We will look for series solutions to (1) around (at most) regular singular points, which without

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1 ENM 511 J. L. Baai April, 1 Frobeiu Solutio to a d order ODE ear a regular igular poit Coider the ODE y 16 + P16 y 16 + Q1616 y (1) We will look for erie olutio to (1) aroud (at mot) regular igular poit, which without lo i geerality will be located at. The otatio adopted below cloely follow that i the ote of Carchidi (hadout ad o web ite) ad i Hildebrad (Sec. 4.4 with hi R16 1 multiplyig y ). If fuctio P 16 ad Q16 are regular aroud, i.e. if they poe a Taylor erie epaio aroud, the y 16 ca alo be epreed i a Taylor erie aroud. Subtitutig the Taylor erie for P 16, Q 16 ad y 16 ito (1) ad requirig that each term of like power of um to zero (each coefficiet of a Taylor erie for zero i zero) lead to the ukow coefficiet of the erie for y 16. The method of Frobeiu i a eteio of thi idea to equatio with regular igular poit ad build o what we kow about equidimeioal equatio (ee hadout o equidimeioal equatio). If (1) ha a regular igular poit at, the the followig limit eit: p o lim P 16 ad q lim Q 16 () o Note that the eitece of thee limit implie that (1) ca be rewritte i the form: P $ Q y $ + y y (3) where P $ 16 ad Q $ 16 are regular aroud, i which cae (3) i a geeralizatio of a equidimeioal equatio. Give thi latter obervatio ad what we kow about Taylor erie olutio about regular poit, we look for Frobeiu erie olutio (1) or (3):

2 Ê Ê y y ; a a (4) The epoet i determied from the (quadratic ice (1) i a d order ODE) idicial equatio: 16 1 o 6 f + p qo (5) which lead to two value of. By covetio, we order 1 ad uch that Re16 1 > Re16. I priciple, each root give a Frobeiu erie (we mut be careful about pecial cae jut a with equidimeioal equatio). The coefficiet of thoe erie are give i equatio (.9) with (.7b) of Carchidi ote, where p ad q are the Taylor coefficiet of the repective erie for P 16 ad Q to do the tediou algebra. 16. Net we ummarize 3 pecial cae before we take advatage of Maple Oe olutio to (1) ca alway be epreed i the form of (4): 1 1 Ê Ê (6) y a a 1 16 The ecod olutio i determied accordig the followig 3 cae: Cae 1: Coider - ot a iteger (icludig whe 1 ad are comple cojugate 1 cae 1 iclude Carchidi cae a) ad c): Ê Ê (7.1) y b b 16 Cae : Coider 1 Ÿ o (Carchidi cae b): Cae 3: Coider y y l + b o (7.) 1 Ê - i a poitive iteger (Carchidi cae b): y cy l + b (7.3) 1 where the cotat c ca be zero or ozero. Ê

3 The trategy to olve a particular problem i to firt determie 1 ad from (5) ad go o to fid the coefficiet a for y116i (6) by ummig the coefficiet of like power of to zero. The ecod olutio i foud i a imilar way uig the form for y16 give by cae 1, or 3. Two (hadwritte) eample appear below. Oe ca alo ue Maple. For eample, to determie Frobeiu olutio to Beel equatio of zeroth order the followig code ca be ued: # Defie Beel' equatio of zeroth order: > L(u):^*diff(u(),$)+*diff(u(),)+(a^*^)*u(); L( u ) : u( ) + + u( ) a u( ) > aume(a, real, a>); # Obtai a erie olutio (for J(,a) ad Y(,a)): > Order: 7: dolve({l(u)},u(),erie); u( ) _C a~ 1 64 a~ a~ 6 6 O( 7 ) + l( ) a~ 1 64 a~ a~ 6 6 O( 7 ) a~ 3 18 a~ a~ 6 6 O( 7 ) _C # Or, Maple ca olve the equatio ymbolically: > ol:dolve({l(u)}); ol : u( ) _C1 BeelJ (, a~ ) + _C BeelY (, a~ ) Although Maple ha doe tediou algebra for u i determiig the Frobeiu erie, oe till hould determie the patter i the coefficiet, i.e. the recurio relatio. For eample, the erie for J16 ca be epreed compactly a: k k+ - J Ê 1 k k1! k + 6!

4 Hadout of 1/9/1 (ote: for the equidimeioal equatio oly 1 Ÿ o i a pecial cae) Equidimeioal ODE (Euler equatio) have a regular igular poit, which i the followig eample i located at o, i.e. u ~ (ee pp i Wylie ad Barrett more o igular poit whe we dicu the method of Frobeiu). Sice the epoet ca be egative, the olutio ca be igular at. Furthermore, whe a epoet i a repeated root of the characteritic equatio, olutio alo eit of the form u m ~ l16, m 1,,..., M -1, where M i the multiplicity of the root. Eample for variou cae follow. 1) Lu 16 u - u + u 1 6 L The characteritic equatio - 3+ ha root 1 1ad. Ã u 16 c 1 + c (o igularity at ) ) Lu 16 u + 5u + u 1 6 L Ã u c + c (igularity at )

5 3) Lu 16 u - u + u L 1 1 Hece, 1 i a double root ( M ). Note that: L 1 1 l if 1 Sice L L 1 it follow that l i alo a olutio. The complete olutio i: u 16 c 1 + c l To ee that thi olutio i ot igular at ote that lim l lim l - 1 lim - lim - if > --1