On General Solutions of First-Order Nonlinear Matrix and Scalar Ordinary Differential Equations

Transcription

1 saartvlos mcnirbata rovnuli akadmiis moamb 3 #2 29 BULLTN OF TH ORN NTONL DMY OF SNS vol 3 no 2 29 Mahmaics On nral Soluions of Firs-Ordr Nonlinar Mari and Scalar Ordinary Diffrnial uaions uram L Kharaishvili cadmy Mmbr orian Naional cadmy of Scincs BSTRT This aricl conains h formula of nral soluions for paricular classs of firs-ordr nonlinar mari and scalar ordinary diffrnial uaions 29 Bull or Nal cad Sci Ky words: nral soluions firs ordr mari scalar nonlinar mari and scalar ordinary diffrnial uaions Firs-Ordr anonical Nonlinar uaion Problm Samn L us considr an uaion d p h F ] 2[ ] [ p d whr = F=F ar ivn n n marics wih coninuous lmns on h inrval h=h is a ivn arbirary admissibl n n mari funcion is an unknown n n mari Hr and vrywhr an admissibl funcion will b calld any funcion in rspc o which opraions prsnd in h aricl ar valid on h whol inrval Dfiniion Th soluion of uaion will b calld mari funcion = dfind on h inrval subsiuion of which in uaion is admissibl as a rsul of which w h idniy Dfiniion 2 L b an arbirary fid poin of h inrval and b arbirary fid consan of mari n n Mari funcion dfind on h inrval and dpndin on arbirary consan of mari n n will b calld h nral soluion of uaion if is a soluion of uaion saisfyin h iniial condiion = Th basic problm consiss in consrucin h nral soluion of uaion 2 Rular Marics Main Thorms To consruc h nral soluion of uaion w shall nd a mari funcion of rular mari Dfiniion 3 Mari R=R wih coninuous lmns i j n will b calld a rular mari if hr iss n n mari funcion Rd dfini coninuous and coninuously diffrniabl wih rspc o on r i j h inrval saisfyin h condiions: whr is a uni mari p Rd= R Rd; Rd Rd Rd Rd Rd 29 Bull or Nal cad Sci

2 6 uram L Kharaishvili Thorm Basic Thorm f h mari = is rular and h admissibl mari funcion = saisfis h condiion h d dfd hn is a soluion of uaion Proof W hav ph d dfd F i Ph h F Thorm is provd L h= whr = = ar arbirary admissibl funcions From Thorm i follows Thorm 2 L h mari = is rular = = ar arbirary admissibl scalar funcions and i f hr iss h admissibl mari funcion d dfd 2 hn mari funcion 2 is h nral soluion of h uaion p F 3 Proof From formula 2 i follows d dfd onsunly s Thorm mari funcion 2 is a soluion of uaion 3 L b an arbirary fid poin of h inrval and b an arbirary fid consan of mari n n ssum ha = Thn from formula 2 i follows d d Fd Thorm 2 is provd From Thorm i follows Thorm 3 L h mari = is rular = = ar arbirary admissibl funcions and i f hr iss h admissibl mari funcion d dfd hn mari funcion 4 is h nral soluion of h uaion p [ ] F 5 Proof From formula 4 i follows d dfd onsunly s Thorm mari funcion 4 is a soluion of h uaion 5 L b an arbirary fid poin of h inrval and b an arbirary fid consan of mari n n ssum ha = Thn from formula 4 i follows Thorm 3 is provd d dfd 4

3 On nral Soluions of Firs-Ordr Nonlinar Mari and Scalar Ordinary Diffrnial uaions 7 3 riria of Rulariy Thorm 4 ririon of Rulariy L i j b an arbirary n n mari wih coninuous and coninuously diffrniabl lmns i j i j n on h inrval L k k P a a a whr m k ar arbirary coninuous and coninuously diffrniabl scalar funcions L d P L R P P rular Hr h do sands for h drivaiv d/d Proof W hav a m and Rd P Rd P and Rd P RP R Rd Thn mari R is Thorm 4 is provd orollary ririon of Rulariy L k a a d R and Rd Thn mari R is rular 4 pplicaions 4 nral Soluions of Firs Ordr Nonlinar Scalar Diffrnial uaions L us considr h uaion p-a =f 6 whr a=a f=f = = ar ivn arbirary admissibl scalar funcions f n= from Thorm 2 i follows Rsul f a=a f=f = = ar arbirary admissibl scalar funcions and i hn h nral soluion of h uaion a f 7 Rmark n his cas For ampl h nral soluion of h uaion whr a=a f=f = f c fd a ar arbirary admissibl funcions c fd Rmark n his cas = - f h nral soluion of h uaion f a a d a d c f d 8

4 8 uram L Kharaishvili f n = from Thorm 3 i follows Rsul 2 f a=a f=f = = ar arbirary admissibl scalar funcions and i hn h nral soluion of h uaion a f a 9 fd c For ampl h nral soluion of h uaion sc a f a whr a=a f=f = ar arbirary admissibl funcions arcsin fd c Rmark 2 n his cas =sin 5 Nonlinar Mari Diffrnial uaions ampls ampl L whr ar arbirary admissibl funcions L and i From orollary i follows ha mari is rular if d L us considr h mari funcion d W hav s [] d Hnc d d d and mari is rular L us considr h uaion 3 whr ar arbirary admissibl funcions From Thorm 2 i follows 2 Fd c For ampl if cons F w hav

5 On nral Soluions of Firs-Ordr Nonlinar Mari and Scalar Ordinary Diffrnial uaions 9 [] is From h uaion 3 i follows P onsunly h paricular soluion of h uaion p whr ar arbirary admissibl funcions cons ampl 2 L whr ar arbirary admissibl funcions L i Hnc mari is rular if d L us considr h mari funcion d W hav s [] d i d and mari is rular L us considr uaion 5 whr is an arbirary admissibl scalar funcion f hn from Thorm 3 i follows ha h nral soluion of uaion 5 Fd d d i 2 Fd For ampl l F W hav

6 uram L Kharaishvili matmaika pirvli riis arawrfivi skalaruli da mariculi vulbrivi difrncialuri anolbbis zoadi amonasnbis Ssab araisvili akadmikosi saartvlos mcnirbata rovnuli akadmia saiasi dadnilia zoadi amonasnbis formulbi pirvli riis arawrfivi skalaruli da mariculi vulbrivi difrncialuri anolbbis krzo klasbisatvis RFRNS uram L Kharaishvili 27 Bull or Nal cad Sci 75 : Nicolai Kudryashov 998 JPhys: Mah n M Ndljkov D Rajr 2 Novisad J Mah 3 4 D Majiros 978 Ukrainskii Mam Zhurnal 3 2 Rcivd pril 29 onsunly s [] if ] [ ] [ Hnc h paricular soluion of h uaion p whr ar arbirary admissibl funcions n conclusion w mus noic ha h rlad problms ar invsiad in [2-4]