On a Certain Class of Linear Differential Equations

On a Certain Class of Linear Differential EquationsAuthor(s): Thomas CraigSource: American Journal of Mathematics, Vol. 7, No. 3 (Apr., 1885), pp. 279-287Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2369274 .

Accessed: 17/05/2014 06:08

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Mathematics.

http://www.jstor.org

This content downloaded from 195.78.108.152 on Sat, 17 May 2014 06:08:24 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=jhup

http://www.jstor.org/stable/2369274?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


On a Certain Class of Linear Differential Equations.

BY THOMAS CRAIG.

The subject of linear differential equations whose coefficients are singly or doubly periodic functions of the first kind has been studied by Picard* and

Floquet,t and it has been shown that such equations always admit of at least one integral which is a periodic function of the second kind.

The case where the coefficients are periodic functions of the second kind does not seem to have been attempted by any one-anad indeed in the general case the problem seems almost impossible of solution-though when the mnulti- pliers are roots of unity the solution (at least for singly periodic functions) call be easily led back to the case where the coefficients are periodic of the first kind.

In the following paper I have determined the conditions which are necessary in order that a linear differential equation shall admit of an integral which is a periodic function of the third kind-in the beginninig I limit myself to singly periodic functions.

I shall start with the case of the linear differential equation of the second order,

1. 2+ P1 + P2Y = 0,

where the coefficients Pi and P2 are uniform functions of x. What must be the form of these functions in order that (1) may have an integral which is a periodic function of the third kind ?

Suppose y = F(x) to be such an integral, and F(x + e)- = 6x+?o where X and 20 are constants; then if we change x into x + cA we must have that F(x + c') is an integral of

2. dy + dy dX -p x + (a~ i) dx + P2 (X ? ()= 0.

* Crelle, Vol. XC. Sur les equations diff6rentielles lineaires ; coefflcients doublemnent p6riodiques. f Annales de l'Ecole Normale, Feb. 1883. Sur les 6quations differentielles lin6aires a coefficients

periodiques. Ibid. May, 1884. Sur les equations differentielles lineaires a coefficients doublement p6riodiques.



280 CRAIG: On a Certain Class of Lineare Differential Equations.

Writinig then in (2) y = F(x + ), = e-Ax4AoF(x) we have, writing for brevity F instead of F(x),

dX2 + [E 2% + p, (x + E)]d+ [ -2_1)1 (x + (o) + P2 (x + o)] F = 0,

also d-2- + (x) dF F . dx2 +p dx)~+~0

Subtracting we have

[-2 + /1 (x + ?) - 1, (x)] 1Fdx + Ex2-Xil (x + c) + 1), (x + &)-p2 (x)] F = 0, which iiiust be satisfied identically; therefore

3 p1(x + () p1), (x) + 22

p2 (X + (a) =P2 (X) + ?Pi(X) + 2

and obviously 4 i1P(x + nw) = 1(x) + 2nX

P2 (x + nc) = p)2(x) + nXp1 (x) + n2X2.

The first of equations (3) is obviously satisfied by P1 (x) = 2- anid the second (0

'2 ,2

by _p, (x) = 2 If we use these values for pl and P2 we have the equation

(replacing X by - y) d2y 2jix dy +t2x2g = o

dX2 "' dz 2

of which the general integral is

e, !Cw cos( ( j )+C2sin(x x E)]

. . aI,Z~~~2 422 4,7 ~~~~~4,w and on giving ya the value iz.e., C =- - (0this is obviously a periodic function

of the third kinid. Changing x into x + o and it is easy to verify that

(x + 4) e -{ C, Cos (x ? + 41/(0) + C2 sin (x ? +V() }

satisfies the equation

d-y 2f- (x + ) dy + Po (X + ) Y 0. dx2 (0 dx (02 x~2 O

The most general values of p.), and p2 are easily found. Suppose q(i)(x) to be a singly periodic function of the first kind, i. e. Op(j) (x + ca) = (i' (x) and give i the values 2, 3 ... n and j the values 1, 2 . . . n. It is obvious now that the most general values of P1m and p2 are given by

?)I (x) =p (2) (x) + --2- 2 (0

I-)2 (x) q)2,,) (x + (2x(~ 2 x2 (0

(



CRAIG: On a Certaitn Class of Linear Differential Equ6ations. 281

Take as another illustration the equation of the third order

d3 Y d2y +Idy d3 + P1 d2 +I92i +P39- 0.

