Boyce/DiPrima/Meade 11 th ed, Ch 4.1: Higher Order Linear ODEs: General Theory

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1 Bo/DiPima/Mad h d Ch.: High Od Lia ODEs: Gal Tho Elma Diffial Eqaios ad Boda Val Poblms h diio b William E. Bo Rihad C. DiPima ad Dog Mad 7 b Joh Wil & Sos I. A h od ODE has h gal fom d d P P P d d W assm ha P P ad G a oios al-vald fios o som ival I = ab ad ha P is owh zo o I. Dividig b P h ODE boms L d d P G Fo a h od ODE h a iall iiial odiios: d d d d d d g

2 Thom.. Cosid h h od iiial val oblm If h fios ad g a oios o a o ival I h h xiss xal o solio ha saisfis h iiial val oblm. This solio xiss hogho h ival I. g d d d d d d = f

3 Homogos Eqaios As wih d od as w bgi wih homogos ODE: If a sols o ODE h so is lia ombiaio Ev sol a b xssd i his fom wih offiis dmid b iiial odiios iff w a solv: d d d d d d L

4 Homogos Eqaios & Wosia Th ssm of qaios o h vios slid has a iq solio iff is dmia o Wosia is ozo a : Si a b a oi i h ival I h Wosia dmia ds o b ozo a v oi i I. As bfo i s o ha h Wosia is ih zo fo v oi i I o i is v zo o I. W

5 Thom.. Cosid h h od iiial val oblm If h fios a oios o a o ival I ad if a solios wih W fo a las o i I h v solio of h ODE a b xssd as a lia ombiaio of : d d d d d d

6 Lia Dd ad Idd Two fios f ad g a lial dd if h xis osas ad o boh zo sh ha f g fo all i I. No ha his ds o dmiig whh f ad g a mlils of ah oh. If h ol solio o his qaio is = = h f ad g a lial idd. Fo xaml l fx = six ad gx = six osx ad osid h lia ombiaio si x si xos x This qaio is saisfid if w hoos = = - ad h f ad g a lial dd.

7 Examl A h followig fios lial idd o dd o h ival I: Fom h lia ombiaio ad s i qal o zo Evalaig his a = = ad = = w g Th ol solio o his ssm is Thfo h giv fios a lial idd f f f

8 Examl A h followig fios lial idd o dd o a ival I: Fom h lia ombiaio ad s i qal o zo Evalaig his a = = ad = = w g Th a ma ozo solios o his ssm of qaios Thfo h giv fios a lial dd f f f f

9 Thom.. If { } is a fdamal s of solios of o a ival I h { } a lial idd o ha ival. Covsl if { } a lial idd solios o h abov diffial qaio h h fom a fdamal s of solios o h ival I L

10 Fdamal Solios & Lia Idd Cosid h h od ODE: A s { } of solios wih W o I is alld a fdamal s of solios. Si all solios a b xssd as a lia ombiaio of h fdamal s of solios h gal solio is If a fdamal solios h W o I. I a b show ha his is qival o saig ha a lial idd: iff

11 Nohomogos Eqaios Cosid h ohomogos qaio: If Y Y a sols o ohomogos qaio h Y - Y is a solio o h homogos qaio: Th h xis offiis sh ha Ths h gal solio o h ohomogos ODE is wh Y is a aila solio o ohomogos ODE. g d d d d d d L g g L Y L Y Y L Y Y Y Y

12 Bo/DiPima/Mad h d Ch.: Homogos Diffial Eqaios wih Cosa Coffiis Elma Diffial Eqaios ad Boda Val Poblms h diio b William E. Bo Rihad C. DiPima ad Dog Mad 7 b Joh Wil & Sos I. Cosid h h od lia homogos diffial qaio wih osa al offiis: L a a a a As wih sod od lia qaios wih osa offiis = is a solio fo vals of ha ma haaisi olomial Z zo: a a a a L haaisi olomialz B h fdamal hom of algba a olomial of dg has oos ad h Z a

13 Ral ad Uqal Roos If oos of haaisi olomial Z a al ad qal h h a disi solios of h diffial qaio: If hs fios a lial idd h gal solio of diffial qaio is Th Wosia a b sd o dmi lia idd of solios.

