On Linear Differential Equations Whose Fundamental Integrals are the Successive Derivatives of the...

On Linear Differential Equations Whose Fundamental Integrals are the Successive Derivativesof the Same Function (Concluded)Author(s): Thomas CraigSource: American Journal of Mathematics, Vol. 8, No. 2 (Feb., 1886), pp. 97-103Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2369292 .

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Integrals are the Successive Derivatives of the same Function. 97

the X's are all different, then n - ka will be equal to unity, and yi will be the only known integral. The number of known integrals will of course depend upon the number of X's which are equal to X1. It is easy to finid the order of the equation which can be no further reduced. We know that 2, is the greatest of the X's, at least it is not less tllan any succeeding X. By continuing the process of reduction long enough we must come to an equation of which

y1 and dy,

are integrals; other known integrals such as Y2, dy2 Y3 dY3, etc., may or may dx' dx' ,myo a not exist according to the relative values of the X's. The equation having y, and dyl for integrals is readily found to be

dn-xi?2y + ?i?ly ,I

dxA? I ++ + + qA+ * -X1+2y -

?.

The coefficients ql, q2. . . are of course derived by known processes from

P11u P12' *- *Pln. As we know at least two integrals of this equation, viz. Y, and

dy, we can reduce it one degree more, giving

dx ~ ~ xn11 dn n-Aj + I da i,-Al Y

dex _ A1 +1+ dxn ` 1 + ***+r-I+13

The degree of this equation can no longer be reduced, at least in the sense in which the word 'reduction' has been heretofore employed. We have then n-k k= n - X, + 1, or k X-1, i. e. n-X1 + 1 is the degree of the equation of the lowest order which can be obtained by the above process of reduction. Of course if we make the substitution

Y yifzdx

we find an equation in z of order n - %j, none of whose integrals are known. Suppose now that the coefficients

Pll X P12, * **Pln of the original equation are uniform simply periodic functions of the first kind having ca for period. Then one or more of the integrals will be uniform periodic functions of the second kind having for multipliers the different roots of the fundamental equation. If the roots of the fundamental equationi are all simple, then we have as fundamental integrals of the given differential equation n uniform periodic functions of the second kind each possessing its own multiplier, which is different from all the others.

VOL. VIII.



98 CRAIG: On Linear Differential Equations whose Fundamental

Suppose A = 0 to be the fundamental equation, and e1, 82, . . . Et its distinct roots, and vj, v2, . .t. v the orders partiing from which the minors of A cease to be all zero for e (i - 1, 2, .. . t); then it is known* that the differential equation admits v1 + v2 + . . . + Vt distinct integrals, which are uniform periodic functions of the second kind. Again, it is known that, in order that the differ. ential equation may admlit of n integrals which are periodic functions of the second kind, it is necessary and sufficient that each root of the fundamental equation A= 0 shall annul all the minors of A up to the order equal to the degree of multiplicity of the root. Finally, those integrals of the equationl which are not periodic functions of the second kind are of the form of poly- nomials in x, the coefficients of the different powers of x being periodic functions of the second kind, all having the same multiplier. In the case in hand we have the differential equation

dny dn-ly d - 2y d- + P] dx + P12 dx- + . + -piY = 0,

the coefficients of which are uniform periodic functions of the first kind, having ca for period. The integrals of the equationl are

d,, dyl d 1\

81 1-yi Y i d y d y_ _ _

YB czx dx2 d1x1\

dy2 d2y2 d2 1Y2

Y2' dx dx2 dx'_

__ 2ya daJl1yca ya ldx' dx2 dx* ,A-I

Since the derivatives of Yi, Y2, . . . y. must be functions of the same kind as the y's themselves, it is clear that the fundamental equation cannot have more than a distinct roots. If the equation have ac distinct roots c1, c2, . . . ?,, these will of course be the multipliers of yl, Y2' . . Y., and consequently of their derivatives; i. e. in this case the equation has n distinct integrals which are periodic functions of the second kind. The degrees of mnultiplicity of the roots are respectively 1, 2 2, *. * * .a

and obvious, -1 makes each minor of A up to the order XI vanish, s2 makes each miinor vanish up to the order X2, and so oni. The case when the fuindamental equation has fewer than a roots is included in the above general theorem; as an illustration, however, suppose that the equation has roots

El 82, * * * 6a, I <X

* Floquet: Annales de I'IEcole NormaIle, February, 1883, page 48.



