Reduction of Higher Order Linear Ordinary Differential Equations into the Second Order and Integral Evaluation of Exact Solutions

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1 Mthemtic Aeter, Vol. 4, 04, o., Reductio o Higher Order Lier Ordiry Dieretil Equtios ito the Secod Order d Itegrl Evlutio o Ect Solutios Guw Nugroho* Deprtmet o Egieerig Physics, Istitut Tekologi Sepuluh Nopember Jl Arie Rhm Hkim, Surby, Idoesi (0) Ahmd Zii Deprtmet o Egieerig Physics, Istitut Tekologi Sepuluh Nopember Jl Arie Rhm Hkim, Surby, Idoesi (0) Purwdi A. Drwito Deprtmet o Egieerig Physics, Istitut Tekologi Sepuluh Nopember Jl Arie Rhm Hkim, Surby, Idoesi (0) Abstrct Higher order lier dieretil equtios with rbitrry order d vrible coeiciets re reduced i this work. The method is bsed o the decompositio o their coeiciets d the pproch reduces the order util secod order equtio is produced. The method to id closedorm solutios to the secod order equtio is the developed. The solutio or the secod order ODE is produced by rerrgig its coeiciets. Ect itegrl evlutio is lso coducted to complete the solutios. Keywords: Higher order lier ordiry dieretil equtios, ect itegrl evlutio, reductio o order, decompositio o vrible coeiciets. MSC umber (00): 4A05, 4A5, 4A0.. Itroductio It is well-kow tht the well-posed problem or the lier dieretil equtios hs bee settled d completed by mes o uctiol lysis []. However, the cocepts will ot be very useul util the eplicit solutios re produced. They re cpble to describe the detil etures o the systems [,]. They my lso help to eted the eistece, uiqueess d regulrity properties o the solutios which re obtied rom qulittive lysis [4]. *Correspodig Author: guw@ep.its.c.id, guw@gmil.com, guwzz@yhoo.com

2 7 Guw Nugroho, Ahmd Zii d Purwdi A. Drwito Thereore, methods or solvig lier dieretil equtios with vrible coeiciets re importt rom both physicl d mthemticl poit o views [5]. Especilly or the secod order ordiry dieretil equtios, with ohomogeous physicl properties, such s i wves propgtio i o-uiorm medi d vibrtio wves with isotropic physicl properties. Sice tht speciic problem ttrcts my mthemticis d physicists, the methods to obti ect d pproimte solutios or secod order equtio re tckled systemticlly d some iterestig results re produced []. Oe cse is the method o dieretil trser mtri to hdle some physicl problems which is computtiolly milder th the previous lytic methods d the method is lso pplied to the higher order ODEs [7]. Also some pproimte methods c be eteded to hdle olier equtios [8,9]. Despite the cocetrted reserch d reports o the problem, the closed-orm solutios or the higher order d secod order ODEs with vrible coeiciets remi oe o the importt re o dieretil equtios [0]. Eve it is recetly climed tht the problem is ot solvble i geerl cse []. I this work, the method or obtiig ect solutios to the secod order equtios is coducted by rerrgig the coeiciets. The solutio o the secod order equtio will be implemeted s bsis or tcklig the higher order equtios. The coeiciets o the equtios re decomposed i order to reduce their order. The reductio is cotiued util the secod order equtio is produced d solved. The eplicit epressio the c be determied by the proposed ect itegrl evlutio i order to complete solutios. Filly, we give ilustrtios o itegrl evlutio by emples.. Solutios or Secod Order Dieretil Equtios Sice the secod order dieretil equtio c be trsormed ito the Riccti clss, we begi rom the ollowig sttemet, Theorem : Cosider the secod order lier ODE with vrible coeiciets, y y y 0 The coeiciets d c be split ito ew uctios,,,, 4, 5, d. By determiig the ew uctios,,, 4, 5, d, the closed-orm solutio is obtied s,

