A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD

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1 Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method for solvig differetil equtios (D.E.). The e pt or e p substitutio is other powerful method. These two commo methods do ot, however, ppl to m D.Es of phsics tht pper i Electromgetism (E & M) theor, i Qutum Mechics, d i other pt subfields of phsics. It should be reclled tht e coefficiets of secod order, ordir D.E. re costt, i.e. " + + Q( ), where,, re costts. ' p or e substitutio pplies ol whe the A geerl D.E. of phsics c be cst i the followig form (for d order D.Es): " + P( ) ' + Q( ) G( ), where the coefficiets P() d Q() re ot costts. Emple: the Legedre equtio is " ( ) ' + l( l + ) It c be rewritte s " ' ( l + ) l +, with l( l + ) P( ), Q( ), d G( ). p The seprtio of vribles is ot pplicble to this equtio, or is the e substitutio. A mightil importt method tht usull pplies to lier d order D.Es. is ow s the Frobeius method. It simpl cosists of ssumig tht the substitutio + will led to solutio. If it does, s is usull the cse,

2 Diol Bgoo () solvig the differetil equtio is reduced to () fidig the vlue of d (b) fidig the vlues of the coefficiets ', to.' Note well tht if +, is power series! However, if is egtive or is ot iteger, the + is ot power series -- eve though it is series. The Frobeius method is therefore somous with series method but ot power series method. A power series method is just specil cse of the Frobeius method. Oe should therefore strt with the Frobeius method (i.e. + ) whe solvig give D.E. for which it is eeded. Oe would strt with ol if oe is sed to use power series or if oe ows tht leds to solutio]. We discuss the Frobeius method below through emple pplictio to solvig the D.E. ( + ) " + ' + As per the Frobeius method, we ssume tht +. The ' + ( + ) d ( )( ) + " + +. Substitutig for,, d i the bove D.E. leds to ( + )( + ) + ( + ) + ( + ) ( )( + ) + ( + ) Oe sets the sums of the coefficiets of the vrious powers of equl to ero d solve for d the. I geerl, this leds to recurrece or recursio reltio i.e.,

3 Diol Bgoo () ( ) +, ( ),... ] where the vlue of the coefficiet with ide is obtied i terms of vlue of the coefficiet with ide -, -, or +, +, + 3 ]. The equtio obtied b settig the sum of the coefficiets of the lowest possible power of equl to ero is clled the idicil equtio. Before writig dow these equtios for the coefficiets, chge of ide is ofte doe. Oe m void this chge of ide -- i which m people me errors -- b costructig the tble below: Term Term Term 3 Term Notes Sum of Coefficiets (-) - ( ) ( + ) ( + ) - + ( ) ( + )( + ) ( + ) + ( + )( + ) + ( + ) + ] + ( + - )( + ) ( +) - Geerl ( + - )( + ) + ( +) + ] + -, + The idicil equtio is d the recursio reltio is, + ] ( + )( + + 3) The solutios of the idicil equtio, b virtue of Fuchs theorem, determie the et step. The qudrtic equtio, hs discrimit so d Fuchs Theorem: A differetil equtio hs series solutio -- b the Frobeius method -- subject to the followig coditios: ) If the idicil equtio hs equl roots (i.e., if ), the ol oe solutio of the D.E. is possible b this method. 3

4 Diol Bgoo () b) If the idicil equtio hs roots differig b iteger (i.e., if iteger ), the oe solutio is possible for the lrgest of the two vlues d. The other vlue of m ot led to solutio b Frobeius method. c) The idicil equtio hs roots whose differece is ot iteger; the two solutios re possible (i.e., oe for ech root). Applig this theorem to the cse of d, where the differece of the two is iteger, the we ow tht we c get series (ot power series) solutio with, the lrgest of the two vlues. With the Frobeius tble show bove, leds to: ( ) + ( ) + ( ) + + ] 7 5] or This mes. A geerl, is give b ( )( + ) + ] Recursio reltio With, ll the coefficiets with odd idices re lso ero. i.e., direct cosequece of the recursio reltio.. This is ( + ) + ] 6 3( 6) + ] A chllegig stge i pplig the Frobeius method is fidig simple ptters stisfied b the coefficiets. These ptters re geerll such tht the resultig series, +, c be idetified usig ow fuctios. There re two ws, i the cse of ( )( + ) + ], to get simple ptter.

