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~5{r{ The Theory of Ordinary Differential Equations J. C . BURKILL

@

UNIVER S I T Y

S e D FR S

M A TH E MATICAL

TEXTS

General Editors A lexander C. A itken DSc FRS D. E. Rutherford DSc DrMath

:;,.

OLIVER

AND

BOYD

LTD

UNIVERSITY MATHEMATICAL TEXTS GENERAL EDITORS ALEXANDER C. AITKEN, D.Sc., F.R.S. DANIEL E. RUTHERFORD, D.Sc., DR. MAm. DBTEIWlNA!nS AND l\IATIUCBS , A. C. Aitken, D.Sc., F.R.S. STATISTICAL l\IATD&MATICS A. C. Aitken, D.Sc., F.R.S. Tn£ TJIEORY OF ORDINARY DlFFBRBNTIAL EQUATIONS J. C. Durkill, Sc.D., F.R.S. RussiAN-ENousu l\IATDIWATICAL VocABULARY J. Durlak M.Sc., Ph.D., and K. Brooke B.A. WAVES • C. A. Coulson, D.Sc., F.R.S. ELECTIUCITY • C. A. Coulson, D.Sc., F.R.S. Pno.JECDY£ GnouETRY T. E. Faulkner, Ph.D. INTEGRATION , R. P. Gillespie, Ph.D. PARTIAL DIFFERENTIATION R. P. Gillespie, Ph.D. R£.u. VAlliABLE J. M. Hyslop, D.Sc. INFINITE S&nms J. l\1, Hyslop, D.Sc, lNTEORATION OF OnniNARY DlFFBRENTIAL EQUATIONS E. L. lnce, D.Sc. lNTnoDUCDON TO TOE TunonY OF FD.'ITE GnouPs \V. Ledermann, Ph.D., D.Sc. GnnuAN-ENoLisn l\IATIIEliATICAL VocABULARY S. Macintyre, Ph.D. and E. Witte, M.A. ANALYTICAL GnouBTnY OF ToRE£ DmENSIONS \V. H. McCrea, Ph.D., F.R.S. TOPOLOGY E. l\1, Patterson, Ph.D. FuNCTIONS OF A CoMl'LEX VARIABLE E. G. Phillips, M.A., M.Sc. SPECIAL llnLATIVITY \V, Rindler, Ph.D. VoLUME AND lNTEoRAL • W. W. Rogosinski, Dr.Phil., F.R.S. VnCTOn l\I&TIIODS • D. E. Rutherford, D.Sc., Dr. Math. CLASSICAL l\lnciiANICS D. E. RutltcrCord, D.Sc., Dr. l\latl1. D. E. RutllerCord, D.Sc., Dr. l\latll. FLUID DYNAMICS SPECIAL FUNCDONB OF l\IATDEMATICAL PDYBICB AND CUB1118TRY I. N. Sneddon, D.Sc. TENSOR CALCttLUS • B. Spain, Ph.D. To&onY OF EQuATIONS • H. \V. Turnbull, F.R.S.

THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS

J. C. BURKILL Sc.D., F.R.S. FELLOW OF P&TERDOUSE, AND READER IN MATIIEUATICAL ANALYSIS IN TilE UNIVERSITY OF CAllDRIDCE

OLIVER AND BOYD EDINBURGH AND LONDON NEW YORK: INTERSCIENCE PUBLISHERS, INC.

1962

@

FmST PunLJSIJED

1956

SECOND EDITION

1962

Copyright 1962 J. C.

BvnKILL

PRINTED IN IIOLLAHD DY NoV• DQRITtiA0S DRUKKERIJ,

VOORIIEEN BOI!!KDRIJICKERIJ OEIIROEDEIIS IIOI'ISIWA, ORONII« o, every solution of the equation y' + ay =f(m) tends to the limit lfa as m-+ co. If, however, a < o, only one solution tends to lfa. 4. Sketch the solutions of each of the equations 1

(a)

y'

+ y = m'

1

(b) y'- y

= -;;·

5. Sketch the solutions of each of the equations (a)

y' =

1 111

+y

1

-

1,

1 (b)

y' = 1 - m1

-

y1 '

What relation is there between the two sets of curves?