3 Reductio o Higher Order Lier Ordiry Dieretil Equtios 77 d y d C d e d C7 d C d e d C where d C4 C d e d C d C5 d. Proo: The bove equtio c be rewritte s, y y y 0 Suppose tht, c be rerrged s, 4 0 to produce, Ce d d (). The bove equtio (b) with, y, d. Equtio (b) c be rerrged s, (c) Set, to get, 4 0 (d) Let,, equtio (e) will become, 0 or 5 0 () with, 5 d. Repet the procedure (c d) to produce, 0 or (b)

4 78 Guw Nugroho, Ahmd Zii d Purwdi A. Drwito where the reltios, , 7 5, d re hold. Let,, the solutio or is, 4 5 (c) 4 d C Substitutig the bove equtio ito,, to get, d C d C 5 d C 5 Tke,, the solutio or 5 c be obtied s, 5 5 Recll the deiitio o 5 d, substitute 4 d equtig with (d) to orm, Let,, the bove equtio c be writte s, Suppose tht, 0 or, the solutio or d re the, d C e d C d d C4 C e d C d C 5 Note tht, 4, with d or (d) () (b) (c) 4 is epressed by (c).

5 Reductio o Higher Order Lier Ordiry Dieretil Equtios 79 Thereore, equtio (b) becomes, 7 0 or 5 0 The solutio or (4) is the, 5 d or C e d C 7 The solutio or y is deied s, (4) d (4b) C e d C d 7 7 y C e d C d d C d e d C7 where d C d e d C d C4 C d e d C d C5 (4c) d. This proves theorem.. Cses o Order Reductio Cosider o homogeous third order lier dieretil equtio with vrible coeiciets below, y y y y (5) 4 Lemm : Equtio (5) is reducible ito secod order equtio d hs closedorm ect solutios. Proo: Let, 5 b (5b) 5 The, the equtio c be rewritte i the ollowig orm, 5 Set, y b y y y 5 4 b b (5c) Thus, the ollowig reltio is obtied, b y y b y y 5 4 5

6 80 Guw Nugroho, Ahmd Zii d Purwdi A. Drwito Multiply by rbitrry uctio to geerte [8], b y y b y y (5d) Suppose tht the ollowig epressio is stisied, the, b Ce b d Let C, equtio (5d) is rewritte s, d b b y y b e y Suppose tht, d b e y u, d y ue d b Thereore equtio (5d) c be epded s, 4 d d d d d b b b b b b 5 ue u e u e ue u e 5 b b b b u (5e) (5) Dieretite the bove equtio oce gi d set the ollowig reltio, d d b b b 5 e e 5 b b 0 Now ssume tht b is give, the 5 b c be determied rom () s, () (b) b 5 Substitutig ito (5c) to give the epressio o b s uctio o 5. Perormig the resultig epressio ito (5b) to geerte 5. Thereore equtio (5) is reduced ito, u u u Let, u v, thus the bove equtio be trsormed to the secod order ODE, v v v (c) 7 8 9

7 Reductio o Higher Order Lier Ordiry Dieretil Equtios 8 The, by the pplictio o theorem, equtio (c) is solvble i closed-orm. The o homogeous prt is covered by tkig homogeous solutio o (c) s prticulr solutio i the ollowig orm, u (d) where is prticulr solutio rom equtio (c). The solutio or is stted s, 7d 7d e e 9d d (e) The combitio o (e) with () d (d) will produce the il solutio. This proves lemm. Lemm : The ourth order lier dieretil equtio, y y y y y 4 5 is reducible to third d secod order equtios d hs closed-orm solutios. Proo: Suppose tht, 5 b, b b 5 7 d b b (7) Thereore the equtio become, b b y y y b y y Multiplyig by rbitrry uctio to give, b b y y y b y y (7b) Let, 4 the, b Ce b 4 d Equtio (7c) is trsormed s, 4 d b b b y y y b e y Let us ssume tht, 4 d b e y u, d y ue 4 d b 7 (7c) (7d)

