Iteratioal Mathematical Forum, Vol. 9, 14, o., 967-976 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/imf.14.449 Series Solutio of Secod-Order Liear Homogeeous Ordiary Differetial Equatios via omplex Itegratio W. Robi Egieerig Mathematics Group Ediburgh Napier Uiversity 1 olito Road, EH1 5DT, UK opyright 14 W. Robi. This is a ope access article distributed uder the reative ommos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract A method is preseted, with stadard examples, based o a elemetary complex itegral expressio, for developig, i particular, series solutios for secod-order liear homogeeous ordiary differetial equatios. Straightforward to apply, the method reduces the task of fidig a series solutio to the solutio, istead, of a system of simple equatios i a sigle variable. The method elimiates the eed to maipulate power series ad balace powers, which is a characteristic of the usual approach. The method origiated with Herrera [3], but was applied to the solutio of certai classes of oliear ordiary differetial equatios by him. Mathematics Subject lassificatio: 3B1, 3E 34A5, 34A3 Keywords: Series solutio, ordiary differetial equatios, complex itegrals 1. Itroductio Oe of the commoest meas of seekig a solutio of a liear homogeeous ordiary differetial equatio (ODE) is to attempt to fid a ifiite series solutio.
968 W. Robi This is a well uderstood process, but ca still be a messy busiess whe attemptig to develop the recurrece relatio after substitutig the assumed form of the ifiite series ito the ODE [7, 8, 9]. I this paper we will itroduce a method for fidig power series solutios to ODE by direct itegratio i the complex plae. (All cotour itegrals that occur below are assumed evaluated i the couter-clockwise (positive) directio.) The basis of this itegratio method, due to Herrera [3], is the elemetary result [4] that, if is a iteger ( ) πi, = 1 d =, otherwise (1.1) where is a complex variable ad a fixed poit withi the closed cotour. The relatio (1.1) is used to derive a itegral expressio [3] a m ( m k) = m πi ( k) m k + 1 ( ) d (1.) for the coefficiets of a assumed power series expasio for a solutio, f (), to our ODE, that is (see, for example, [7, 8, 9]) f ) = am( m= m (1.3) The basic idea is best illustrated by a simple example, with further details preseted as the paper is developed. We will solve the defiig equatio of the (egative) expoetial fuctio, that is, we seek a series solutio, about, of the first-order equatio + (1.4) with the superscript umbers givig the order of the derivative. Assumig the +1 series solutio (1.3), we divide through equatio (1.4) by (with becomig the ew dummy variable i the series solutio) ad itegrate roud the closed cotour, avoidig ay sigularities of f (), to get f d + + 1 + 1 d (1.5) ad compare the powers i the deomiators of the itegrads of (1.5) with that of (1.) to get two equatios for the dummy variable m, oe for each value of k (oe ad ero, respectively), that is
Series solutio 969 ad m k + 1 = m 1+ 1 = + 1 or m = + 1 (1.6a) m k + 1 = m + 1 = + 1 or m = (1.6b) Havig idetified the appropriate values of k ad m, we use (1.) agai to rewrite (1.5) as, after cacellig ( + 1) a + 1 + a,,1,,3,k (1.7) With a a arbitrary costat, we recogie the recurrece relatio for ae, but with ew dummy variable, so that after the third part of the solutio process, solvig (1.7), we would write = ( f a 1) ( ) = (1.8) +1 (Note that varyig the idex i shifts the recurrece subscript accordigly.) The above procedure ca be followed i each situatio where we are sure a power series solutio exists for (first- ad) secod-order liear ODE (at least) ad it becomes apparet, whe a few examples have bee worked through, that the solutio of such liear ODE i series is reduced to the solutio of oe equatio i oe ukow repeatedly. I the above simple example o explicit use was made of (1.1), but the derivatio of (1.), preseted i sectio below, relies o (1.1). The method itself was itroduced by Herrera [3] to produce series solutios for oliear ODE, but its applicatio to liear