An algorithm for solving second order linear homogeneous differential equations

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t l 9 l t 6 A c a d e m r cP r e s sI n c . ( L o n d o n ) L t d
J. J. Kovacic
been implementedalso in the uaplE computer algebra system,see.for example.Char er al. ( 1985) .by C a ro l y n Smi th (1 9 8 4 ). This paper is arranged so that the algorithm may be studied independentlyof the proofs. In section l, parts I and 2 are necessary understandthe algorithm. parts 3 and to 4 are devoted to proofs. In the other sections,part I describesthe algorithm, part 2 contains examples,and the remaining parts contain proofs. Sincethe first appearance this paper as a technicalreport. a number of papers have of appearedon the same problem: Baldassarri(1980),Baldassarri& Dwork (lg7g), Sinser (1979.19819g5). t, Special thanks are due to Bob Caviness and David Saunders of RPI for their encouragement and assistance during the preparation of this paper.
l. Introduction In th i s paper we pr e s e n t a n a l g o ri th m fo r fi n d i ng a " cl osed-form" sol uti on of the ql dill-erentialequatiofl -1"'+ +by, where a and b are rational functions of a complex va ri a b le. x .pr ov ideda " c l o s e d -fo rm"s o l u ti o ne x i s ts.The al gori thm i s so arrangedthat i f n o so l u t ic r n f ound. t h e n n o s o l u ti o n c a n e x i s t. is Th e f r r s t s ec t ionm a k e s p re c i s e h a t i s m e a n t b y " cl osed-form"and show s that there w a re fo u r pos s ible c as e s .T h e fi rs t th re e c a s e sa r e di scussedi n secti ons 3. 4 and 5 T re sp e ct iv ely . he las t c a s ei s th e c a s ei n w h i c h th e g i ven equati on has no " cl osed-form" so l u ti o n.I t holds pr ec i s e l y h e n th e fi rs t th re ec a s e s l . w fai In the s ec onds ec t i o nw e p re s e n tc o n d i ti o n sth a t are necessary each of the three for ca se s.A lt hough t his m a te ri a l c o u l d h a v e b e e n o m i tted. i t seenas desi rabl eto know i n a d va n c cwhic h c as es re p o s s i b l e . a Th e a lgor it hm in c a s e sI a n d 2 i s q u i te s i mp l ea n d can usual l ybe carri edout by hand. p ro vi d ed t he giv en e q u a ti o n i s re l a ti v e l y s i m p l e . H ow ever. the al gori thm i n case 3 i n vtl l ves quit e ex t ens i v e o m p u ta ti o n s It c a n b e p ro grammedon a computerfor a spepi fi c c . d i ffe re nt ialequat ion w i th n o d i fl i c u l ty , In fa c t. th e author has W orked through ;everal cxa mp lesus ing only a p ro g ra mma b l ec a l c u l a to r.O nl y i n one exampl ew as a coi nputer n e ce ss ar yand t his wa s b e c a u s en te rme d i a te u m b ers grew .to 20 deci mal di gi ts. more . i n th a n th e c alc ulat orc o u l d h a n d l e . F o rtu n a te l y ,th e necessary condi ti ons for case 3 are quite strong so this casecan often be eliminated from consideration. Th e a lgor it hm does re q u i reth a t th e p a rti a l fra c ti onexpansi onof the coeffi ci ents the of d i ffe rent ialequat ion b e k n o w n . th u s o n e n e e d sto factor a pol ynomi al i n one vari abl e o ve r the c om plex num b e rs i n to l i n e a r fa c to rs .O n ce the parti al fracti on expansi ons are kn o u 'n. only linear alg e b rai s re q u i re d . Usi ng t he M A CS Y M A c o mp u te r a l g e b ra s y s te m .see.for exampl e.P avel l e& W ang (1 9 8 5 ). B ob Cav ine s s a n d D a v i d S a u n d e rs of R enssel earP ol ytechni c Insti tute p ro g ra m m edt he ent i re a l g o ri th m (s e eSa u n d e rs(1981)).Meanw hi l e.the al gori thm has
I.I.
LIOUVILLIANEXTENSIONS
T he goal of th i s p a p e r i s to fi n d " c l o s e d - form"sol uti onsof di fl ' erenti al equati o' s. B ' a " c los ed- f or m"s o l u ti o n w e me a n . ro u g h l y . o ne that can be w ' ri ttendow n by a fi rs1-)' cl rr c alc ulus s t ude n t. S u c h a s o l u ti o n m a y i nvol ve esponenti al s.ndel i ni te i ntegral s and i s olut ions of p o l y n o m i a l e q u a ti o n s . (As w e are consi deri l g functi ons of a compl cr v ar iable.we n e e d n o t e x p l i c i tl ym e n ti o n tri gonometri cfuncti ons.they can be w ri ttcn rrr t er m s of ex p o n e n ti a l s .N o te th a t l o g a ri thms are i ndefi ni te i ntegral s and hence i tre allowed. )A m o re p re c i s e e fi n i ti o ni n v o l v e sthe noti on of Li ouvi l l i an fi el d. d DEFtNtrtclN. F be a differentialfield of functionsof a complex variable -x that conrai's Let C( x ) . ( T husF i s a fi e l d a n d th e d e ri v a ti o noperaror' (: drd.r) carri esF i nto i tsel f F i s ). said to be Lioutillian if there is a tower of differentialllelds C('x):FolFt9" s u c h t h a t . f o r e a c hi : either Fi:Fi-,(r) 1 .. . . , t 1 . w h e r ex ' l x e F , _ ,
I n t h i s p a p e r w r - p r e s e n ta n a l g o n t h m f o r f i n d i n g a " c l o s e d - l o r m " s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n . 1 " ' + . r I " + b r ' .w h e r e a a n d h a r e r a t i o n a l f u n c t i o n so f a c o m p l e x v a r i a b l e r . p r o v i d e d a " c l o s e d - [ o r m " s o l u t i o n e x i s t s .T h e a l g o r i t h m i s s o a r r a n g e d t h a t i i n o s o l u t i o n i s f o u n d . t h e n c ntr solutitrn an exist.
l. The Four Cases In the first part of this section we define preciselywhat we mean by "closed-form" s olut ion. I n t h e s e c o n dp a rt w e s ta te th e four possi bl e cases that can occur. Thesecascs ar e t r eat ed in d i v i d u a l l y i n th e l a tte r s e cti ons.The thi rd part i s devoted to a bri cf des c r ipt ion th e G a l o i s th e o ry o f d i ffe re n ti al of equati ons. s theory i s usedi n the prc-rofs Thi of t he t heor e mso f th e p re s e n tc h a p te r a n d those of secti ons and 5. P art 4 contLri ns 4 a pr oof of t he t h e o re ms ta te di n p a rt 2 .
.1.S.tmbolit'('ontputution (1986) 2, 3 43
Hale Waihona Puke Baidu
An Algorithm for Solving SecondOrder Linear Differential Equations Homogeneous
JERALD J. KOVACIC
JYACC Int'..919 Third Ar('tlue Irlen'York, NY 10022,Lr.S.A. (Received Ma.t' 1985) 8