11

EXISTENCE OF SOLUTIONS

§4

6. Verify that Ute process of successive approximation or § 8 applied to the equation y' = ky yields the known solution. Curry out the same verification for the pair of simultaneous equations y' = z, ::' = - y 7. Find the solution, for :z:

y'

(y ~

= 0, :: =

1, when :z:

=

0).

0, of the equation

= max (:z:, y),

y(O) = 0.

8. Find the solutions, as far as Ute terms in :z:l, or the equations (i) y'

= zs + siny,

(ii) y' = :z:z,

y(O)

=

0;

y(O) = 0, ::(0) 1.

=

='=:r+y,

9. Discuss the behaviour ncar the origin of solutions of the equation

(am- bl ::/= 0), distinguishing the cases (b - l)s

+ 4am >

=0

0,


R 1 , in 0 ;:;;; :t ~a, and F(:z:, y) is continuous In (:r, y) for 0 ~ :r ~a and all y. Given that y 10 y 1 are solutions In 0 ~ :z: $ a of

y' = F(:r, y) respectively wiUt y 1 (0) ~ y1 (0), prove tlmt y 1 Show that the equation y' = 1

+ y 1 + :r1

(:r ~ 0),

+ R 1(:z:)

> y 1 in 0
I z(x) I for x > x 0 , so long as z(x) does not van-ish. The following calculation illustrates the use of,a comparison differential equation for estimates of magnitude of solutions. If, in (Y), as x-+ oo, g(x) -+ -a11 (a > 0), then, for arbitrarily small positive 11• any positive solution of (Y)

satisfies the inequalitieB elo-.,)z

< y(x)
efa-ql:r: for all

sufficiently large .r.

The other inequality is proved similarly.

12. Zeros of solutions. If (Y) has a solution (not identically 0) with more than one zero, theorem 9 shows that there must be an interval in which g(x) > 0. THEOREM 11. A finite value ~ cannot be a limit point of zeros of a solution u(x) of (Y), unless u(.r) = 0. PROOF. Suppose ~=lim Xn, where u(.rn) = 0. Since u(x) is continuous, u(~) = 0. Also '("') _

.

U c ; - l1 m

.,,. ... (

u{.rn) - tt{~) _ 0 • Xn- ~

By the uniqueness theorem, tt(.r) = 0. 'l'IIEOREllt 12. Tile zeros of two linearly independent solutions of (Y) interlace i.e. between two consecutive zeros of one lies a zero of the other. PROOF. Observe that, if two solutions both vanish at a point, their Wronskian is 0 and they are linearly dependent {i.e. one is a constant multiple of the other). Suppose that tt 1 (x), u 2{x) arc linearly independent solutions of (Y), and that oc, fJ are consecutive zeros of ul(.r). From ui' + gu1 = 0, u4' + g1t 2 = 0, we have ui' u 2 - u 1 u~' = 0. Integrate from oc to fJ and we have

28 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS § 12

Hence Since tX, pare consecutive zeros of u 1 (x), ~(tX) and ul(p) have opposite signs. Therefore u 2 (tX) and u 2 (p) have opposite signs, and so u 2(x) vanishes at least once between tX and p. Interchanging the roles of u 1 and u 11 , we see that their zeros interlace. THEOREM

18.

If 0 < m < g(x) < M for a ~ x ~ b, and, if x0, x 1 are consecutive zeros (lying in (a, b)) of a solution of (Y), then n

v'M < x1

-

Xo


0, the series for F(a, b; c; 1) converges and F(c)r(c- a- b) F(a, b; c; 1) = F(c- a)r(c- b)•

t Gillespie, lntegratUm,

§ 88.