Integrals are the Successive Derivatives of the same Function. 99

belonging to the functions Yi , * 8y

Then it is obvious that we have 1 + X2 + * *+

integrals, which are periodic functions of the second kind. Similar remarks apply of course in the case of coefficients which are doubly

periodic functions of the first kind, except that in this case we have two fundamental equations corresponding to the two sets of multipliers.

Resuime now the original system of equations of the nth order, satisfied by yi and its At - 1 derivatives, viz.

dncY' + p da-1 + + = dixn }1i dXn- +pi. Y

dn+ yi + dnyi +. +PI dx = de+' d de* dy _ 0

don +\i dn+Ai-2y .+ in y 0.

dxn+Ai 1 +Pn+,\dv -pi. dx i

From the first and third of tllese we derive after easy reductions the equation If (n-1) + It (n-2) + . . . = 0i

or, writing P12 (2) P13 _P(2) 7j77- P22 p P23

(n-1-2+* ( + p2)Y -? $Y/ P}j22 yi ) +2nY

simllarly y fl) p22 + . +p2nY )

(P,)2 My(fl+l) + P(2)Y(n) + ... + p = 0,

y(u+Ai-3) +p(2)yn+A-4) + ., . +_p2)y(i3)

0.

The integrals of the equation

P(2) dn ~1Y + n(2)d Y + + p2)yJ= dx n-1 L2dx1t- * n, 0

are easily seen to be

dY1 2y1 dAl- 3Yi dI dx2 dx *X*AI-3 t

dY2 d2~y2 dA2 3 Y2 Y2, dx dx 2 dx 2 3

dya d2y,'. dA 3 ya

dx dx2 dxAa3-



100 CRAIG: On Linear Differential Equations whiose Fundamental

We have thus n - 2a integrals of the equation of the order n - 1, and these are the same as the known n - 2a integrals of the equation pM/) p(1) I(n -2) +(l3) I(n -3) +...+p2Y ?

P22 Y I1923 Y -P2n) Y=O or say y(n-2) + p (n-3) + . +. +pfIy = 0.

There remain now 2a - 1 unknown integrals of p'2) and 2 (a - 1) unknown integrals of p(l)'.

From (Pi)a we can form again the equation p (2)y(n_2) + }2)Ay(n -3) +

. . . + p2n'Y =o

or say (n-2) + 23) (u- 3) + . . . +jA3y 0

the integrals which are easily seen to be

dy1 d2y1 d& -4yi 'd dX2 dXAl-4

d1Y2 d2y2 d A2-42J

dx dx2 dxA2-4

4a d2Ya da-4ya lYa I d;c X ;g *** tA-4 dx, dx2 dx'\a

These n - 3a integrals are the known integrals of y(n-3) + p(14)(n-4) + +

-p41)y _

mentioned above. There remain now 3a - 2 unknown integrals of the new equation of order n - 2 and 3 (a - 1) integrals of the equation of order n - 3.

This process can of course be continued just as in the earlier case, where the coefficients were p(l); it is hardly worth while, however, saying anything more about it.

We may go back, however, to the original system once more, and by com- bination of the first and fourth, the secolnd and fifth, etc., and making some easy reductions, obtain

yl(n-1) + p3)y(n-2) + ** + Jpy = 0,

y(n-l) + (32)(n- 2) + .. . +P2(3)Y 0,

y(?+ Ai-4) + 4,32)y(n+?*-5) + * 3 + 2 n -4) - 0

where 1= 3)+= p{/', (3p = + p 7, etc. The integrals of

dn-I' _n-2_ .



