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"K., j, ~} : *** Q A S. $ º (Iſbe (Anibergitp of Chicago * 7 s. # S. ‘. . ; * * { (~ a * * * On The Solution Of Certain Types Of Linear Differential Equations In Infinitely Many Variables A DISSERTATION SUBMITTED TO TEIE FACULTY OF TEIE OGDEN GRADUATE SCIEHOOL OF SCIENCE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PEIILOSOPEIY DEPARTMENT OF MATHEMATICS BY WEBSTER GODMAN SIMON Private Edition, Distributed By TEIE UNIVERSITY OF CHICAGO LIBRARIES CHICAGO, ILLINOIS Reprinted from AMERICAN JOURNAL OF MATHEMATICS, Vol. XLII, No. 1, January, 1920 (Iſbe (ſinibergitp of Chicago On The Solution Of Certain Types Of Linear Differential Equations In Infinitely Many Variables A DISSERTATION SUBMITTED TO THE FACULTY OF TEIE OGDEN GRADUATE SCIEHOOL OF SCIENCE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PEIILOSOPEIY DEPARTMENT OF MATHEMATIOS BY WEBSTER GODMAN SIMON Private Edition, Distributed By TEIE UNIVERSITY OF CHICAGO LIBRARIES CHICAGO, ILLINOIS Reprinted from AMERICAN JOURNAL OF MATHEMATICS, Vol. XLII, No. 1, January, 1920 H.) On the Solution of Certain Types of Linear Differential Equations in Infinitely Many Variables. BY WEBSTER G. SIMON. The main purpose of this paper is to prove the existence of certain types of solutions of particular kinds of linear differential equations with periodic coefficients in infinitely many variables. As a means to this end the existence of exponential solutions is established for certain types of linear differential equations in infinitely many variables with constant coefficients. The starting point is the existence theorem given by von Koch,” and a generalization of Poincaré's theorem + concerning the development of the solu- tions of the differential equations as power series in a parameter p, when the functions appearing in the differential equations are themselves power series in p. Then our work is very similar to that of the finite case, the finite determinants becoming infinite determinants which together with all their first minors converge absolutely. The type of determinant used in this paper is more general than the normal determinant, and is the following.S. The determinant 1 + aii, 012, 0,132 a21, 1 + q22, 0.282 0.31, aa2, 1 + asa, e e º & & e o e 5 5 2 is such that there exist two sets of positive constants S1, S2, ' ' ' , T1, T2, ' ' ' , CO which are of such a nature that aij | < S.T., and > ST, converges. This 3-1 is a type of infinite determinant given by von Koch,S which together with all its minors converges absolutely. - - Using the methods thus indicated, we find that many of the phenomena of the finite systems are carried over into the infinite systems of differential equations.| t *Won Koch, Ofversigt af Kongliga Vetenskaps Akademiens Förhandlingar, Vol. 56 (1899), pp. 395-411. # Poincaré, “Les Méthodes Nouvelles de la Mécanique Céleste,” Vol. I, chapter II. § Von Koch, Acta Mathematica, Vol. 24 (1901), pp. 89-122. Hereafter this paper will be denoted by K. |See Moulton and MacMillan, AMERICAN Journal, OF MATHEMATICs, Vol. 33 (1911), pp. 63-96. 27 \ 28 SIMON: On the Solution of Certain. Types of Linear I The differential equations which we shall consider are of the general type CO (1) a => 0\} (t)ay, i = 1, 2, co, JF1 where w, is the derivative of as with respect to the independent variable t. Concerning this system of differential equations we shall establish two existence theorems. Theorem I. Suppose the system (1) satisfies the following hypotheses: (H,) The 9, (t) are analytic functions of t when |t| < R. t (H,) Positive constants, S, S., . . . , T1, T2, . . . , easist such that | 643 (t) | < S.T., t | < R, and S.T., + S.T., + · · · converges. (H,) The wº (0) = 8, where the 8, are constants such that 8, T., + 8,T, converges. - Then there eacists a unique system of functions CO º e (2) wi = 8. --> 39t, i = 1, 2, co, f=1 which satisfy the system (1), and which converge for |t| < R. Although this theorem was established by von Koch,” it will be proved briefly here because the notation and the results are essential for the later parts of this paper. From (H,) we have (3) 09(t) = a + aft + . . . , i, j = 1,2,... . . .o. Upon substituting the series (2) in (1) and equating coefficients, we get CO d(0) G(0) * i j 8. 5 r Bº = º, j=1 CO }/3(2) = (1) Q(0) (0) (2 (1) (4) < 28; ; (e. 8, +a; 8; ) CO X. - º (x+1)6?” => x aº”8%, where 89–8. j=1 k=0 We see that the formal solution is unique. * Loc. cit. Differential Equations in Infinitely Many Variables. 29 To prove the convergence of (2), consider the system of differential equations (5) &= Sºx Tjči, * = 1, 2, • * - CO . JF1 * CO Upon substituting & = 8, --> yºtſ in (5) and equating coefficients, we get f=1 CO y;” – Six Tºyſ", j=1 CO 2y? = Sax Tºyº” (6) < Y; j=1 ! Y; 2. 