . . . < tm = to + h

Iti - ti-II < min (<5, t), i = 1,2,' .., m

Then the approximate solution will be given by the recursion formula

X(t) = Xi- I + (t - ti-JF(Xi- I, ti-I)

for ti- I :::;;; t:::;;; ti' where Xi = X(ti)'The details of the proof proceed as before.

(1)


3. The Lipschitz Conditions

We shall next prove the fundamental inequality. The necessaryextension of the Lipschitz condition is given by the following definitionsand lemmas.

Definition 1. A continuous scalar function f(X, t) defined in some(n + I)-dimensional region R is said to satisfy a Lipschitz condition on Xfor kif

If(XI , t) - f(X2, t)1 ::;: kl Xl - X21

for (Xl' t), (X2, t) in R.

Lemma 1. Let R be convex in the components Xl' x2,· • ., Xn of x,·i.e., let the n-dimensional region defined by (X, t) in R, t = to, be convexfor each to. Then if each of the partial derivatives [df(X, t)l/dx" i = 1,2,· .., n, exists and is bounded by N in R, f(X, t) satisfies a Lipschitzcondition on, X in R with constant k = nNe

Lemma 2. Let R be imbedded in a larger region D and let every pointof R have a distance from the boundary of D ~ <5. Then if f(X, t) and[df(X, t)l/dx" i = 1, 2,· . ·,n, are bounded by M and N respectively' inD,f(X, t) satisfies a Lipschitz condition on X in Rfor

k = max (27, nN) (2)

.Proofs. These are analogues of the lemmas of Chapter 1, § 3, the onlydifference being in the value of k. By the mean-value theorem forseveral variables

(3)

the partial derivative being evaluated at some point of R (for Lemma 1)or of D (for Lemma 2). But

nI lx, - x/I::;: niX - X'I

i=1(4)

whence the proofs proceed as before.

Definition 2. A vector function F(X, t) is said to satisfy a Lipschitzcondition on X for constant k if

IF(XI , t) - F(X., t)1 ::;: kl Xl - X21 (5)

SYSTEMS OF DIFFERENTIAL EQUATIONS '17

Lemma 3. A vector function F(X, t) satisfies a Lipschitz condition onX if and only if each of its components j;(X, t) does (where the constantsmay be different).

Proof. (1) If k holds for F(X, t), then

Ih(X1, t) - j;(X2, t)1 ~ IF(X1, t) - F(X2, t)1 ~ klX1 - X21 (6)

(2) If k holds for each j;, then

IF(X1, t) - F(X2, t)1 = L~I{h(Xl' t) - j;(X2, t)}2]i

n

~ ! Ih(X1, t) - j;(X2, t)1 ~ nkl~ - X21 (7)'=1

Finally we shall need the following:

Lemma 4. If a vector function X(t) is differentiable over some intervalt1 < t < t2, then wherever IX(t)1 =1= 0, [dl X(t)ll/dt exists and

Idl :~t)11 ~ Id:~t)I (8)

Proof. If IX(t)I =1= 0, since IX(t)1 = [i~1 x j 2(t)r, it follows that

~ dX j

kXj-dl X(t)I_ i=1 dtdt - IX(t)!

Furthermore, in any case

IX(t + dt)1 - IX(t)I < X(t + dt) - X(t)dt - dt

(9)

and from the existence of both derivatives it is clear that the same inequality will hold in the limit. Hence the lemma.

4. The Fundamental Inequality

We can now state the fundamental inequality for the vector equation(2) of § 2:

Theorem 2. Let F(X, t) be continuous in some (n + I)-dimensionalregion R and satisfy a Lipschitz condition on X with constant k. LetX1(t) and X2(t) be two approximate solutions of(2) of § 2 over some intervalIt - tol ~ b with errors E'1 and E'2 respectively. Set

P(t) = X1(t) - X2(t); p(t) = IP(t)I; E' = E'1 + E'2 (1)


Then

Ip(t)1 < eklt-tol Ip(to)I + i (eklt-tol - I) (2)

for It - toI< b.