8 8 Guw Nugroho, Ahmd Zii d Purwdi A. Drwito Epdig equtio (7b) s, d d d d b 4 b 4 b 4 b 5 ue u e u e u e 5 b b b b d d d b 4 b 4 b 4 4 ue u e u e b d d b 4 b 7 ue u e b b 7 b b u 5 Perormig the ollowig reltio, d d d b b 4 b b 4 b 5 e e 7 e 5 b b 7 b 0 Suppose tht b d 7 re give, the 5 0 b 5 c be determied orm (8) s, (8) (8b) b 5 Substitutig (8b) ito the secod reltio o (7) to give b s uctio o. The et step is implemetig ito the irst reltio o (7) to produce 5. Thereore, the ourth order equtio is reduced ito, 5 u u u u Let, u v, thus the bove equtio c be trsormed to the third order equtio, v v v v (8c) The, by the pplictio o theorem d lemm, equtio (8c) will hve closed-orm solutios. This proves lemm. It is iterestig to ote tht, by iductio, the procedure c be pplied to y order higher th two d the cosidered equtios re trsormed ito the secod order equtios. Theorem : Higher order lier dieretil equtio is reducible ito the secod order equtio d hs closed-orm solutios.

9 Reductio o Higher Order Lier Ordiry Dieretil Equtios 8 4. Remrks o Itegrl Evlutio It is importt to ote tht the itegrls which pper i the ect solutios re usully pproimted i series orm []. The solutios cosequetly re o loger ect. I order to resolve the problem, ow the ollowig itegrl is cosidered, d B e d (9) By settig, d gd (9b) B e d R Q e Equtio (9b) c be dieretited oce to give, e R Q e R Q e R Q ge d gd gd gd Rerrgig the bove equtio s, R g R e Q g Q The solutio o R is the epressed by, gd R e e e Q g Q d C Let, (9c) gd gd gd (9d) gd e Q g Q 4 The, R is evluted i the ollowig, gd gd gd R e 4 d e 4 d ge d C Suppose tht rom equtio (9e), e gd C where C is lso costt. e C (9e) (9) The epressio or d e gd gd e Thus, equtio (9) will become, is writte s, (0)

10 84 Guw Nugroho, Ahmd Zii d Purwdi A. Drwito gd d d R e 4 d e 4 d e d C C (0b) d Without loss o geerlity, set 4d l e, d the epressio o 4 is obtied s, d 4 l e (0c) The solutio or Q is cosequetly obtied rom (9e) s i the ollowig reltio, gd Q e C 4 e gd d Substitutig (0) to get, 4 (0d) gd d Q e C e d C Equtios (9b), (0b) d (0d) will give the evlutio s, d gd d d 4 C C e d R Q e e C e d C (0e) where 4 is determied by (0c). Equtio (0e) c be dieretited oce d rerrged to be, d d e d e d 4 () Now suppose tht e d L rbitrry costt. The reltio o is the give by, e d Let, equtio (b) will the produce, d e Let d e d d 4 l e L, with is, the bove equtio will the become, (b) (c)

11 Reductio o Higher Order Lier Ordiry Dieretil Equtios 85 d d d e e e Equtio (d) c be rerrged s, d e d e The solutio or is d (d) e d (e) The solutio or is d e The step is ow peromig the itegrtio o () to give, () d d d e d L e l e () Rerrgig () d substitutig (), d e d d e d d e e d The polyomil equtio or e d is the, (b)

12 8 Guw Nugroho, Ahmd Zii d Purwdi A. Drwito d d d e d e e d d d e e d e e or K K K K K K with K. Without loss o geerlity let to get, d e 4 K K K K K K K K K 0 By usig the cubic ormul, (c) (d) 4 K K K K K M, (e) K K K 9 K K K 4 K K K K K K N K K K K K K K K K K 7 K K K 4 ()