4. CO l (A + 1)Yº = Sax Try?", where yº) = \) i. j=1 On comparison of (6) and (4) we see that y? - | 8°l, i, j = 1, 2, co. Hence & dominates as for every i. From (5) we have (?) #4-#4- -e. On taking {(0) = 0, we have (8) & — 84 = Sté, i = 1, 2, ' ' ' Co. Therefore each equation (5) reduces to (9) é’ = Cé + K, where C = S, T, +- S.T. -- . . . , K = 8, T. -- 82T2 + . . . . From the theory of a finite system of differential equations we know that (9) has a unique analytic solution for | t| < R. On combination of this fact with (8), we see that the solution (2) of (1) converges when |t| < R. Now we define as a fundamental set of solutions of (1), a set such that every solution of (1) can be expressed as linear homogeneous functions with constant coefficients of the elements of the set. Then if we denote by pij the elements of the fundamental set, it follows that the determinant of the pij converges and is not zero for all |t| < R, and conversely. Furthermore it OO - ſ:04, (t) dº, * : has the form A = Aoe’ +1 for all |t| < R, where A, is the value of the determinant A when t = to. If, for example, we take the solutions defined by bw (0) = 1 and bºy (0) = 0, i=A j, we see from (8) and (9) that - S;T: S;T . g $ |*|=1 ++ (e"—1) , and lºw ==#| (e° — 1) |, i. #4 j. 30 SIMON: On the Solution of Certain Types of Linear We see that the determinant of the pij is absolutely convergent. Theorem II. Suppose that the system (1) satisfies the following hypo- theses: (H,) The 64; are eaſpansible as power series in a parameter p, which converge for all real |t| < R if | p | < p. (H2) For p = 0, the 643 = ahy, where the aij are constants. (H,) The 64; satisfy all the hypotheses of Theorem. I uniformly with respect to p if | | | < p - Then the solutions of (1) can be ea pressed as power series in p, which converge for | p spo 3 p and for all real |t| < R. CO In Theorem I we showed that as –3. => 89t, |t| < R, p | < p, j=1 where n3" = Pº (8”, . . . , 8-0). OO In particular 89 => a; 89. The a;’ are power series in p. Hence j=1 - OO OO CO gºaº - 38%.”, 89 => 38%.” which converge for | p | < p. Then by k=1 j=1 k=1 a well-known theorem in the theory of double series, we infer that 3% = CO 3.8% | p | < p. In a similar fashion we proceed step by step, and show . tº-1 that the 8" are also power series in p, which converge for | p | :)" ki, ka, tº * g. , kr 5 and suppose that this minor is one which is different from Zero. Then the system (13) admits the solution f il, * & & ir \º (i. i. e e e ir ) (Mo) * * * il, tº 6 e ir (X0) (i. tº e g i.) ºr, T ki, k * @ e kr š, H + (i. * e ge i) $. - i. tº gº tº ir (No) (i. tº e G i.) depending on r independent parameters ä, éº, and admits no others. To sum up our results we have the following (15) Cl; += * See K, p. 104. # See K, p. 108. # See K, p. 109. 32 Smon: On the Solution of Certain Types of Linear Theorem III: If the 64; of (1) are constants and if Ao is a root of A(A) = 0 such that all minors of order r–1, but not all of order r, vanish for A = A, then there exist r independent solutions of (1) of the form (11). Now the characteristic equation, since it is an integral transcendental function, may have no finite roots, it may have a finite number of finite roots, or a denumerably infinite number of roots. In the first case the ci are all zero, in the second case, a finite number of solutions of the form (11) exists and in the last case an infinite number of the form (11) exists. This is the case we shall study, for this infinite set constitutes a fundamental set of solu- tions as the following discussion shows. For each X that is a root of A(A) = 0 there is a solution of (1) of the form (16) aij = cije"j*, i, j = 1, 2, oo. We know that the system (1) admits the fundamental set of solutions bij, i, j = 1, 2, . . . co, where bis(0) = 1, bij (0) = 0, i A j. Consequently CO - wi, -> Aº’hii, i = 1, 2, . . . Co. k-1 First let us assume that the aj are all distinct. Then it follows immediately that the system (16) constitutes a fundamental set. For consider wº, bºy, i = 1, 2, . . . co, j = 2, 3, ' ' ' Co. CO > A (1) k; q, tº º e k=1 k; $1k, $12, $11, $12, ' ' ' CO = 4 (1) e e º 34%, $22, ' ' ' A: º * • e is Choose the initial conditions such that A* = 1, which is no actual restriction since we are merely determining the arbitrary constant which enters, and we can multiply our solution by a constant, not zero, when we are through if we choose. Thus acti, pig, i = 1, 2, co, j = 2, 3, ' ' ' co, constitute a fun- damental set. Similarly we show that aii, a 42, $43, constitute a fundamental Set, where j = 3, 4, . . . Co. Continuing in this manner and passing to the ſimit we see that the aii, i, j = 1, 2, . . . co, constitute a fundamental set. A result of this determination of the arbitrary constant is that cis = 1. Next let as = a1, and as # all, j = 3, 4, . . . Co. There are two cases that may arise here, viz., all of the first minors of A(A). vanish when a = a1, or not all vanish. Consider the former case. Then (15) assures us that we can Solve for the ci; in terms of two of them. On choosing the notation so Differential Equations in Infinitely Many Variables. 