Here we must have

PI (X+ C) =PI (x)+ 3x

P2 (X + cJ) = P2 (x) + 2X191 (x) + 3X2 p3 (X + C)) -P3 (X) + 2P2 (X) + 2pI (X) + 3,

these are satisfied by the values

3)X 3 k2VX2 _3_3 pi (x) Cv P2 () 922 3 (X) CV

Replacing 2X by - we have the equation d3y 3,ux d2y + 3'tx2 dy Y13x8

dx3 co dx2 (02 dx (o3 J the general integral of which is

y = e T{C cos ( /3 L)+ C2 sin (x 3) + C3}

472 which for a = is a function of the third kind. The general integrals of these

two particular equations of the second and of the third orders are thus periodic functions of tne third kind-this does not hold in the case of an equation of

this form of any higher order: for example take the equation

d4y 4)X d3y + 6jt2x2 2y I3x 3 dy +4 4

dx4 (o dx3 (02 dx2 (08 dx (O -

the general integral is

y e2- [A cos (n,x + a) + B cos (m2ax + )], A, B, a, 3 being constants and ml, m' the roots with their signs changed of the

equatioii +2 + + 3=12 . 0) (02

The general integral in this case is not a periodic function of the third kind-

but each of the particular integrals jjX2 jjX2

e2w cos (mix + a), e 2 cos (m2x + a)

is such a function if x be properly determined. The general form of the coeffi-

cients in any case is readily found. Take the equation of the ftth order

P (x)-dty ? + dl l

-2 dx X7Ji7 + Pjn1 F12 jU'2_-- + + )nq =

0.



282 CRAIG: On a Ceertain Class of Linear Differential Equations.

If F(x) be an integral of P (x) 0, then F(x + ca) will be an integral of P (a + () 0.

Expressing the niecessary conditions for this and writing F(x+ c) = e-xF(x) wve find

p, (x + 1() =p (x) + nl2

pI2f(X + Ica) II21(x) + (n -I 1) 12 2(2

(n-i+a)(n-i+ a-i).i. 2).-i+(n-i+1)_

IJi (X + 1(w) =pi (X) + (n- i + 1) 7p-1(X) + 1.2 + 1 i -22 (X)

+ .(n -i+a)(n -*+a 1).- (n -i+I) I L xatpi -(X)

a ! n.n-1 n-2.n-i+l i f

P)n (X + lz) Ppn (X) + l"I-)n1 (X) + i22 Pn -2 (X) + + In;Xn where I is any integer; the geneeral term may be written briefly

a-O a = o

These conditions are all satisfied if we write n.n-1 nif l+1 Aix

p- (x)-~ iJ(Oi *_ni..s n.n -1 IA2x2

t. e. 1 ) (x) P2 (x) = 12 (2 etc. (0~~~~. 0

the general values for the coefficients are now easily seen to be m)x

pi (x)- p(n) (x) + P +

(0 1.2 (0~~~2 X P2 (X) = A2f) (X) + (n -1) 49n (x) x + n- A t

( 0 1.2 (02

pi(x)-p)Q (x) + (n ( + 1) 2()X +

(0a.n i+a - I . n-i+1 a(0 a

+ + . _X a ! (O)a

n.n- I... n_i+1 ),Yz

... qb (x) +

PN? (t2 n (t) + z+-02 (on X2+ -- ''



CRAIG: On a Ceetain Class of Linearl Differential Equation,. 283

or briefly v 'n-i+aIn-i+1 (x)