ODE, as developed below, proves remarkably efficiet ad the methodology should be more widely kow, which is oe of the mai purposes of the curret work. The paper is orgaied as follows. I sectio, for completeess, we provide a derivatio of (1.), as Herrera [3] simply stated formula (1.) outright. The, i sectio 3, we provide a selectio of examples of the applicatio of the Herrera method to the productio of power series solutios to some of the basic equatios of mathematical physics [1,, 4, 6, 7, 8, 9]. I the fial sectio, sectio 4, we discuss the problem of higher-order recurrece relatios i the series solutio of secod-order liear homogeeous ODE. I particular, the series solutio of the secod-order liear homogeeous ODE with costat coefficiets requires some care, due to the possible occurrece of a three-term recurrece relatio. Also, at the ed of sectio 4, we cosider (very briefly) a applicatio of the Herrera method to a third-order ODE.. Derivatio of the Basic Formula Our approach is basically a formal oe. Suppose we start with the power series expasio of f () about the o-sigular poit, that is [7, 8, 9]
97 W. Robi f ) = am( m= m (.1) valid withi a assumed o-ero radius of covergece. If we differetiate (.1) k times, we fid that or ( k) m am( ) ( m k) = m= k m k ( k) k m = a( ) + am( ) ( k) ( m k) m= k m k (.) (.3) k + 1 Dividig through (.3) by ( ) ad itegratig roud a closed cotour cotaiig while avoidig ay sigularities of f (), we get, from (1.1) ( k) k + 1 ( ) d = a ( k) d = a πi ( ) ( k) (.4) so that a ( k) = πi ( k) k + 1 ( ) d (.5) O chagig dummy variables ( m) i (.5), we have equatio (1.). Fially, lookig back, we fid ( = ) m = k, k + 1, k +, k + 3, K, while k,1,,3, K. This completes the derivatio of (1.). We ow apply the Herrera method to derive power series solutios to some stadard secod-order liear homogeeous ODE. I the rest of the paper we will always take, as the equatios we cosider are kow to have such power series [1,, 4, 6, 7, 8, 9] ad we ca trasform back to the origi by a chage of idepedet variable ayway. Further, we will cosider the problem to be solved o obtaiig the recurrece relatio for the series solutios; the equatios ivolved beig well-kow, the process of solvig the recurrece relatios is widely available i may textbooks ad moographs (icludig those refereced below). This philosophy will be take to apply to covergece issues also. 3. The Solutio of Some Stadard Secod-Order Equatios The first of our examples, ivolves solvig the Airy equatio [1]
Series solutio 971 Assumig (1.3), we divide through equatio (3.1) by closed cotour, to get () (3.1) +1 ad itegrate roud the () f + d 1 f d (3.) ad compare the powers of the deomiators of the itegrads of (3.) with that of (1.) to get two equatios for the dummy variable m, oe for each value of k (two ad ero, respectively), that is m k + 1 = m + 1 = + 1 or m = + (3.3a) ad m k + 1 = m + 1 = or m = 1 (3.3b) Havig idetified the appropriate values of k ad m, we use (1.), agai, to rewrite equatio (3.), after cacellig ad re-arragig, as the recurrece relatio ( + 1 1)( + ) a + a,,1,,3,k (3.4) Settig i (3.4), we must have a 1 (there are o egative idices) ad therefore a also. With a ad a 1 arbitrary costats, we have [1] the recurrece relatio for the Airy fuctio. The third ad fial step, to solve the recurrece relatio (3.4) ad obtai the power series solutio explicitly, is wellkow [1] ad we move o to our ext example. As a secod example, we cosider the Bessel equatio of order ero, that is () + + (3.5) Assumig (1.3), we divide through equatio (3.5) by closed cotour, to get +1 ad itegrate roud the d + d + + 1 () d (3.6) ad compare the powers of the deomiators of the itegrads of (3.6) with that of (1.) to get three equatios for the dummy variable m, oe for each value of k (two, oe ad ero, respectively), that is m k + 1 = m + 1 = or m = + 1 (3.7a) ad