33 that (; #) is not zero, we get two independent solutions of the form Qil - c;1991*, Qi 2 F cize”, * = 1, 2, e Q & CO, which with $44, j = 3, 4, . . . co, constitute a fundamental set; for - A (1 1 2) (2) & Cº º a'il, 212, $13, ' ' ' A: '#11 + A*@ia, A: $11 + A. $12, $18, 1. 2 (2) tº e > 221, 222, $28, ' ' ' | T Aºbal + A*$22, A: '#21 + A. $22, $23, tº e 5 e g 5 ſº & 5 e & e e s a tº e e º e º e 5 tº e s e e º e s a tº 5 e & 5 e and the Aº, Aº may be taken to be zero, and A9 = Aº = 1. So these solutions, we see, constitute a fundamental set. - - Next let us assume that not all the first minors of A(A) Vanish when a = a, - as ; and let us choose the notation so that the first minor (i) is not zero . Then to get a solution associated with al, we make the transformation y = b,a, + b,a, + . . . , y' = b,aſ-H bº. -- ' ' ' and, if possible, deter- mine the b's in such a manner that y' = ay. On Substituting these equations in (1) we get b. [alia, + aiza, + . . . ] + baſa,121-H aczº, + . . . ] + ' ' ' ' = aſb,a, + baa, + . . . ]. * This equation must hold for all initial values of the a's. Therefore it is an identity in them; and the b’s satisfy the following set of equations ! | (a, -a)b, + a,b, + gº tº e = 0, (17) t 4 aqab, + (a,a – a) be + · · · = 0, In order that these equations have a solution for the bºs not all Zero, we get as a necessary and sufficient condition, just as in the case of c's in (13), A(A) = 0. This condition is satisfied, for we have assumed that when a = a1, the characteristic equation is satisfied. Furthermore we have chosen the nota- tion such that the minor (i) is not zero when a = a1. Then we get for the bºs by = mibi, j = 1, 2, . . . co, I where my = º . For convenience we shall take b. = 1. Then the equations 1 become - ſº a 1/ - 9,- G1/12 * & a' = d21), -- (aas – dºma)a's -- (aas – aims) as + . . . , (1)- e gº © & tº © e tº tº © g e tº tº e g º 5 a. H aniyi -- (ana – anima)a's -- (ans – anima) as + . . . , \- 3 34 SIMON: On the Solution of Certain Types of Linear Next we make the transformation ye= diy, + dº, -i- ' ' ' , y' = day. -- d.º.-- . . . and determine, if possible, d, d, . . . , such that y' = aya + y, Then, as in the foregoing, we get the following, infinite set of equations for the determination of the d’s ſ (a, - a)d, + acid, +- asids + · · · = 1, (17): 0 + (asa — a 21m2 — a) de + · · · = 0, On setting a = — 1/A, we get an equivalent set of equations for the deter- mination of the d’s, the determinant of which is 1 + \al, Adel, 1 + A (ass — asima), ' ' ' A, (A) = |0, 1 + A (ass – maasi), ' ' ||= (1 + Aa1) A (ass – daima), By making use of the equations (17), the reader will convince himself that A, (A) is identically equal to A(A), for it is obtained from A(A) first by inter- changing the rows and columns of A(A) and then multiplying the elements of certain rows by certain quantities and adding them to the corresponding elements of other rows. Furthermore there is a first minor of A1(A) which is not zero when a = a1. This minor involves the elements of the first row, since all the other first minors vanish. The determinant of the coefficients of the equations, omitting the first one, is zero for a = a1, since when a = a1, A(A) has a multiple root A. There- fore we can solve for the ratios of d2, ds, . . . . Then let us substitute the values of the d’s thus obtained in the first equation of (17). First, we know that the sum will converge *; for the d’s are proportional to the first minors of Ai (A), and the sum of the products of the elements of a row and the cor- responding first minors converges. Secondly, since the d’s which we have determined carry an arbitrary factor, we can finally determine them so that the first equation is satisfied. Therefore our equations have been reduced to (1”) y' = a1y, y' = y, + aya, a = a 'y, + a 'y, + a.a., + . . . . . . . . where the a; are the transformed as of the original equations. And on solv- ing the systems (1/) and (1”) and putting in place of y, and y, the cor- responding value of a., and aca, we see that the solutions of (1) associated with at are . avia – cine”, aia - (cia + toia) 8*.