5 Diol Bgoo () ) Log w of fidig ptter: 6 ( + ) ( + ) Ptter verifictio We the ssume tht follows this ptter: or for ll, follows ( + ) the ptter: B virtue of mthemticl iductio, if from this ( + ) ssumptio we c prove tht + lso follows the ptter, the we c coclude tht the ptter is rigorousl followed b ll eve-umbered coefficiets. The odd oes re ero. Pluggig the ssumed ptter ito the recursio reltio leds to: + ( + )( + + ) + ] The deomitor of this frctio is D ( + )( + + ) +. Crrig out the opertios d rerrgig terms leds to D ( )( α + ) + α + α α + where we set + α. So, α + α α( α + ) α, D. But α +, so + ( + )( + + ) is of the form ( + ). Hece we proved b mthemticl iductio tht, eve,. ( + ) ) Short w of fidig ptter ( )( ) ] ( )!

6 Diol Bgoo () The et step i fidig ptters is to epress ll i terms of! (We recll tht.... ) 3 5 ( + ) + + ( + ) ( + ) ( + ) ( + )!. Similrl, ou c show, b mthemticl iductio, tht ( ± ) i prticulr, ( ) ( )! ( )! So, usig d ( ) ( )! ( ) +, we get + ( ) ( ) +! + ( ) si( ) ( ) +! si is the Frobeius solutio of " + ' + ( + ) + Plese see our tetboo for the solutio of the Legedre d severl other D.Es. b the Frobeius method. Note well tht the ivolved equtios do ot hve costt coefficiets. Eve equtios with costt coefficiets c be solved b the Frobeius method --- it is, however, th pt time esier to solve them with e or e p substitutio! If ou too or re tig differetil equtio course, of relevc i phsics d egieerig, the ou will eed to grsp the Frobeius method for tht course. Recpitultio: Ke Steps of the Frobeius Method ) Chec to see tht the ordir D.E. is lier d hs t lest oe coefficiet tht is ot pt costt. Use e or e p if ll coefficiets re costt.] 6

7 Diol Bgoo () ) Set +, get ' d ". Substitute i the D.E. 3) Costruct the Frobeius tble d set equl to ero the sum of the coefficiets of ech power of the vrible. ) Solve the idicil equtio d ppl Fuchs theorem. 5) Obti for pplicble roots of the idicil equtio b obtiig ptters or compct forms of coefficiets. Etrct from the resultig solutio ow fuctio s doe bove b idetifig or etrctig the power series epsio of si(). Sigulrities of D.E. Let D.E. be i the stdrd or coicl form " + P( ) ' + Q( ) Sigulr Poits t fiite vlues of, i.e., A poit ( is fiite) is ordir poit for this D.E. if both P ( ) d ( ) Q re fiite whe ( is ot ifiit). For the Legedre D.E. below,,, 5,, etc., re regulr poits. " ' ( l + ) l + If either P ( ) or ( ) Q go to ifiit whe, the is sigulr poit for the D.E. For the Legedre D.E. bove, -, + re sigulr poits s P() d/or Q() go to ifiit whe goes to - or to +. There re two ids of sigulr poits (lso clled sigulrities): regulr d irregulr sigulr poits, respectivel deoted below s R.S.P d I. S. P. R.S.P: If P ( ) or ( ) Q whe 7

8 but ( ) P( ) d ( ) Q( ) Diol Bgoo () re both fiite whe, the Sigulr poits t ifiit is regulr sigulr poit for the D.E. I.S.P: If P ( ) or ( ) Q whe d either ( ) P( ) or ( ) Q( ) irregulr or essetil sigulr poit for the D.E. To loo for sigulr poits t ifiit, we set P( ) ( ) ) if d whe, the is (d therefore ). Q re both fiite whe, the is ordir poit of the D.E. b) if either diverges, but respectivel o fster th d, the is regulr sigulr poit. P( ) c) if ( ) diverges fster th or Q diverges fster th, the is irregulr (or essetil) sigulr poit.. We list below some e D.E. of phsics d their sigulr poits. R.S.P I.S.P Legedre D.E. ( ) " ' + l( l + ), +, - Chebshev D.E. ( ) " ' +, +, - Hermite D.E. " ' + α - Simple Hrmoic " + w - Oscilltio D.E. " + ' + Bessel D.E. ( ) 8

9 Diol Bgoo () The bove itroductio o d idetifictio of sigulr poits is of gret importce. Ideed, whe two differetil equtios hve the sme sigulr poits, the, s ou will see i dvced courses, their solutios re relted. With prctice, oe c prcticll write dow the solutios of some differetil equtios usig the solutios of other tht hs the sme sigulr poits. END NOTES: If oe hs prtil differetil equtio to solve, the oe begis b pplig the seprtio of vribles s discussed i coectio with the Lplce equtio. The, the resultig ordir D.E. re solved usig the Frobeius method if the hve some o-costt coefficiets. 9