1, 2, 3.. . n. It remains now to show that a differential equation whose coefficients are of the kind just found possesses at least one integral which is a periodic function of the third kind. It will be quite sufficient to do this for equations of the second order. The equation

dx2 + PI 1X- + P2- = 0

has for coefficients Pi (x) = Pi (x) + 2-v

P2 (X) = P2 (x) + At- (X + i2

where P1 (x) and 12(x) are periodic functions of the first kind having ( for their -_\2 +Atl

commlon period. Assume y= e2w q +(x),

where +p (x) is a simply periodic function of either the first or second kind. Substituting this value of y in the above differential equation gives

d2y 22x dy ( Px 22 x> dX2 ipi dx mKP2 P1mc + w;v22P

Introducing now the above values of pi and_p2 this becomes

dlo+ rld( + P2p p= o. dx'2 dx 2

Now Picard has shown that this equation always possesses at least one integral which is a periodic function of the second kind (becoming of the first kind for the multiplier- 1). We conclude therefore that it is always possible to find at least one integral of our given differential equation which is a periodic function of the third kind.

Next we seek to determine the maximum number of integrals of this kind that our equation mnay possess. In the differential equation

dny dn -ly

d + PI de,w- + . +z - - 0 . AX2

Assume y = eD P where .P (x) is, in general, a periodic function of the second kind, and give 1v, the value

ai

=o - n-i+aln-i+l 2axfPi_)a(X)

a=O



284 CRAIG: On a Certain Class of Lineacr Differential Equations.

where cpi-a is a periodic function of the first kind. The equation now becomes

@ _ d8(l5 + n-1 d P '

t dn2+ ***+ dp O.=?

The solution of our problem is therefore led to the investigation of a linear differential equation whose coefficients are singly periodic functions of the first kind. The theory of this equation has been completely developed by Floquet and it is only necessary therefore to quote his results.

Denote by F1 (x), F2 (x) . . . Fn (x), n. distinct solutiolns of 1D = 0. If the variable describes any path from x to x + X the uniform functions F(x) become F(x + co) while the coefficients qp resunme their original values, consequently the functions F1 (x + c), F2 (x + ca) . . . F,2 (x + &) constitute another fundamental system of integrals of c1= 0.

We have then F, (x + ()_All F, (x) + -412 F2 (X) + ***+ Aln FPYit (X)

F2 (x + () A21F1 (x) + A22F2(x) +... + A2,aFn (x)

Fn (x + ca)A,1 F1 (x) + An2F2 (x) + ... + An, Fi, (x)

where the determiinant

A11 A12 * . * Al?

A21 A22 . . . A2n

Anl An12 ...Ann

does not vanish.

Suppose now that F = 0 admits as integral a periodic function of the second kind, say F (x), where F (x + co) - RF (x), we have then

F (x)niF1e)+n2F2(x) + . . . + nThn Fn (x), where the constants z cannot all vaniish. Now since F (x + c) = F (x) ve have immediately

F1 (x) E Aini + F2 (x) 5Ai2ni +... + Fn (x) EMin ni - ni Fi (X). This requires that the constants nz satisfy the equations

(All - ,)ni + A2122 + . . . + AnlZn - 0

12n, + (A22 - ?-2+ +AnOn 0



CRAIG: On a Certain Class of Linear DiJ'erential Equations. 285

consequently A11 - E A21 . . .A

A12 A22- Aw,2 -

Aln A2n . . . Ann

The equation A 0 is called the fundamental equation, and its roots, which are the multipliers of the periodic functions of the second kind, are evidently all different from zero. It is also well known that the roots of the fundainental equation are independent of the choice of the fundamental system of integrals, a fact which was proved by Fuchs.* Floquet has shown that the general integral of this equation is of the form

4'11 (X) + X4'12 (X) + . . . + Xg1 - "ll (X)

+ 4'21 (X) + 422 (X) + . . . + X2 1412 (X)

+4'ml (X).+ X4m2 (X) + * + X' 42mp (X),

where '4k is a periodic function of the second kind whose multiplier is s. In this '1c2 . m. . E are the distinct roots of A = 0 of the orders ?cl, 2 ..m

respectively. The general conclusions arrived at by Floquet concerning the equation

(D = 0 are as follows: I. Let el c2 . . . Em be the distinct roots of the fundamental equation A 0;

let Xj denote the order parting from which the minors of A cease to be all zero for - =

10 <v = 0 admits as distinct integrals

%I1+X;2 + Xi,*;t

periodic functions of the second kind and no more; 20 There exists a fundamental system of solutions consisting of

1l + /%2 + * * * + ;tm

periodic functions of the second kind, and also m- (XI + X2 + * * * + +m)

expressions each of which is of the form of a polyinomial in x with coefficients which periodic functions of the second kind all having the samne multiplier;

30 The multipliers which appear in this fundamental system either as elements or as coefficients in the elements are equal to the different roots Ei, 82 . . .m of the fundamental equation. * Crelle, Vol. LXVI, Zur Theorie der linearen Differentialgleichungen mit veranderlichen Coefficienten.