*, i = 1, 2, ' ' ' Co. * F. Riesz, “Les systèmes d'équations linéaires a une infinité d'inconnues,” p. 34. Differential Equations in Infinitely Many Variables. * 35 Then by a precisely similar argument to the foregoing we can show that a'il, wiz, hij, * = 1, 2, co, j = 3, 4, co, constitute a fundamental Set. - When for a given value of a the characteristic equation has a root of higher multiplicity, discussions similar to the foregoing must be made. We should expect to treat next the case that A(A)= 0 has a root of infinite multiplicity. But such a case cannot arise. For we have seen that A(A) is an integral transcendental function of A. Therefore it can be expanded in a Taylor’s series in the neighborhood of any finite A. Suppose that Åo were a finite root of infinite multiplicity. Then A(A) Vanishes together with all of its derivatives when A = Ao. Hence A(A) vanishes at every point in the neighborhood of Ao. Then it is identically zero. But this is not true, for when X = 0, A(A) = 1. Hence A(A) has no root of infinite multiplicity. If a = 0, we can not make the transformation which carries the system (12) into the system (13), and the determinant of (12) diverges. Then we do not know whether the equations (12) have a solution for the cy not all Zero; but each special case must be considered as it arises. To sum up our results we have Theorem IV. If and only if A(A) has an infinite number of roots A, the system (1) has a fundamental set of solutions each of whose elements is of the form a'i j = e°j*/ij (t), i, j = 1, 2, co, where a1 = – 1/A; and the Wii are polynomials in t of degree at most (n — 1), and n is the order of multiplicity of the root Aj. III Now we shall assume that the 64; of Theorem I are periodic with the period 27. We have seen that the system (1) has, as a fundamental set of Solutions, the set bij (t), where (18) ºpis (0) = 1, $43 (0) = 0, i # j, i, j, - 1, 2, • • CO . Let us make the transformation (19) aci + e”yi, where a is an undetermined constant. Then the equations (1) become CO 3 F1 36 - SIMon: On the Solution of Certain Types of Linear Since the system by constitutes a fundamental set of solutions of (1), any solution of (20) can be written : º CO (21) y: = e^*> Ajipij (t), i = 1, 2, • * * CO . j=1 We now inquire whether it is possible to determine the 43 and a so that the y, defined by (21) shall be periodic with the period 27. From the form of (20) it is evident that necessary and sufficient conditions that the yi be periodic with the period 27 are - (22) y; (2+) —ys (0) = 0, i = 1, 3, ' ' ' Co. On imposing these conditions on (21), We get (28) #4, sºu(?-)-ºw (0)] = 0, i=1,2, *. j=1 Then we make the transformation (24) e-*at = g/a — 1, assuming of course that e^at # 1. Then the equations (23) become (25) $4,000 (2x) + A.I. --wºu(ºr) –11–0, j=1 i =A= j, i = 1, 2, ' ' ' Co. In order that these equations have a solution other than that in which the Aſ are all zero, it is necessary that the determinant of the coefficients be Zero. On writing the diſ (2+) simply bºy, the determinant is - 1 + o (p11 – 1), Gºbl2, tº € tº (26) D(G) = oºzi, 1 + q (bag — 1), ' ' ' | = 0. This determinant, we see from the fact that the determinant of bij converges absolutely, is an integral function of a. Equation (26) is called the funda- mental equation associated with the period 2"r, and does not admit o' = 0 as a root. - - As in the characteristic equation in II, the fundamental equation may have no finite root, a finite number of finite roots, or a denumerably infinite number of finite roots. - As in the case of constant coefficients, if the fundamental equation has no roots, the system (1) has no solution of the form (19), if the fundamental equation has a finite number of finite roots, the system (1) has a finite number of solutions of the form (19), and finally if the fundamental equation has an Differential Equations in Infinitely Many Variables. 3? infinite number of finite roots, the system (1) has a fundamental set of Solu- tions, each of the elements of which is of the form (19). - The discussion of the form of the solutions follows precisely the same lines as that in the case of constant coefficients except when go is an n-fold root of (26) and not all the (n — k)th, k > 0, minors of D (or) vanish for a = oo. Now let us assume that or = go is an n-fold root of (26) and not all the first minors of D (or) vanish. Then there is only one solution of (1) of the form as = e”yi, where the yi are expressed as in (21). And let us choose the notation so that a minor corresponding to the elements of the first column is not Zero. Then we take as a new set of solutions (??) air - 6°1'yū, a y = $ty (t), i = 1, 2, . . . oo, j = 2, 3, . . . oo. This set of Solutions can be shown to constitute a fundamental set in a pre- cisely similar fashion to that in II. Then we make the transformation * = ea'a, i=1,2,... . . . As above we get OO (28) z = e^*[A16°4′ys, + X Agº (i)], i = 1, 2, . . . oo. j=2 Necessary and sufficient conditions that the 2, be periodic with the period 2tr are 2: (2T) – 24 (0) = 0, i = 1, 2, . . . oo. On imposing these conditions on (28). we get CO (29) Alſe “*”ys, (2+) — ya(0)] +S Aſbú (2+) e-ºar — $4;(0)] = 0. j=2 - After setting e^* = aſo – 1, the equations (29) become - - CO (30) A1 (1 — g/g,) yú (0) +340 by + A*[1+ w(ºw–1)] = 0, 7- k # j, k = 2, 3, . . . co, i = 1, 2, . . . Co. The fundamental equation for the equations (30) is (1 – 0.70,) yia (0), Gºbia, ' ' ' (31) (1 — o/o.)