56 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS § 29

PnooF. If Un is the nth term in the series for F(a, b; c; 1 ), Un Un+ 1

=(1+n)(c+n)= 1 +c-a-b+1 (a

+ n)(b + n)

n

+

o(~)· nil

Convergence is shown by Gauss's test. Then, from Abel's limit theorem, t F(a, b; c; 1) =lim F(a, b; c; a:)

=...1-0

=~~r(b 9~1-b) J:tb-1(1-t)c-b-l(1-a:t)-a dt, from {28.1) r{c) J1 .111-1( )c--o-6-ldt - F(b)F(c-b) o ,- 1 - t ' since this last integral exists, and (1 - a:t)- 0 or :: < O, it vanishes at C = oo or C = - oo respectively. \Ve arrive thus at solutions of the equation, or which the following are typical, where C is the interval of the rcnl axis specified. If p > 0, q > 0, C is (0, 1 ); p > 0, z < 0, C is (- oo, 0). If p < 0, q < 0, :t < 0, no single segment of the real nxis meets

62 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS § 33

the requirements Cor C, but we can take o contour composed of the port or the real axis Crom - co to - 15, then o circle of rndius (j and centre the origin, und then returning from - IJ to - co. These indications do not profe88 to give the complete solution of the example, but they lend up to the next section.

33. Choice of contours. In the general case of § 82, there arc various possible types of contour C. The condition [fP(C)Jc = 0 is satisfied if C is closed and fP(C) returns to its initial value after describing it. In this case, C must contain at least one of the points ex,. inside it, for if not it would give only the trivial solution w = 0. Another possibility is to make C go to infinity in one or more directions for which fP(C} -+ 0; as fP(C) depends on z, these directions will depend on the values of z. When C goes round the point cx1 counter-clockwise the power (C - cx1)k1 is multiplied by e2'Zft1 • We can therefore define a C for which [fP(C)Jc = 0 by taking a loop round each of cx1 and «:a twice in opposite directions (a doubleloop contour), as shown in the figure.

Fig. 1.

For clearness in the diagram, the parts of the contour are drawn out separately; they can in fact be circles described twice round « 1 and «a together with segments of the line joining them. By taking double-loop contours round cx1 and each of «2, ••• , «,. in turn we obtain n - 1 independent solutions of the equation and these solutions have the advantage of being valid for all values of z. These n - 1 solutions may be expected to be independent; a general formal proof of independence (e.g. by the

§34

CONTOUR INTEGRAL SOLUTIONS

63

Wronskian criterion of§ 7) would be formidable. If it is possible to deform one contour cl continuously into another C2 without passing over any of the points ex1, then integrals along cl and c'i! yield the same solution of the differential equation; if such a deformation is not possible, the values of the integrals arc in general different. The reader will sec that it is impossible to deform one of the double-loop contours defined in the last paragraph into another without passing over points ex1• To construct an nth independent solution of the equation valid for given values of z, choose a direction in the 4, plane for which the real part of (z + k0 )C is negative, and take as the contour of integration, say, one coming from infinity in that direction, encircling ex1 (and no other ex) and returning to infinity in the same direction. (Fig. 2)

Fig. 2.

It is possible to define each of the n solutions by a contour of this type instead of taking n- I double-loop contours and only one of this type. 34. Further examples of contours. General principles governing choice of contours have been laid down in § 88. Some details have still to be clarified - for instance, we have still to show how to find n independent solutions when the ex's arc not all different. The procedure to be followed will be seen more readily from a study of particular examples than from description in general terms. As a first example it is instructive to see how the technique of § 82 would yield the known solution of the linear equation with constant coefficients.

64 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS § 34 E:mmple 1.

+ ... + b1w' + b0ro =

bAwiAI

We lind that

0.

Jc ~tP(C)t¢ is a solution if

fc ~tP(C)R(C)~ = o, where R(C)

Suppose that (C Then

/1)'

= bAC" + ... + b1C+ b0• is a factor or R(C).

fc ~

P(C)R(C)dC

Ar

=o A1

/1)' + ... + c- p + p(C), regular at C= fJ, and Cis a contour enclosing fJ

iC

P(C)

=

·-~-!+l"cJC, zf e=•c c-!-!k(1 - C>l"tiC along appropriate contours. 8. Find solutions In series of the differential equation

y" - 'J.ry'

+ 2Ay =

0.

Investigate also solutions of the Corm fceb 1u(l)dt, where Cis a suitable contour. Show in particular that, if). eo e-1 1+9~~:1 t-A-1 dl

J o


- !, C can be taken to be the segment (- 1, 1) of the real axis as in § 44; more generally, a figure-of-eight or an infinite contour will serve.