97 W. Robi m k + 1 = m 1+ 1 = + 1 or m = + 1 (3.7b) ad m k + 1 = m + 1 = or m = 1 (3.7c) Havig idetified the appropriate values of k ad m, we use (1.), agai, to rewrite equatio (3.6), after cacellig ad re-arragig, as ( + 1 1) a + 1 + a,,1,,3,k (3.8) Now, settig i (3.8), we must have a 1 (there are o egative idices) ad therefore a 1 also. So there are o odd powers i the power series. With a a arbitrary costat, we recogie i (3.8), for the recurrece relatio for the Bessel fuctio of order ero. The third ad fial step is to solve the recurrece relatio (3.8) ad obtai the power series solutio explicitly. As this is well-kow [8], we take this step as read ad cosider our ext example. So, for our third example, we solve the cofluet hypergeometric equatio (also called the Kummer equatio) [7] f () + ( γ ) α (3.9) where α ad γ are costats. As before, we divide through equatio (3.9) by ad itegrate roud the closed cotour, to get () f + d + ( ) f γ d + 1 d α d (3.1) 1 +1 ad compare the powers of the deomiators of the itegrads of (3.1) with that of (1.) to get four equatios for the dummy variable m, oe for each value of k (two, oe, oe ad ero, respectively), that is m k + 1 = m + 1 = or m = + 1 (3.11a) ad m k + 1 = m 1+ 1 = + 1 or m = + 1 (3.11b) ad m k + 1 = m 1+ 1 = or m = (3.11c) ad m k + 1 = m + 1 = + 1 or m = (3.11d) Havig idetified the appropriate values of k ad m, we use (1.), agai, to rewrite equatio (3.1), after cacellig ad re-arragig, as ( + 1)( + γ ) a + 1 ( + α ) a,,1,,3,k (3.1)
Series solutio 973 With a a arbitrary costat, we recogie the recurrece relatio for the power series solutio of the cofluet hypergeometric equatio [7] ad, as before, we termiate the example here. Aother importat equatio, which is our last example i this sectio, is the hypergeometric-type equatio [6] () ( a + b + c) + ( d + e) + λ (3.13) with a, b, c, d, e ad λ costats. Agai, applyig the method as before to (3.13), we get, sas details, a three-term recurrece scheme for the a, that is c( 1)( + ) a + ( b + e)( + 1) a + ( [( 1) a + d] + λ ) a (3.14) + + + 1 = I this case, the three-term recurrece relatios (3.14) ca be reduced to two-term recurrece relatios i two separate ways. First, the requiremet that c (3.15) i (3.14), leaves us with ( b e)( + 1) a + ( [( 1) a + d] + λ ) a (3.16) + + 1 = For example, (3.16) is satisfied by the Laguerre equatio, the geeralied Laguerre equatio ad the Bessel polyomial equatio [6]. Secodly, if i (3.14) we are left with b = e (3.17) c( 1)( + ) a + ( [( 1) a + d] + λ ) a (3.18) + + = For example, the Hermite, Legedre ad hebyshev equatios [6] satisfy the requiremet (3.18). Fially, for both the Romaovsky ad Jacobi polyomial equatios [6] we are left, still, with the full three-term recurrece relatio(s) (3.14), as this time either b or e (or both) is oero. However, the recurrece relatios will, ideed, termiate provided, for some iteger λ = λ = [( 1) a d] (3.19) + ad polyomial solutios will emerge. The relatio (3.19) is the well- kow [6] restrictio o λ for the extractio of orthogoal polyomial solutios from (3.13).
974 W. Robi Note that, all the above ODE may be solved idividually. This cocludes our brief survey of power series solutios to importat ODE from mathematical physics. 4. oclusios ad Discussio It must be said that the facility with which the Herrera method produces power series solutios to liear homogeeous ODE is othig short of remarkable. All problems ivolvig balacig powers i adjacet series i the stadard approach (see, for example, [8]) simply melt away, to be replaced with the solutio of simultaeous simple equatios (alog with the usual algebra, of course). I priciple we have, i fact, the geeral solutios to our equatios, as the secod solutio is always obtaiable (if ecessary) from the stadard costructio [8] of a secod solutio from a kow solutio. I the rest of this sectio, we discuss, through examples, the problem of dealig with the occurrece of three-or-moreterm recurrece relatios. The existece of such recurrece relatios is wellkow ad geerally uavoidable, as show by the geeral theory [8]. (For a brief discussio of three-term recurrece relatios see referece [].) As our first example, ivolvig the occurrece of three-or-more-term recurrece relatios, we cosider the classic secod-order liear homogeeous ODE with costat coefficiets (κ ad ω ), that is [5, 8] f () + κ + ω (4.1) Applyig the method as before (agai sas details) to (4.1), we do ideed get a three-term recurrece relatio (ad ot a easier to solve two-term recurrece relatio, which yields a closed-form solutio). O makig the substitutio κ = e g( ) (4.) however, we fid that (4.1) is reduced to where () g + ω g (4.3) ω = ω κ (4.4) Applyig the method, equatio (4.3) ow yields the two-term recurrece scheme