!/21(0), 1 + q (qaa – 1), • * * Fº Making use of (21) and taking A, -1, equation (31) becomes º 38 SIMON: On the Solution of Certain Types of Linear *A*– (q – 1), a biz j=1 OO = D (or) = 0. *A*–4,0–1), 1+2(e–), T" So we have !/11(0), gºia, . = 0 (32) D(o) F. (1 — o/o.) D1 = (1 — a ſo.)|yai(0), 1 + q (qaa – 1), . . . . - v. Since o = 0, is an n-fold root of D (a) = 0, D1 (o) has the factor (1 — o/a,)”. Since (27) constitutes a fundamental set, any solution can be expressed in the form CO (33) (C# = Baeºlºgia + 3Bºiſ, t = 1, 2, ' ' ' , Co. j= Now let us make the transformation, corresponding to (19), to get a Second Solution associated with al. (34) wia = 8*(yia + iyū), i = 1, 2, • • OO . On imposing the condition that a is shall satisfy the system (1), we find since e°1°yia is a Solution, CO . (35) 9%, -i- dayiz –399 (), -ya. i = 1, 2, Co. From the form of (35) we see that sufficient conditions that the yia shall be periodic with the period 2" are . (86) ya(?-)-y,(0) -o-; Biſºv(ºr)—ºu(0)] j —2tyi, (0) = 0. On substituting e-ºff = 0.1/0, -1, (36) becomes OO (37) — 27 (or, − 1) yi, (0) +xbºw- Bºſí + o-, (bº. — 1] = 0, k # j. The condition that these equations be consistent is, since ori A 1, 911(0), oiºpia, ' ' ' D, (q) = |yai (0), 1 + qi (b2a – 1), ' ' | = 0. e e º ºs º e º is e 5 5 e o In (32) we showed that D, vanishes (n − 1) times when q = q. By hypo- theses not all the first minors corresponding to the elements of the first column Differential Equations in Infinitely Many Variables. 39 are zero for a = 0,. Hence we can solve equations (37) for B2, Ba, ' ' ' , terms of ya (0). Consequently in this case we get a second solution associ- ated with a, which is of the form (34). In a similar manner we can go ahead step by step and get the following group of solutions associated with a a'i, H 6*.*Wii, a'iz = 8*[yia + iyúl, Øiºn = e°1*[y in + lyin-1 + · · · + 1/ (n *s- 1) ! t”y;,]. If a, is a triple root of D (a) = 0, such that all first minors are zero, but not all the second minors are Zero, the solutions associated with an are a's, H e”:'yū, wia = 8*yia, ris = 8*[y is + i (yū -H yie)]. All the sub-cases can be treated, as they arise, by the methods given here. IV Now let us assume that the system of differential equations (1) satisfy the hypotheses of the second existence theorem, and in addition the coefficients in the power series expansions of the 64; are separately periodic with the period 2m. That theorem tells us that the solutions of (1) can be written in the form CO (39) Qij =32, (t) p", i, j = 1, 2, Co. And we shall take the initial conditions such that CO (40) ru(0) = x*(0)}=c, whence a; (O) = Cij, w; (O) = 0, k = 1, 2, . . . Co. where the cij are constants such that cis = 1, and their determinant is abso- lutely convergent. These conditions coupled with the fact that if the deter- minant of a set of solutions converges and is not zero when t = 0, it converges e OO and is not zero for every value of t for which >0s, converges, show that the $:1 - System (40) constitutes a fundamental set of solutions. Now we inquire if we can find solutions of (1) of the form (41) a’s – eq’ſs, where the yi are periodic with the period 2", and a is a constant which remains 40 SIMON: On the Solution of Certain. Types of Linear to be determined. After making the transformation (41), the differential equations and their solutions become CO CO -. * º (k), klar, d = © e tº (42) y'.-- a!/? -āſaw +3.9% pºlyi, i = 1, 3, CO , OO CO $/i - –34,eºlº (t) +º, (t) pººl, * = 1, 2, ' ' ' Co. j=1 º, c=1 * On imposing the conditions that the yi be periodic with the period 27, viz., yı (2+) —yº (0) = 0, we get CO CO e (49) 0-34ſ,”(*)-cu-sº, (2+) pººl, i = 1, 2, ' ' ' Co. After setting e-" = o/a — 1, the equations (43) become (4) 0–34,225 (8-432;(2)”)—ouſe-1)]–9. i = 1, 2, . . . Co. To avoid the trivial case where the A3 are all Zero, we must set the determinant OO (45) D(o,r) – ſº (ºr) +3*(w)º-cu(g-1)]–0, CO which is a condition on the undetermined constant a. Since the wº + sº pk Hori are the elements of a fundamental set, and cis-1, it follows that D(a,p) converges absolutely for all finite o’s. If the fundamental equation is satisfied when a = ao, it is also satisfied when a = ao + v V-1, where v is any integer; but all distinct solutions of the differential equations can be obtained by taking v = 0, since the ratios of the A3 are the same for v = 0 as for w = p. And for p = 0 the