BESSEL FUNCTIONS

§46

83

46. Application of oscillation theorems. In this section the variables are real. Bessel's equation a;3y" xy' + (;z;S - vs)y = 0 is reduced to normal form by the substitution y = va;-l, giving

+

v"

+ (1 -

4v:a4x:a

1) v =

0.

This equation is then satisfied by v = ;z;!J.(;z;). Since the coefficient of v tends to 1 as m -+ co, theorem 18 of § 12 gives at once THEOREM 24. If ot, is the rth positive zero of J.(;z;), then, as r-+ co,

(i) !X.r+1 - !X.r ,...... n, and (ii) !X.r ,...... rn. The next result uses only Rolle's theorem and makes no appeal to the work of Chapter III. THEoREM 25. The zeros of J.(;z;) and J.+l(;z;), other than a; = 0, interlace. PRooF. The relation J•+l(;z;) = QJP

-.!!.. {J.(;z;)} rk

QJP

(see § 48, example 2), deducible from the generating function, shows, by Rolle's theorem, that between two zeros of J.(m)fa;P lies at least one of J.+l(a:)fa;P. Similarly the other result of the same example d ;z;P+lJ.(m) = rk {m"+lJ.+l(m)}

shows that between two zeros of m"+lJ.+l(m) lies at least one of m"+lJ.(m). Since the zeros of J.(x) and J.+1(m), other than m = 0, are the zeros of the functions discussed in the last two paragraphs, the theorem follows.

84 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS § 46

We next apply the ideas of § 18 and § 14 about eigenvalues and eigenfunctions. The function u(x) = xlJ,(i.x) {40.1) satisfies the equation

u"

+ (,ta - 4v:~ 1) u =

o.

Consider solutions which vanish at x = 0 and x = I. If v > - !, u(O) = 0 for all ).. The vanishing of u(1) means that J,(.t) = 0, that is to say, the eigenvalues arc the zeros of the Bessel function. If An,, ).,. arc two values of A (not necessarily eigenvalues) and um(x), U 8 {x) the corresponding functions as defined by {40.1), then

" + (12

Un

An -

4v24x9 - 1) Un = 0,

1\lultiply these equations respectively by U 8 , "m• and subtract. Then integrating from 0 to 1 we have [u:n(x)u8 (x) - Um(x)u~(x>JA

=

(l; - A~n)

1

fo UmUntk.

(46.2)

If v > - !. the expression in square brackets on the left-hand sidcof(46.2) vanishes for x = 0. It also vanishes for x = 1 if An, and ).,. are eigenvalues. So, if m =1= n, we have from (46.2)

and therefore

JxJ,(An,x)J,().,.x)tk = 1

0

0

(m =I= n).

as

BESSEL FUNCTIONS

§46

'fo evaluate this integral for m = n, let An be the nih eigenvalue and in (46.2) replace A.n by a continuous variable ;., taking values tending to An• Then the equation (46.2) gives

J u,.unlk 1

(.A.; - .A.!)

0

= -u,(l )u~(l)

= -

J.(A,.)AnJ~(An)•

- J.(A,)

An - Ap

J'(1 )

-+ •

lin •

And so

E:Mmples.

1. By writing ro = vzi, transform the equation dSW rl%1

+ elm =

k(k

+ 1) z1

w

into one of Bessel's type and write down its solution. 2. Prove that the equation ro" + :w = 0 can be solved by Bessel functions of order

± !·

8. Prove that Bessel's equation may be written in either of the

forms

+v+ 1} {.!!..th - :r:!:.} y + y {~ dz :r:

0• t!- v ~ 1} ~~ + ;} y + y = 0. =

Hence show that J.+l(:r:) = -

(~ -

J._,(-1:) =

(~ + ;) J.(-1:),

;) J.(-1:),

86 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS § 46 4. Prove that J,.(m)

d)n = (- 1)":1:" {1 \; ;& J 0 (m),

5. Prove that, If a > 0, I'

J

co 0 e-••J,(bt)IP-'dt

b'T(p + v)

= 2"(a'+b')IIP+t>lT(v

Deduce the value of

6. Ifv

+ v > o,

J;

+ 1) F

(p+v 1-p+v b' ) 2' -2-;v+ 1 ; a'+b' •

e-••J0 (bt)dt.

