Chapter 3 Higher Order Linear ODEs

Transcription

1 ht High Od i ODEs. Hoogous i ODEs A li qutio: is lld ohoogous. is lld hoogous. Tho. Sus d ostt ultils of solutios of o so o itvl I gi solutios of o I. Dfiitio. futios lld lil iddt o so itvl I if th qutio k k ilis tht k k. Ths futios lld lil ddt o I if holds fo so k k ot ll zo. Dfiitio. A gl solutio of o o itvl I is solutio of o I of th fo h lil iddt solutios o bsis of. E.. Show tht futios lil ddt o itvl. Solutio. E.. Show tht futios lil iddt o itvl. Solutio. t k k k I th k k k Tkig ± w gt k k k k k k k k 8k Thus w ust hv k k k E.. Solv th th-od difftil qutio iv " Solutio. T s solutio. Th w gt th htisti qutio Its oots ± ±. will ov tht lil iddt lt. So gl solutio is

2 A iitil vlu obl osists of d iitil oditios K K K Tho. If th offiits of otiuous o so o itvl I d th th iitil vlu obl hs uiqu solutio o I. I E.. Solv th iitil vlu obl " " Solutio. T. Th w hv. Its oots. Th lil iddt. Th gl solutio is Its divtivs " Thus w hv " Th sw is Th oski of solutios of is dfid s th -th od dtit Tho. Suos tht th offiits of otiuous o o itvl I. Th solutios of o I lil ddt iff I. Futho if I th I h if I th lil iddt o I. E.. Pov tht i E. w do hv bsis. Poof. Tk th lil iddt.

3 Tho If th offiits of otiuous o o itvl I th hs gl solutio o I. Tho If th offiits of otiuous o o itvl I th fo v solutio Y of is th fo Y wh fo bsis of o I d suitbl ostts.. Hoogous Equtios with ostt offiits A hoogous li qutio with ostt offiits Its htisti qutio is s I. Distit Rl Roots If ll oots of l d difft th Now w ov thos solutios fo bsis of. Th oski of thos solutios is solutios of. Th gl solutio is E. Solv th difftil qutio " Solutio. Th oots of th htisti qutio ± d th gl solutio is Tho. Solutios of with l o ol fo bsis of iff ll oots of difft. s j Tho. Solutios of with l o ol lil iddt iff ll difft. s j

4 s II. Sil ol Roots If hs i of ol ojugt oots α ± ωi th hs two lil iddt solutios osω d siω. E.. Solv th iitil vlu obl " " 99 Solutio. Th htisti qutio is. It hs th oots ± i d. H gl solutio d its divtivs Aos B si Asi B os " Aos B si Fo th iitil oditios w obti A B A 99 A B Th sw is os si s III. Multil Rl Roots If hs l oot of -th od th osodig lil iddt solutios E. Solv th difftil qutio v iv " Solutio. Th htisti qutio is. Its oots. Thus th gl solutio is s IV. Multil ol Roots If hs i of ol doubl oots th osodig lil α ± ωi iddt solutios osω d siω d osω d siω. Th osodig gl solutio is α [ A A osω B Bsiω]. Nohoogous i ODEs A ohoogous li qutio: d it osodig hoogous qutio A gl solutio of is of th fo h wh h is gl solutio of d is solutio of.

5 If th offiits d i otiuous o I th gl solutio of ists d iluds ll solutios. A iitil vlu obl osists of d iitil oditios K K K with Ud thos otiuit ssutios it hs uiqu solutio. I. Mthod of Udtid offiits hoos suitbl of. A.Bsi Rul. hoosig i th Tbl. B. Modifitio Rul. Multilig b wh j is th sllst ositiv itg s.t. o t of is solutio of.. Su Rul. If is su of futios i th tbl th is hos s th su of osodig futios. j j E. Solv th iitil vlu obl 7 " " Solutio. Th htisti qutio. It hs th til oot. H gl solutio of th hoogous qutio is. h Now w t b ul B. Th 9 8 Substitutio of ths ito th qutio 9 8 Thus givs gl solutio s ist th iitil vlus to dti th ostts 7 ] [ ] [ ] [

6 H th sw of th obl is Mthod of Vitio of Pts d d d wh fo bsis of th osodig hoogous qutio with oski obtid fo b lig th jth olu of b th olu j d T ] [ E. Solv th ohoogous Eul-uh qutio l " > Solutio. T. Th w hv. Its oots. Th th bsis of th osodig hoogous qutio Now w fid th dtits i 7 Notig tht l / l th si th offiit of. is l l l 9 l l l l l l l d d d d d d Th sw is l

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