equation (45) reduces to (46) Do = | gº (2m) — ci, (a — 1) | = 0. When p = 0, we have the case of constant coefficients which we treated in II. Now we shall determine the cij and the a% = A^* of the solutions as we did there. But having the cij we might also determine the a of the solu- tions by means of (46). Since for given initial conditions the solution of the differential equations is unique, and since the initial conditions are the same in the two cases, the solutions obtained by the former method and the solution obtained by means of (46) are the same. We recall that the cº, are not all zero if and only if the characteristic equation has finite roots. Therefore for every root of the characteristic equa- tion there is a corresponding root or of (46); and conversely, for every root of (46) there is as corresponding root of the characteristic equation. There- Differential Equations in Infinitely Many Variables. 41 fore if the characteristic equation has no finite roots, then (46) has no finite roots, if the former has only a finite number of finite roots, then the latter has only a finite number of finite roots, then (46) has an infinite number of finite roots which yield independent solutions of the system (1). Now since the solution (41) converges for all p sufficiently small, includ- ing zero, the failure to find an a for p = 0 is not due to clumsy analytic methods, but shows that the system (1) has no solution of the form (41). From the theory of implicit functions we know that if (46) is satisfied when a = ao, then we can solve for a as power Series in p, provided p is sufficiently small. Therefore the existence of a root of the characteristic equation is both necessary and sufficient for the existence of a solution of the system (1) of the form (41), and the existence of an infinite number of roots of the charac- teristic equation is necessary and sufficient for the existence of a fundamental set of solutions of (1), each of the elements of which is of the form (41). We shall study in detail the case that the characteristic equation has an infinite number of roots; and we shall use only those values of a obtained from (45) which for p = 0 reduce to the values of a obtained from the characteristic equation. . - - When e^* = 1, the transformation e^* = c/o — 1 can not be made. Then as in the case of constant coefficients when a = 0, we can make no gen- eral statement about the solution, as the determinant of the As diverges then. V Solutions when the aº are distinct and a" — aft =# 0 mod V-I. The part of (45) which is independent of p is CO D-locue."—oug-1) |-|c, III(1 1–eº,” “) O v J C 7 * 7 ſt 7 + 6 j=1 1 — 6-2atr If (45) were an identity in p, its roots would be the roots of (46), viz., a = a'. Let us assume that we have the general case in which it is not an identity in p, and set (47) a = a " + 8. Then we get (48) D (o, u) = Do + p.F.(8), p.) = CO - e-26. T — 6–2 (a (*), Fö,)tr / o–2a: . (9) tr | 04 j | (1– 16" ld 16 ) (1 + 1. 6 J e-2(a)(*)+6.)T 1 — e-2(a(*), Fö,)t II 1 – e-2(a(*), F6.) It k; + ph'i (8k, p.) j # k. 42 SIMON: On the Solution of Certain Types of Linear where F. (8, p.) is a power series in p and 6k, converging for | 8 | < co, p | < p > 0. Since by hypothesis no two of the aº differ by an imaginary integer, the expansion of (48) as a power series in 8k and p contains a term in 8k of the first degree and no term independent of both p and Šk. Therefore we know by the theory of implicit functions that (48) can be solved uniquely for 8; as a power series of the form (49) 3 = º(u), which converges for | p || > 0 but sufficiently small. Now we substitute this value of a = aſ” + 8; in (44), and get an infinite number of linear homogeneous equations for the A3 whose determinant con- verges and is zero, but the first minors of that determinant are not all Zero, since by hypothesis, the roots of Do = 0 are all distinct and no two differ by an imaginary integer. Consequently these equations determine uniquely the ratios of the A3 as power Series in p, which converge for u sufficiently small. On Substituting these ratios in (42) we have the particular solution yik, t = 1, 2, co, expanded as a power series in p. Hence we may write it CO (50) yi. => y?(t)p'. j=1 Since the periodicity conditions have been satisfied, CO yū.(t +2+) —yº.(t) =>[yº (t + 2*) – yº (t)]uj = 0, j=1 for all p sufficiently small and for all real t. Therefore 9% (t + 2T) —yº (t) = 0, j = 0, 1, 2, co, whence it follows that the yº, j = 0, 1, 2, . . . co, are separately periodic. A solution is found in a similar fashion for each a". VI Solutions when no two of the alſº are equal but when aº – aſ” = 0 mod V-1. Suppose that when p = 0 the characteristic equation has two roots such that a' and aº differ by an imaginary integer and that none of the other a'ſ” are congruent to dº” mod V-1. Then we see from (45) Differential Equations in Infinitely Many Variables. 