+ ! > p + v > 0, prove by making a tend to 0 in example 5

that co 21.-IT(lp + !v) o J.(t)IP-'dt = T(!v- il' + 1).

J

(It may be assumed that, for large values of t, I J,(t) where K is a coDBtant. This will be proved in § 49.)

I < Kt-!,

CHAPTER IX

ASYMPTOTIC SERIES 47. Asymptotic series. An asymptotic series is a series which, though divergent, is such that the sum of a suitable number of terms yields a good approximation to the function which it represents. The idea is most readily grasped from an example. Example. Find an approximation for large positive values or x to the solution or the equation

y'-y=

-~ X

which tends to 0 as or~ co. The equation bas an irregular singularity at inrinity. If we carry out the process of finding a series in powers of 1/z, we obtain 1 1 Y -- -Z _ -z2

+ z2 _21 _

•••

+ (_ 1 )n-1 (n-1)1 + • • ., Z"

which diverges for all values of z. The equation, being linear and of the first order, can be integrated by a quadrature, and the solution which tends to 0 as z ~ co is found to be

y

= /(z)

=

foo eo-• dt. I1J

'

If we integrate this expression Cor /(z) by parts, we see its relation to the divergent series. 'Vc have, after n integrations by parts, f(z)

1 1 2! =--+ - - ... + z z1 z2

(n-1)1

Joo eo-•

z"

m P~

(-1)n-1-- +(-1)"nl

Now nl

J eo-• oo

-

m l"~

nl Joo

dt < z•>+l 87

Ill

eo-•dt

nl

= zn+l -.

-dt.

88 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS § -48 So the sum of n terms of the series is an appronmation to f(111) with an error less tl1an the numerical value of U1e (n + 1 )th tenn. For a given volue of 111 the terms of the series decreose in absolute magnitude until the nth term where n is the integer next Jess than 111. If 111 is Jorge, we can, by stopping at an corly term in the series, obtain an approximation of bigh aecumcy. (It lll is only 20, the sum of 4 terms gives f(IIJ) with an error Jess than 1/101 ).

48. Definition and properties of asymptotic series. The formal definition of an asymptotic series was given by Poincar~

(1886).

Let S 11 (z) be the sum of the first (n S(z)

+ 1) terms of the series

= Ao + Al + ... +A:+ ... , z z

Let R,.(z) = f(z)- S11 (z). Then, for a given range of tug z, say ot :::;;; arg z :::;;; {J, the series S(z) is said to be an asymptotic expansion of f(z) if, for each fia:ed n, lim z"R,(z) = 0. 1•1-+00 We shall write f(z) "'S(z). This definition applies to a power series in 1/z which converges for sufficiently large I z 1. say for I z I > R. For then there is a constant M, depending on R only, such that for all values of arg z

I Rn(z) I
t fn) J oo e-rv-! (1

, ! (2.v} o

+

iv ) -! dv. 2x

( 4!>.1)

92 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS § 48

The general term of the binomial expansion of (1

' -! + 2a: w)

is

( -1)" 1 . 3 ... (2n- 1) (iv )" 2"nl

2m

and the remainder after the term in v" is less than K(vjm)"+l, where K depends only on n. The contribution to the in tegml on the right-hand side of ( 4!).1) of the term in v" is

_!_)"1 . 8 .. . (2n - 1) F(n + *) (- '.kl: nl ( - .i)" ].2 . 32 . .. (2n- 1 )2 yn, ~

=

8x nl and t he contribut ion of the remainder term is less thnn Kfm"+ 1, where again 1( depends only on nand is independent of m. H ence the integr·al on the r ight-hand s ide of ( Po· Then tp(p) is called the Laplace transfonn of f(t) and is usually written ~{f(t)} or ,97(/). The Laplace transform has the following properties. {1)

.!'R(/1

+ ••• + /n) = .!l'(/1) + ••• + !t'{/n)•

(2) .!l7(cf} = c!t'(f), if c is constant. These two properties show that !t' is a linear operator. (3) ff{e- 01 /(t)} = tp(p + a). (4} If !t'{/1 (t)} = tp 1 (p) and !t'{/9 (t)} = tp11 (p), then tPt(P )tpa(P)

s;

= !t' { ft(u )/2(t - u )du}.