Proof. Wherever P'(t) exists and pet) =F 0, by Lemma 4 p'(t) exists and

Ip'(t)1 < /P'(t)/

By the definition of approximate solution

/P'(t)/ = IX1'(t) - X2'(t)1 < IF(X1(t), t) - F(X2(t), t)1 + £

and by the Lipschitz condition on F

IP'(t) I < kl X1(t) - X 2(t) 1 + 4: = kp(t) + 4:

whence afortiori we havep'(t) < kp(t) + £ (3)

From here on the proof follows exactly as in Chapter I, since we need tointegrate (3) only over intervals in which pet) =F 0, and therefore p'(t) isdefined except for a finite number of points. There is the additionalsimplification that in the present case pet) is nonnegative.

5. Existence and Properties of Solutions of the System

From the fundamental inequality our basic results follow exactly as inChapter 1. We omit the proofs of the following theorems; they differonly formally from those of the simpler case.

Theorem 3 {Uniqueness). Let F(X, t) be continuous and satisfy a Lipschitzcondition on X in some neighborhood of the fixed point (Xo, to). Thenthere can exist at most one solution of

dXdt = F(X, t) (1)

such that X(to) = Xo.

Theorem 4 (Existence). Let F(X, t) be continuous and satisfy a Lipschitzcondition on X in the (n + I)-dimensional region R

IX - Xol < b, It - tol ~ a (2)

Let M be the upper bound of IFI in R. Then there exists a solution X(t)of (1) defined over the interval

It - tol < h = min (a, ~) (3)

(We omit the analogues of Theorems 5 and 6.)


Theorem 7. Let F(X, t) satisfy the conditions of Theorem 4. Thenthere exists a region S

IX - .Kol ~ a' < a, It - tol ~ b' < b (4)

with the following property; if Xo is a vector such that IXo - .Kol < a',then there exists a unique solution X(t) of(1) with the initial value X(tJ = Xo,defined for It - tol ~ b'.

Theorem 8. Let S be any region with the properties of Theorem 7.Then the solution X(t) of (1), considered as a function of its initial valueXo = X(to), is continuous in Xo and t simultaneouslY,for (Xo, t) in S.

The proof of the differentiability of the solutions with respect to theirinitial values is more difficult; we give it in full.

Theorem 9. Let F(X, t) satisfy the conditions of Theorem 4 for R

IX-.Kol~a, It-tol~b (5)Write

(6)

i = 1, 2,' . " n

; = 1

i =F 1

(7)


We shall prove that

lim P.(t1xlO, t),Ll21.o.....0

i = 1, 2,' . " n (8)

,#

(9)

(~,

(10)

exists uniformly for (Xo, t) in S. By Theorem 8, x~t) [hence P.(!1xIO' t)]is continuous in ~O' X20,' • " xnO' t simultaneously for (xo, t) in S.Therefore by virtue of the uniform convergence of (8) it will follow that

ax.(Xo, t) l' (A- _ )~ = 1m P. u;(;IO' toXIO Ll:r:u.....O

has the required properties.By applying the fundamental inequality to the two solutions

XI" • " xJ and (Xl + ~, XI + !!al ,' • " Xft + t1xft)' we obtain

V~I + t1x1" + ... + !1Xftl '5:. etbl~ol, It - tol '5:. b'

By the theorem of the mean in n variables, we may write

dt1x.(t)dt = fa(~ + !1~, XI + t1x1,' • " Xft + t1xft; t)

- f~XI' XI" •• xft; t)xft a

= I ~f~~, XI" • " Xft; t) !!a1: + 'fJ~t)i-IOX1:

where

(11)

uniformly for (XO, t) in S. Then by (7) and (10) we may write

dp.{t1xlo, t) ~ a r A-d = k ~ fa(x l , XI" • " Xft; t)P1: + ~,<uXIO' t)t i-IOXle

(12)

where

But by (9)

(13)

(14)

Hence by (11)

uniformly for (XO, t) in S.

lim I'.(t)1 = 0Ll:l:to.....O

(15)

SYSTEMS OF DIFFERENTIAL EQUATIONS 3t

Now consider the system of differential equations in n unknownsqi' i = 1, 2,· . ., n

dqi n d-d = I ~ fi(xl(t), x2(t),· . ., xn(t); t)qle

t k=l aXle

with initial conditions

(16)

i = 1

The right-hand side of (16) is obviously continuous in the q's and iscontinuous in t by definition. By § 3, Lemma 1, it satisfies a Lipschitzcondition on the q's. Hence (16) satisfies the hypotheses of Theorem 4.Now the system (PIt P2'· .., Pn) satisfies (16) with an error

€ = n max I'i(t)I, i = 1,2,· .., n, (XO, t) in S (17)....