43 (51) D(a,p)= CO - 1 — 62a1(*)T - 2 1 — e”; “T - ſºul [1–Hºl. III–Hºl-tº-", j= where as in (47) we have set a = a "+ 8,. The term of lowest degree in 8, alone is found by expanding the first bracket and turns out to be of the second degree. To get the terms in p alone we suppress those involving 8, after which we get a factor p from each of the first two columns. So we see that in general the term of lowest degree in p alone will be in this case of the second degree. Hence we have - 1 — e2a1(*)T 2 * 1 — e”; “)" (52) D 3– | Cij | [1 - 1 --> =#|| II [1 - 1 * #: + 8, 9.5', (8, p.) + 8°F, (8, p.) = 0. In a similar manner if p of the aº are congruent to aſ” mod V-1, then the term of lowest degree in 8, alone is of degree p, and in p, alone it is of at least the pth degree. The problem of the form of the solution of (52) is one of implicit func- tions. Writing the first terms explicitly we have 84-H kiö, u + Kozu, + terms of higher degree = 0, where kii, koa, ' ' ' , are constants independent of 8, and p. On factoring the quadratic terms we get (53) (8, - dip) (8, - dep) + terms of higher degree = 0. If d, and de are distinct, there are two solutions, and these have the form * (54) 811 = dip -- pºp, (p), 8.2 = dau + p^P, (p), where P, and P. are power series which converge for p sufficiently small. In this case the solutions are found as in W. But if d, and d. are equal, the character of the solution is in general quite different and depends upon the terms of higher degree than the second. In general it will be a power series in + Va. This case we shall consider in detail. We see from the form of (53) that the expansion of a, as a power series in VP will contain no term in Vp to the first power, but will have the form a1 = a!" + Opº/* + aſ "p + aſº/ºpº/” — . . . . ſº Suppose that this expansion has been obtained from equation (45). * Chrystal, “Algebra,” Vol. 2, pp. 358 ft. 44 SIMON: On the Solution of Certain Types of Linear Then since a, is not a multiple zero of D, not all the first minors of D are zero when a = a,. The ratios of the A3 will be determined from (44). If pA is a non-vanishing first minor corresponding to an element in the first column of D, it follows from the form of (45), remembering that we have a; (t) = cue”, that solving (44), we get pºA; PA ſº tº G & 4.- :4, Aj = PA 41, j = 3, 4, 3 CO 5 where A = A(0) + AG/*p/* + A®p. + gº e º ſº tº 5 Aj := Aº + Aſºº + A}, + tº e º dº tº e On substituting these series for the A3 in (42) we find that the yū are devel- opable as series of the form $/ii = y? + yº/ + vº'ſ + . . . . . i = 1, 2, ' ' ' Co. So we see that in general the yi, carry terms in VP, although the term in VF is absent in the expansion for a 1. However, if all the first minors corresponding to the elements of the first column are zero, and if there is a first minor distinct from Zero corresponding to the elements of the second column, the results are precisely the same. But suppose that all the first minors corresponding to the elements of both the first and second columns are zero. Then suppose that a first minor corres- ponding to an element of the kth column is not zero. Then it follows from the form of (45) that when a = af, it will carry the factor p?; and let this minor be denoted by u’A. Then solving (44) we get - PA2 pºA where A1, A2, A3, do not in general vanish when p = 0. It follows from the first two equations that Ak must carry p as a factor, since the A3 are finite for p = 0. Hence in this case the yi, have the same form as before. Simi- larly the yia have the same properties. The solutions associated with a!", a *, ' ' ' , are found as in the pre- ceding case. If there are several groups of a? in which these congruences exist the discussion must be made for each one separately. *As 4k, A, -º-A, j = 3, 4, e ſº CO . pua p”A p"A . VII Solutions when aº is a multiple root. Now suppose that two of the aſ' are equal and only two, and that there are none of the congruences treated in VI. Let us choose the notation so that a* = a ". From our work in the theory of linear differential equations Differential Equations in Infinitely Many Variables. 45 in infinitely many variables in constant coefficients and from IV, the solutions are of the form OO 0 OO (54) as ==Aſſeus", * +3*] or CO 0 CO (55) as + A, ſcue” +3*** + A2I(cia + tesi) e” +º] OO OO +34,[cue,”--> a ºpk]. k=l if After setting e-ºar = g/g – 1, the fundamental equation becomes either CO (56) D(a,p)=|[g(cue”,” +3.2% (2n)tº) – cu(?-1)] ||=0 or (5) p(x,a)- ſate,”-Hº (*)º-ca(a_1)] [g((c., +?rea)* +3*(*) –c.,(2–1)], |=0. For p = 0 both of these, by the theory of infinite determinants, reduce to (58) D,-] cuſ [1–2(1–sº.")]. I [1–2(1–sº,”)]. 