(5) A continuous function is uniquely determined by its Laplace transform. The proofs of {1 ), {2 ), (3) arc easy. To prove (4 ), we have, by inverting the order of integration in the repeated integral,

98 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS

.P

{f:

ft(u)fa(t - u)du}

=I: I; II(u)f (t=I; /{u)du I: (t =I: ft(u)du I: e-fl 1dt 1

2

e-91 / 2

u)du u)dt

fr9(u-+11lf 2 (v)dv

= 9't (p )tpa(P ). The property (5), which is essential in justifying the usc . of Laplace transforms, needs a more substantial investigation. This will be given after we have explained the manipulative detail. The following table is a short list of transforms of common functions.

f(t)

tp(p)

1

-p

eo'

1

1

p-a

tn-l

1

(n- 1)1

P"

tn-leol

1 (p- a)"

(n- 1)1

sin at cos at

t sin at 2a

2~3 (sin at- at cos at)

a pll+a: p p2 +as p

+ all)ll 1 (pll + all)ll

(pll

99

THE LAPLACE TRANSFORM

The method of solution by transforms. l\Iultiply the differential equation (1) by e-P 1 and integrate from 0 to co (assuming that p can be chosen so as to make the integrals converge). Integrating by parts and using the initial values of y(t) and its derivatives, we have

J: e-P y'dt = - y + p J: e-Pt ydt, J: e-Pty"dt = -Yt- PYo + p2 J: e-P'ydt, 1

0

and, generally, for s ;2;; n,

s:

e-PI yhldt

= - Ya-1 - PYa-2 - ••• - p•- 1 Yo+ p'

J: e-Piydt.

So y will satisfy the equation (1) with the given initial conditions if (aopn + alPn-1 + ... + an)9'{y(t)} = Yo(aopn- 1 + a1pn-s + · · · + an-1) + Yt(aoPn-2 + a,pn-3 + • •. +an-:) + ... + Yn-2(aop + at) + Yn-tao + .!f{r(t)} (2) The equation (2) is called the subsidiary equation. A table of transforms is used to find .!f{r(t)} from r(t), and then to find y(t) from .!f{y(t)}. Illustration. Solve the equation

+ 2y =

y"' -By'

8e 1,

given that y(O) = 0, y'(O) = 1, y"(O) = 2. The Laplace transform of the equation is (p1

8p

-

8 + 2).2'(y) = p + 2 + p--. -1

giving 9'( ) -

1

y - (p- 1)1 1 9(p

8

+ (p- 1)1 (p + 2) 1

2

+ 2} + 9(p- 1} + 8(p- 1)1 + (p

1 - 1)1 •

100 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS

From the table of transforms, the solution is y = - -~r·· + ~· + Jk1 + !Pe1• In U1e last step we nssume the fact, still to be proved, that y Is uniquely determined by .!i!'(y).

The uniqueness theorem. To prove the property (5) above, we need a lemma (Lerch's theorem). · LEIDIA. If tp(a:) i8 continuous for 0 ~a: ::::: 1, and

J: m"tp(a:)cl.v = then tp(a:) =

for n = O, I, 2, ••• , 0 for 0 ~ a: ::::: l. PROOF. If the conclusion is false, there is an interval (a, b) with 0 0 (or tp(x) :::;;:: - k < 0) for a~ x:::;;:: b. We proceed to define a polynomial p(.11) for which 0

J: p(x) tp(x)cl.v >

0

and this will contradict the hypothesis. Let c be the larger of ab, (I - a)(l- b), and let q(a:) = 1

+-1c (b -

a:)(a: -a)

Then q(a:) > 1 for a< a:< b, and 0 < q(x) < 1 for 0 < a: < a and b < a: < 1. If we choose a sufficiently large integer m, the polynomial p(a:) = {q(m)}m will take arbitrarily large values in a :JJ1 , /(:JJ, y) = 2.r; if y < - a:', /(:JJ, y) = - 2.r. CHAPTER Jl,

1. A 1e" + A 1e_., 2. Ae-too + Bt!'

+ B 1 cos .r + B 1 sin :JJ -

+ Cxe" + !a:te".

i:JJ sin .r.

