NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
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1 NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Abstract In a rcnt papr [] 5 th indinit intgrals o a crtain typ ar calculatd using som linar homognous dirntial quations o scond ordr with variabl coicints associatd with th intgrals In som simpl cass lik th ampls considrd in this papr th linar indpndnt solutions o ths dirntial quations ar dirctly calculatd rlativ to lmntary unctions In mor diicult cass th powr sris mthod must b usd In such situations it is advisabl to us th algbraic symbolic calculus on computr Eampls o this typ will b givn in a subsqunt papr Bcaus th main ormula rom which th intgrals can b calculatd is not rigorous provd in [] w giv hr a corrct proo basd on th Abl-Liouvill ormula or th dirntial quations o scond ordr For compltnss w giv hr th proo or this ormula and som o its applications ncssary to our work Also w includd two ampls Kywords: indinit intgrals scond ordr linar homognous dirntial quations Abl-Liouvill ormula Mathmatics Subjct Classiication: 6A36 34A3 Abl-Liouvill ormula or scond ordr linar dirntial quations considr hr th scond ordr linar homognous dirntial quation () with continuous coicints a a ( ) and a Lt b a known solution o th quation () Thn w also hav Th dtrminant + a y + a y a y + a + a a () y y ( y ) y y (3) it is calld th ronski s dtrminant o th solutions and o th quation () Thorm Th ronski dtrminant o th solutions and o th quation () satisis th Abl-Liouvill ormula u ( y ) k (4) y y
2 whr k is a constant and u a d (5) a Proo Eliminating th last trms o quations () and () w can writ as a rom which rsults [ y y ] + a [ y y ] [ y y ] y y By intgration it rsults y y y y a a y y u ln k ln + whr k is a constant From this rlation it rsults Abl-Liouvill ormula (4) Linar indpndnc o th solutions o a linar scond ordr dirntial quation ronski s dtrminants ar usd to dtrmin linar indpndnc o th solutions o a linar dirntial quation w hav Thorm Two solutions and o th quation () ar linar dpndnt i and only i Proo I ( y ) y y (6) y ( y ) y y By intgration rom th last rlation it rsults ln C ln y
3 C y whr is an arbitrary constant Thror w obtain C hnc th solutions y and ar linar dpndnt Rciprocally i th solutions y and ar linar dpndnt w hav y C hnc ( y ) y y C C only i y Consqunc Two solutions and o th quation () ar linar indpndnt i and ( y ) (7) Th abov provd Abl-Liouvill ormula allows strngthning th rsult givn in Consqunc Namly Consqunc Two solutions and o th quation () ar linar indpndnt i and only i thr is a valu or which (8) 3 Solutions o scond kind o th linar homognous dirntial quations o scond ordr I is a known solution o th quation () th Abl-Liouvill ormula can b considrd as a non-homognous linar dirntial quation o th irst ordr rom which can b obtaind nw solutions g o th quation () linarly indpndnt with namd solutions o scond kind o th quation () Mor actly w hav Thorm 3 I is a known solution o th quation () and its coicints satisy th condition (5) thn a scond solution g o th quation linarly indpndnt with is givn or vry constant k by th quation o irst ordr and has th orm ( y( ) ( )) u g g k (9) g k y u d () Proo Equation (9) rsults rom dinition (3) o th ronski s dtrminant and rom th Abl- Liouvill ormula (4) Putting quation (9) in th normal orm k u y y and using th wll-known ormula or th solutions o a non-homognous linar dirntial quation o irst ordr with variabl coicints w obtain th gnral solution o th quation (9)
4 d d u k u y d + C k + d C For C w obtain th scond typ solution y g givn by ormula () Dirct proo sk th solution o th quation () in th orm v y () whr v is th nw unknown unction o th quation () Bcaus y v + v and y v + v + v th quation () rcivs th orm v + [ a + a ] v + a [ + a + a ] v + a () Using th rlation () th quation () bcoms v + [ a + a ] v a (3) Considring th nw unknown unction v th quation (3) rcivs th orm bcoms w (4) w + [ a + a ] w a (5) w Dividing th quation (5) with w w w + u Th quation (6) has th solution u k (7) From (4) and (7) it rsults v w a and using th rlation (3) th quation (5) (6) u d + C k d + C (8)
5 Finally rom () and (8) it rsults or C and g y ormula () 4 A nw orm or homognous linar dirntial quations o scond ordr Thorm 5 Th quation () has th solution whr u is a givn dirntiabl unction i and only i th quation has th orm y u y + [ u ] y () and its coicints satisy th condition (5) In this cas th scond kind solution o th quation is givn by th ormula () Proo From (5) it rsults a u a I is a solution o th quation () w hav () hnc u a Substituting ths valus o coicints in quation () this acquirs orm () Convrsly it is obviously that quation () has as solution and its coicints satisy th condition (5) Th last statmnt o thorm 5 it ollows rom thorm 3 5 Computing indinit intgrals by linar homognous dirntial quations o scond ordr I w know two linar indpndnt solutions o th quation () thn rom th abov rsults which ar drivd rom th Abl-Liouvill ormula w can calculat th indinit intgrals o th orm I u d () Namly w hav th ollowing main rsult ar givn unctions irst onc and scond twic tims dirntiabl thn th indinit intgral () can b calculatd rom th ormula Thorm 6 I u and I u u g d + C () [ g g ] g is a scond kind solution o th linar dirntial quation () and C is an arbitrary constant Proo Th ormula () rsults rom th abov ormulas (9) and () rom thorm 3 In conormity with thorm 5 th unctions and g ar th linar indpndnt solutions o th dirntial quation () whr Rmark To calculat indinit intgrals o typ () by this nw mthod must b don th ollowing stps: must b idntiid rom th intgral (); ) Th unctions u and
6 ) Using th unctions u and unction is solution; is dtrmind 3) Th solution o scond kind using powr sris Th ormula () can not b usd to calculation o g intgral to b calculatd; 4) Using th unctions u and g () 6 Eampls th homognous linar dirntial quation () or which th g o th dirntial quation () is obtaind by dirct mthods or bcaus it contains th th indinit intgral () is calculatd rom th ormula giv two ampls that was also considrd in [] In that articl or scond ampl has bn usd th powr sris mthod which is not ncssary To calculat th indinit intgral I d w rwrit th intgral as choos u 3 and 3 I d ( ) Substituting ths unctions in () w obtain th scond ordr linar homognous dirntial quation 3 y + y y with th two linarly indpndnt solutions and g that th intgral is Using th ormula () it rsults I 3 d ( ) + C + C d To calculat th intgral > nπ sin ( ) sin In this cas th quation () taks th orm and y y y n w choos u ln dt Making th chang o variabls t w hav y y ( t) y ( t) and d dt y y ( t) + y ( t) y ( t) + 4ty ( t) hnc th dirntial quation bcoms d ( t) + y y t with linar indpndnt solutions th ormula cos t and sin t Thror g cos( ) and th intgral is givn by
7 d sin( ) sin( ) cos( ) ( sin ( ) + cos ( ) Rmark Using th chang o variabls + C ctg( t w hav ) + C d sin ( ) dt sin t ctg ( t) + C ctg( ) + C Rmark Ths ampls show that by th nw mthod o calculating th indinit intgrals th usual rsults ar obtaind Rrncs: [] O Kiymaz Ș Mirasydioglu A nw symbolic computation or ormal intgration with act powr sris Appl Math Comput 6 (5) 5-4 [] S L Ross Dirntial Equations Ginn-Blaisdkk London 964
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