1= As before we set a = a " + 8, and expand as a power series in 8,. Then we see that the term of lowest degree in 8, alone is of the Second. When the determinant is of the form (56) with a * = a ", the term of lowest degree: in p alone is of the second in general. Then we have the same form as in VI. But if the determinant is of the form (57), the term in p alone is in general of the first degree. In the former case we have a consideration similar to that in WI; in the latter case, in general the solutions for 8, are of the form (59) 8, - pºſ”P(pº/*) 8, a = p +/*P(— p"/*), where P is a power series in Vu, and contains a term independent of p. The discussion of the special cases is made just as in VI. On substituting these expansions for a = a "+ 8, in (44) we solve for the A3 as power series in Vp. These Ai Substituted in (42) give yi, and yi, as power series in Vu. If p of the aft are equal, then for these roots the expansions of D starts with 8 as the term of lowest degree in 8, alone, and except in the special cases corresponding to those mentioned in the foregoing, the term in p alone is of the first degree. Consequently in general for aº = dº . . . = a', we have 84; = e/pºſt P(eſpº/p), j = 0, 1, . . . , p — 1, where e is a pth root of unity. ' 46 SIMON: Certain. Types of Linear Differential Equations. VIII Solutions when there are equalities and congruences among the a'. Suppose that two of the a", for example aº and aº are equal, and that a third one, say a', differs from a ” by an imaginary integer. Furthermore we shall assume that there are no other equalities or congruences among the a 9). Two cases arise here: (a) the solutions are of the form (54) with aft = a "; (b) the solutions are of the form (55) Case (a). In this case we have D,-] cul (1–2(1—eº)] iſ [1–2(1–sºlº)]. j =4 In setting a = a + 8, we find that the term in D of lowest degree in 8, alone is of the third. To get the term in D of lowest degree in p alone we set 1 = 0. Then it becomes evident at once that each of the first columns carry p as a factor, while the remaining ones do not. Consequently the term of lowest degree in p is of the third degree at least. Furthermore since the first three columns vanish when p = 8, - 0 there are no terms of lower degree than the third in 8, and p. Hence in general we see D of the form (60) D = 8: + Y218.1 + yoap.” -- ' ' ' = 0. The problem is now one of implicit functions. The details of the special cases must be treated as they arise. However, we make the general statement that since the roots of the cubic terms of (60) set equal to zero are in general distinct, it follows from the theory of implicit functions that the three values of 8, are in general expansible in integral powers of p. Case (b). In this case we have CO Do == [1 *E*R*. a (1 *-*. gºatºr)]s II [1 — a (1 -*. ezajºr) = 0. j=4 On introducing 8, as before, we find that the term of lowest degree in 8, alone is of the third degree. But when the terms involving p are retained in D, only the first and third columns vanish when p = 8, - 0, and consequently the expansion of D will contain a term in pº alone. Furthermore, since the first and third columns vanish for p = 8, - 0, there will be no terms of degree lower than the second in p and 8,. Hence in general D has the form (61) D= 8 + y,48, p + yoap” + · · · = 0. In the general case in which y, and yos are not zero, there is one solution in integral powers of p and two in powers of Vp. When the roots a;" have higher multiplicities and more congruences among them, we make a similar discussion. W IT A. Webster Godman Simon was born October 3, 1892 at Cincinnati, Ohio. He attended the public grammar and high Schools in Cincinnati and gradu- ated from Hughes High School in 1910. He entered Harvard University in the fall of 1910 and received the degree of A. B. in June, 1914. In the following year he received the degree of A. M. in mathematics. During the academic year 1915-1916 he was instructor in mathematics at Harvard. In the fall of 1916 he entered the graduate school of the University of Chicago, and in March, 1918 took the degree of Ph.D. with mathematics as his major and astronomy as his minor subject. During his years at Harvard he attended courses in mathematics, mathematical physics, and astronomy, under Professors Byerly, Osgood, Bocher, Bouton, Jackson, B. O. Peirce, H. N. Davis, Drs. G. M. Green, and Duncan; at Chicago he studied under Professors E. H. Moore, F. R. Moulton, Dickson, Bliss, Wilcynski, and MacMillan. He feels deeply grateful to all these men for their guidance and interest, especi- ally to Professors Byerly and Osgood. It gives him great pleasure to take this opportunity to express his deep gratitude to Professor F. R. 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