13 Reductio o Higher Order Lier Ordiry Dieretil Equtios 87 With the reltios s N M N (d) is writte s, d s N M N K K K K K K K 4, the root o d e d s s (g) This will solve the itegrl i (9). Thereore, the ollowig theorem is just proved, Theorem : Cosider the ollowig itegrl equtio, d B e d There eists uctiol d which re deied by, d e d d such tht the itegrl B c be evluted s, d e K K K K d e d N M N N M N K K K 4 where K Emples; d e, M d N re deied by (e) d (). Now, the emples o the proposed itegrl evlutio tke rom the itegrl tble re give []. Cosider the itegrl, y e d e where ccordig to theorem the uctios d re d respectively. The compriso re show s i the ollowig,

14 88 Guw Nugroho, Ahmd Zii d Purwdi A. Drwito Figure. The comprio o the kow itegrl ormul gits the proposed itegrl evlutio Figure shows tht the computtios o the proposed itegrl evlutio or the spesiic itegrl equtio re very close to the kow result, eve coicide or certi costt coeiciet. 5. Coclusios The method o reductio o the higher order lier ordiry dieretil equtios is proposed i this rticle. The mi strtegy is to decompose the coeiciets d the process thus cotiued util secod order equtio is obtied d solved. The procedure or solvig secod order equtio d ect itegrl evlutio re lso coducted d developed to complete the solutios. The pper hve ilustrted the ew ide o coeiciet decomposititio to solve the geerl ODEs with vrible coeiciets. It is show tht the method c obti the solutios o rbitrry coeiciets d rbitrry order higher th oe i closedorm. Moreover, the ew ormultio o itegrl evlutio will mke the results re trctble or computer simultios. We pl to coduct the pplictios i our uture works. Reereces [] W. Zhuoqu, Y. Jigue d W. Chupeg, Elliptic d Prbolic Equtios, World Scietiic Publishig Co. Pte. Ltd., (Sigpore, 00). [] E.A. Coddigto, A Itroductio to Ordiry Dieretil Equtios, Dover Publictios Ic., (New York, 989).

15 Reductio o Higher Order Lier Ordiry Dieretil Equtios 89 [] H.I. Abdel-Gwd, O the Behvior o Solutios o Clss o Nolier Prtil Dieretil Equtios, Jourl o Sttisticl Physics, Vol. 97, Nos. ½, (999). [4] J.G. Heywood, W. Ngt d W. Xie, A Numericlly Bsed Eistece Theorem or the Nvier-Stokes Equtios, Jourl o Mthemticl Fluid Mechics, (999), 5. [5] V.A. Glktoov d A.R. Svirshchevskii, Ect Solutios d Ivrit Subspces o Nolier Prtil Dieretil Equtios i Mechics d Physics, Tylor & Frcis Group, (Boc Rto, 007). [] L. Bougo, O the Ect Solutios or Iitil Vlue Problems o Secod Order Dieretil Equtios, Applied Mthemtics Letters, (009), [7] S. Khorsi d A. Adibi, Alyticl Solutio o Lier Ordiry Dieretil Equtios by Dieretil Trser Mtri Method, Electroic Jourl o Dieretil Equtios, No., (00), 8. [8] G. Adomi, Solvig Frotier Problem o Physics: The Decompositio Method, Kluwer Acdemic Publishers, Dordrecht, (The Netherlds, 994). [9] Z. Zho, Ect Solutios o Clss o Secod Order Nolocl Boudry Vlue Problems d Applictios, Applied Mthemtics d Computtio 5, (009), 9 9. [0] L. Shi-D, F. Zu-To, I. Shi-Kuo, X. Guo-Ju, L. Fu-Mig d F. Bei-Ye, Solitry Wve i Lier ODE with Vrible Coeiciets, Commu. Theor. Phys. 9, (00), 4 4. [] S.V. Meleshko, S. Moyo, C. Muriel, J.L. Romero, P. Guh d A.G. Choudury, O First Itegrls o Secod Order Ordiry Dieretil Equtios, J. Eg. Mth., DOI 0.007/s , (0). [] R. Wog, Asymptotic Approimtios o Itegrls, Society or Idustril d Applied Mthemtics, (00). [] G. Petit Bois, Tbles o Ideiite Itegrls, Dover Publictio Ic., (New York, 9).