Series solutio 975 + + = ( 1)( + ) a + ω a (4.5) ad, o ispectio of (4.5), we may take both a ad a 1 as arbitrary costats. As usual [5, 8], give (4.5), there are three cases to cosider. If ω = ω κ >, o replacig a 1 with ω a 1, we get the recurrece schemes for a cos( ω ) ad a si( ω ), so that g( ) = a cos( ω ) + a si( ω ). ω 1 1 () If ω = k <, o replacig ω with ω ad a 1 with ω a 1, we get the recurrece schemes for a cosh( ω ) ad a1 sih( ω ), so that, i this case we have g( ) = a cosh( ω ) + a sih( ω ). 1 (3) If ω = ω κ, we have, o itegratig (4.3) directly, g ) = a + a. ( 1 For each of the above three cases, the solutio, f (), of (4.1) is give, ow, by equatio (4.). I this last example, we see oe of the stadard meas of avoidig three-term recurrece relatios, that is, the ODE is trasformed i the hope that the trasformed ODE has a power series solutio with a two-term recurrece relatio. For other examples of this trasformatio techique, see, for example, referece [8]. Geerally speakig, three-or-more-term recurrece relatios are difficult to avoid otherwise. This fact is made eve more clear i our ext example. With c, K, d costats, the equatio () 1 1 = ( c + d ) + ( c + d ) + ( c + d ) (4.6) is a basic geeraliatio of the Kummer equatio, (3.9). Applyig the method, to equatio (4.6), we get the geeral four-term recurrece relatio (,1,,3, K) ( + + 1 + 1 1 1 = 1)( + ) d a + ( + 1)( c + d ) a + ( c + d ) a + c a (4.7) with, as a 1, a iitialiig three-term recurrece relatio 1 1 = d a + d a + d a (4.8) The occurrece of three-or-more-term recurrece relatios is simply a fact of life. This fact brigs us to our cocludig remarks. It is apparet that the method ca be applied to higher-order ODE tha (first ad) secod-order ODE, the mai problem beig the developmet of the series coefficiets from the higher-order recurrece relatios that are boud to arise with higher-order ODE. Eve for the secod-order ODE, as we have see, three-or-more-term recurrece relatios
976 W. Robi occur ad it is ot clear that ay closed-form solutio for the series coefficiets, eve for three-term recurrece relatios, is available. So, if we tackle third ad higher- order liear homogeeous ODE usig the preset method, it will certaily prove ecessary to elist the aid of some form of computer algebra package i the eumeratio of the power series coefficiets. However, to fiish o a positive ote, we preset a example of a higher-order ODE which does ot have this problem, that is (3) + 4f + (4.9) which has a two-term recurrece relatio, as the reader may verify for himself. Refereces [1] Boyce W. E. ad DiPrima R..: Elemetary Differetial Equatios ad Boudary Value Problems, Fifth Editio. Joh Wiley ad Sos, Sigapore (199). [] Figueiredo B. D. B.: Geeralied spheroidal wave equatio ad limitig cases. J. Math. Phys. 48 (7) 1353. [3] Herrera J..: Power series solutios i oliear mechaics. Brookhave Natioal Laboratory Report 37494. Udated. [4] Hildebrad F. B.: Advaced alculus for Applicatios. Pretice-Hall, New Jersey (196). [5] Jeffrey A.: Mathematics for Egieers ad Scietists, Fifth Editio. hapma ad Hall, Lodo (1996). [6] Nikiforov A. F. ad Uvarov V. B.: Special Fuctios of Mathematical Physics. Birkhauser, Basel (1988). [7] Raiville E.D. ad Bediet P.E.: Elemetary Differetial Equatios, Seveth Editio. Macmilla Publishig ompay, New York (1989). [8] Simmos G. F.: Differetial Equatios with Applicatios ad Historical Notes, Secod Editio. McGraw-Hill, Lodo (1991). [9] Zill D.G.: A First ourse i Differetial Equatios, Fifth Editio PWS-KENT, Bosto (1993). Received: April 3, 14