Hence by the proof of Theorem 5 (as in Chapter 1), lim Pi = qi; i = 1,£-+0

2,· .., n, uniformly for (XO, t) in S. Therefore by (15) and (17), lim PAt)exists, uniformly for (XO, t) in S. This completes the proof. ~Xl0-+0

From Theorem 9 we can immediately deduce the following theoremon continuity and differentiability of the solution with respect toparameters:

,Theorem 10. Consider a system of differential equations in which thefunctions /;. depend upon any number ofparameters PIt· . ., Pm

i = 1, 2,' .., n (18)

If each of the f's has partial derivatives with respect to Xl' x2,· . " Xn;PI' P2" . " Pm continuous in some (n + m + I)-dimensional region R,then the solutions

i=I2···n" , (19)

will have partial derivatives in PI' P2" . " Pm continuous in all theirarguments through whatever part of R the solutions (19) are defined.

Proof. Regard the parameters PI' P2'· . ., Pm as new variables, andadjoin to (18) the additional equations

~; = 0, j = 1, 2,· . ., m (20)

Then all the conditions of Theorem 9 are met by the system (18) + (20);hence the solutions Xi have the required properties with respect to the

(21)


"initial values" of the I-l'S; since the I-l's are taken as constants withrespect to t the result follows.

We shall not need the analogues of Theorems 10 and 13 of Chapter 1.We state the two following theorems again without proof:

Theorem 11. If in some bounded (n + I)-dimensional domain R, F(X, t)is continuous and locally satisfies a Lipschitz condition in X, the solution of

dX- = F(X t)dt '

passing through any point of R may be uniquely extended arbitrarily closeto the boundary of R.

Theorem 12. If R is the domain

X unrestricted

and if F(X, t) satisfies a Lipschitz condition uniformly in every subdomainof the type

t1 < t1' < t < t2' < t2, all X

then the solution of (21) passing through any point of R may be extendedthroughout the entire open interval t1 < t < t2• •

This is an immediate consequence of the fact that the solution can beextended throughout any subinterval of t1 < t < t2•

As an example,. take the extremely important case of a linear system;i.e., a system of the form

dx. n

dtJ = .I ailt)x; + bi(t), i = 1, 2,· .., n (22)

1=1

where au(t) and bi(t) are continuous in some open interval t1 < t < t2•

By § 3, Lemma 1, (22) satisfies the conditions of Theorem 12; hence thevalues of the x's at any point to (where t1 < to < tJ determine a solution(of course uniquely) throughout the whole open interval t1 < t < t2• Thestudy of such systems will occupy the following chapters.

Finally we remark that Picard's method may be applied to the generalvector equation; the generalization of the process of Chapter 1 is perfectlystraightforward. We shall see an application of this in the next chapter.

6. Systems of Higher Order

Consider the differential equation of nth order in one unknowndnx- =ji(x(n-l) ... x' x· t) (1)~n " , ,


This equation may always be replaced by a system of differential equationsof the first order. To do this, we take x and its first (n - I) derivativesas new unknown functions Xl' X 2,' • " X n , which must obviously satisfythe relations

dXI--xdt - 2

dXn_ 1--=xdt n

dXndt = f(xm ' • " X 2' Xl; t)

(2)

It is easily seen that (I) and (2) are equivalent, i.e., that any solution of(2) defines a solution of (1) and vice versa. It is also clear by the existenceand uniqueness theorems that wherever f satisfies a Lipschitz conditionon its first n arguments, the solutions of (I) will exist and be uniquelydefined by the value of X and its first (n - I) derivatives at any point to'All our other theorems admit ofobvious generalizations to the equation (1).

If in particular f is linear in the x's, i.e., if

n

f(xm xn- l ,' • " Xl; t) = ! ai(t)Xi + b(t)i=l

(3)

it is clear that (2) will satisfy the conditions of Theorem 12, hence thatthe solutions of (1) may be extended throughout any interval in which thea's and b's are continuous.

Finally, any system of differential equations of higher order in severalunknowns may be. reduced to a system of the first order, provided onlythe system is explicitly solved for the highest derivative appearing of eachunknown function

d IDIXi _ t( (k) )

dt IDI - J i X; ,t (4)

where k = 0, 1,' . " m; - 1; i, j = 1, 2,' . " n. It is easily seen that11

(4) is reducible to an equivalent first-order system of N = ! mi equations,i=l

and that all our theorems likewise hold for such a system.

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