Integrals are thae Successive Derivatives of the same Function. 101

are obviously d1 d2y1 dd1-4 Y

are i dx' dx2* A 4

dy2 d2y2 dX2 Y2 Y2 X G dx2 ' d X2-4

dya d2Ya d_ -4_,

y/aX dx' dx2 d * 4

We know thus n -3a integrals of _(-)(+p32) (n

2)+ *,*+ p2nY ? and these are the known integrals of the equations

((n-2) + (2) (n-3) + + p(2) y O y(n +-p3() y(n-+4) + I _

There remain then 3a - 3 integrals of the equation of order n - 3, 3a - 2 unknown integrals of the equation of order n - 2, and 3a - 1 unknown integrals of the equation of order n - 1.

Taking the general case, combine the first of the origiinal set with the i + 1st, the second with the i + 2nd, etc.; we have as the result

Y(n -1) + p2i2) Y(n 2) +. + p2n My- =0

where d p,2 . d ,pil dip,3 d'p1l etc. P dxi dxdxP3 dxi

The known integrals of this are dy1 d2y1 dAily

dx dx2 * dA

_-' t

dy2 d Y2 dA2-i-y Y2 dx2' d 2-i-

dya d___a d'acJilya

dx dxa dx-\-l

These are the known integrals now of the set of equations, including the p(i)

equation, y(n-1) + () y(s-2) + ... + p2i) y 0,

y(n-2) + r vl)y(? 3) + + p(i-A)1y -0

y(n-3) + p4(-2)y(n-4) + J +p(i-2)/y -0

y(T-i) + (1 n-i- + . 1 .+ ,nY 0.

The numbers of uniknown integrals of these equations are respectively



102 CRAIG: On Linear Differential Equations 10hose Fundamental

In resume6: we have the two equations

(P) = dYn + Pll d -l

+ -)12 d_ 2y + + 0 dxn dx"-' dx7-

-d Y1 +dP212 den +p23 dn + * + ]22nY = 0,

having the common n - a integrals

dya dka 2ya

dx dxXCJ2 the three equations

(P) = 0,

+ ]2)3 (1) dxn Y1 + 22) d n -2- + + pt2

0

dn-2+ 1dzn 'Y+ +3-- dx n-. 3dXn_ + P3n/ 0

having the nt - 2ac common integrals

dy, d\-3y- Y1' dx dX,\,-3

dya dka-3/ Yas dx dxk-3

and the i + 1 equations (P) =O

d," 1j P2 dn2 ly + + PMi) ;

dxY + p2)dy + + ' = 0,

d12 Yi + p(pi -r 1) dn ,_ - Yj + ***+ p(+1) y = +xn- 33 dn-3 n

dx-i + pl n-

having the n - ia common integrals dy1 dkl--1y

?J1 t * dx Al-i-1

dy, dxk-i-ly dx dxi

From each of these systems we may deduce an equation satisfied only by the



Integrals are the Successive Derivatives of thte same Function. 103

common integrals, i. e. we would have, including the given equation, a new system of equations of the respective degrees i, n -a, n - 2a,I . .. n -a ...

satisfied only by the corresponding group of common integrals. The properties of each of the new equations will of course be exactly the same as those of the original equation.

In all that precedes it has been tacitly assumed, unless statement is made to the contrary, that all the functions Yi, Y2 X ... ya, and consequently their derivatives, are uniform functions; fuirther, if ai denote a singular point of the function, the function will remain finite at the point ai on being multiplied by a deter- minate finite power of (x - ai). The coefficients I'll; p12; Pin are therefore of the form

P11 P12 Pill,

_-i' (x- -aj)2 (x -ai)7

where p Fk(,-l)(x)(x -as)

p denoting the total number of singular points; Fm(x) an integral function of x at most of the degree m; and 4_ (x - a1)(x - a2) ... (x - a,) .*

It is of course perfectly obvious that the coefficients of the various derived equations are of the same form as those of the given equation, formed as they are merely by differentiation, miiultiplication and division. This being the case it follows that the unknown integrals of these derived equations are uniform functions of the same character as the known integrals

Y'dy1 d2y1 dkciyi -d- ' d x kIXk

dy2 d2y2 dk2y2

Xdx- dx2' dx k

2 d4ya d2a * dkayC* Ya, 7J7 dx_x2 dx'-

* Crelle, Vol. LXVI. Fuchs: Zur Theorie der linearen Differentialgleichungen mit veranderlichen Coefficienten.



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