VOL. VII.



286 CRAIG: On a Certain (1Cass of Linear Differential Equations.

II. In order that 1 = 0 may admit of n distinct integrals which are periodic functions of the second kind, it is necessary and sufficient that each root of A = 0

shall annul all the minors of A up to the order which is equal to the degree of multiplicity of this root.

We have therefore in the case of the differential equation d'ny d___1

d Y1 + -P1 dx _1 +*** + p.n ? =t0 a=i

(wherep _ =ni+aln i+l 2 twhelqe pi ~~~~~ 2, a !v *-(x)J

a=o

%It + 22+** m

linearly independent integrals which are periodic functions of the third kind. It remains now to find the remaining n - (ai + a2 + . . . X.) integrals, which, with those just mentioned, constitute a fundamental system of the given equation.

It is easy, and, for present purposes, quite enough to verify wlhat these integrals are. Assume

-AX2 -Ax2

e 20 = i e 2w [3iJ (X) + 2(0i2(X) + x20o,(X) ++ . + x'rl1 (x)]

give Mtj the meaning above assigned to it and let 0k (x) denote a periodic function of the second kinid having Ei for a multiplier. Substitute this value in the differential equation

d4y + idn-ly F 2dn-2y -Y= dxS? +J91 d.fl-i + I) z,- + ***+p2 - dxla dx"- 2dx n-2

and give pl, p2 . . . pn the above values, viz. a =: i

Pi;; ?n-Xi+al|n-Xi+l Ax, (X); a==O

we have then dnYi + - + + * i 2

+ ' =

and this as we know possesses n - (1+ A2+ ***+ Am)

integrals of the form

Y,= E)i (x) + x)i2 (x) + x20i3 (x) + ... + x12 (, (x). The solution of the given differential equation is thus seen to depend in all cases upon the solution of an equation of the form

dny d-1y dli-2y + + Pdl 1 2d+ n2 = 0,



CRAIG: On a Certain Class of Linear Differential Equations. 287

where qpl, p2 . . . Pn are singly periodic functions of the first kind. The results arrived at then are as follows:

I. In order that the linear differential equation

dn + pi dn-1

+ P2 ddn.-2 + ***+ nY = ? 7

may possess integrals which are periodic functions of the third kind, it is necessary and sufficient that the coefficients p be of the form

a = i

,n- i +a I ni +1 2Axo (Pi_f,(4

a=O

i. e. of the form of polynomials in x whose coefficients, apart from certain determinate constants, are periodic functions of the first kind having X for a

period. It. If a linear differential equation has the above form it always possesses

at least one integral whiclh is a periodic function of the third kind. The total number of linearly independent integrals of this kind is always

= 1 + %2 + * * * x7 7

where X, eE and the fundamental equatioll A - 0 have the meanings assigned to them above.

III. There exists a fundamental system of initegrals consisting of

%1 + 2 + * * * + x

periodic functions of the third kind and in addition n -(2 + X2 . + Am)

functions, which are each of the form of [a polynomial in x whose coefficients

are periodic functions of the second kind, all having the same multiplier] x e 2a

i. e. a polynomial in x whose coefficients are periodic functioins of the third kind. 1V. The different multipliers of the periodic functions of the secolid kind

are roots of the fundamental equation. V. In order that the differential equation may have n linearly independent

integrals which are periodic functions of the third kind, it is necessary and sufficient that each root of A - 0 annul all the nlinors of A up to the order equal to the degree of muultiplicity of the root,



On a Certain Class of Linear Differential Equations

Documents

Transcript of On a Certain Class of Linear Differential Equations