{~log (az + 1) } + B sin {~log (az + 1) }. 4. A:JJ + Be" + :JJ1 + :JJ + 1. 5 • .4(1 + 2.r') + B.ry'(l + :a:1 ).

8. A cos

6. At!'+ B..:1 •

+ B) -

ye 1 =

:JJI, :JJ +A)+ (:JJ1 + l)(!z' +B). .r0 + \t- y 0 t - 11 + iZol1 + lt1 , Yo + 21 - Zof - ft1 ,

::e'

=o + ft.

7. tr(Ae"

8. .r(-i:JJ1 9. :re 1

= =

-

+ B sin :.:1 • A cos :JJ + B tan a: + ! see .r.

11. A cos :.:1

12. 18. {A log (1 - sin :JJ) + !A sin a: + B}/(1 + sin a:). 14. {:JJ +sin (t- z)- t cos (f- .r)}/P. 15. (n + 2)(1 + :r)y = :a:A+I + (n + 2)(.40f:•>+ 1 + B). 100

110 THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS

16. zly = n(:r: cos :r:- sin a:) - n 1 (:r: sin m +cos a:). 17. (Sufficiency). There are constants c1, ••• , not all O, such that

c,.,

+ ... + Caa1,. =

c1a 11

0

(1 ~ i ~ n).

l\lultiply the ith equation by c1 and add. We have

J:

+ ... + c,.u,.)1~ = o. 18. (Au 1 + Bt~a)/(u 1u;- u;u1 ). 19. A:re" + Be...,f:r:. 20. 2p.p, + p; = 0. (c1u1

CIIAPTEB IV,

Independent solutions of each of 1-12 are given in finite form when a series is so expressible. 1.

zt,

(1 - :)l. 48 64. 2. 1 + 12::' 5:4 - Ill ::'

+

~( :T l

8. Cf 22 4.. 1 5. :, ::1 + z log ::. co

6. wl

+ z,

+ .... 8 1.8 1.8.5 + 2 ::• - U z' + 2 , 4o • 6 z'

:'(I -

}

- ••• •

z)-1,

="'

= ~ (nl)•'

m1 = w1 log z - {:

1

+ (2~ )' {1 + ~) +

:4

(~ )' {1 + i + ~ + • · ·} ·

zS

7' 1 + 2'":8":4 + 2. 8. 4o. 6. 7. 8 + .... ::'

z'

=+8.4..5+···· =·+4..5.6+··· 8,

101

=

co

:"

:E I t 1 1.2 ... (n -l)'n

w.=w.log: 9. :(1 - :)-1, r

co :" (2 2 2 1) +I-ft.2• ... (n-l)'n i+2+ ... +,._l+n. 1 (1

10. :i, :i(l - !:)i.

- :).

SOLUTIONS OF EXAMPLES

11.

e-•',

e-•1 log z.

ro1

oo

2.8 ) + 5 • 6 • 1 • 2 klzl + • • • •

2

5.1 k:

12. OJ + 4k:• + kt:.~. ~ ( 1 + 18. ro1

111

Z"

= l:o n lk(k + 1) ••• (k + n OCI

- 1)

,

Z"

= : 1-•l: o nl(2

- k)(8 - k) ... (n + 1 - k)

.

2:"(1 + 21+ • • • + n1) · f (nl)' 00

= ro1 log z = !, ro1 = cosh 2yz,

For k = 1, ro1 For k 14. u

= z(1

-

~~:)- 1 ,

00

16

(1 -

(-

~~:)-1

ro1 = !sinh 2y:.

+ u log z.

2:J:I)"

'A~(2n-8)(2n-1)(2n+1)+B

m

1

.O