14 Examl : Disi Ral Roos of Cosid h iiial val oblm Assmig xoial sol lads o haaisi qaio: Ths h gal solio is

15 Examl : Solio of Th iiial odiios ild Solvig H

16 Examl : Gah of Solio of Th gah of h solio is giv blow. No h ff of h lags oo of h haaisi qaio

17 Comlx Roos If h haaisi olomial Z has omlx oos h h ms o i ojga ais. No ha o all h oos d b omlx. Solios osodig o omlx oos hav h fom i i os i os i si si As i Cha. w s h al-vald solios os si l ± im

18 Examl : Comlx Roos of Cosid h iiial val oblm Th 7/ 5/ Th oos a - i -i. Ths h gal solio is Usig h iiial odiios w obai os si Th gah of solio is giv o igh. os si

19 Examl : Small Chag i a Iiial Codiio of No ha if o iiial odiio is slighl modifid h h solio a hag sigifial. Fo xaml la 7/ 5/ wih h os si 7/ 5/ 5/ os 6 Th gah of his solio ad oigial a giv blow. si

20 Rad Roos Sos a oo of haaisi olomial Z is a ad oo wih mlili s. Th lial idd solios osodig o his ad oo hav h fom If a omlx oo is ad s ims h so is is ojga l - im. Th a s osodig lial idd sols divd fom al ad imagia as of o s os os l + im si s s i i i s i os si si

21 Examl : Rad Roos Cosid h qaio Th Th oos a i i -i -i. Ths h gal solio is os si os si

22 Examl : Comlx Roos of of Fo h gal solio of h haaisi qaio is. To solv his qaio w d o s El s qaio o fid h fo h oos of : - = os + isi = i o - = os + m + isi + m = i +m fo a ig m - / = i +m / æ = os + m ö æ è ç ø + isi + m ö è ç ø Lig m = ad w g h oos: i i i i sivl.

23 i i i i Examl : Comlx Roos of of Giv h fo omlx oos xdig h idas fom Cha w a fom fo lial idd al solios. i Fo h omlx ojga ai w g h solios = / os / = / si / Fo h omlx ojga ai w g h solios = / os / = / si / So h gal solio a b wi as i + + +

24 Bo/DiPima/Mad h d Ch.: Th Mhod of Udmid Coffiis Elma Diffial Eqaios ad Boda Val Poblms h diio b William E. Bo Rihad C. DiPima ad Dog Mad 7 b Joh Wil & Sos I. Th mhod of dmid offiis a b sd o fid a aila solio Y of a h od lia osa offii ohomogos ODE L a a a a g ovidd g is of a aoia fom. As wih d od qaios h mhod of dmid offiis is iall sd wh g is a sm o od of olomial xoial ad si o osi fios. Sio. dissss h mo gal vaiaio of aams mhod.

25 Examl Cosid h diffial qaio Fo h homogos as Ths h gal solio of homogos qaio is Fo ohomogos as i mid h fom of homogos solio. Ths bgi wih As i Cha i a b show ha A Y C Y

26 Examl Cosid h qaio si 5os Fo h homogos as Ths h gal solio of h homogos qaio is os si os si Fo h ohomogos as bas of h fom of h solio fo h homogos qaio w d Y Asi Bos 5 As i Cha i a b show ha Y si os 8 8 Ths h gal solio fo h ohomgos qaio is Y

27 Examl Cosid h qaio Fo h homogos as Ths h gal solio of homogos qaio is Fo ohomogos as i mid fom of homogos solio. Ths w hav wo sbass: As i Cha a b show ha Th gal solio is os si os E Y D C Y B A Y C Y 8 si 5 8 Y

28 Bo/DiPima/Mad h d Ch.: Th Mhod of Vaiaio of Paams Elma Diffial Eqaios ad Boda Val Poblms h diio b William E. Bo Rihad C. DiPima ad Dog Mad 7 b Joh Wil & Sos I. Th vaiaio of aams mhod a b sd o fid a aila solio of h ohomogos h od lia diffial qaio L g ovidd g is oios. As wih d od qaios bgi b assmig a fdamal solios o homogos qaio. Nx assm h aila solio Y has h fom Y wh a fios o b solvd fo. I od o fid hs fios w d qaios.

29 Vaiaio of Paams Divaio of 5 Fis osid h divaivs of Y: If w qi h Ths w x qi Coiig i his wa w qi ad h Y Y

30 Vaiaio of Paams Divaio of 5 Fom h vios slid Fiall Nx sbsi hs divaivs io o qaio Rallig ha a solios o homogos qaio ad af aagig ms w obai Y g g

31 Vaiaio of Paams Divaio of 5 Th qaios dd i od o fid h fios a Usig Cam s Rl fo ah m = ad W m is dmia obaid b laig m h olm of W wih. g m = gw m W wh W = W

32 Vaiaio of Paams Divaio 5 of 5 Fom h vios slid m = gw m m = W Iga o obai : m = Ths a aila solio Y is giv b Y = wh is abia. å m= gsw m s ò ds m = W s é ê ë ò gsw m s W s ù dsú û m

33 Examl of Cosid h qaio blow alog wih h giv solios of osodig homogos solios : Th a aila solio of his ODE is giv b I a b show ha g Y = s W m s W s ds ò é ë ê ù û ú m m= å W

34 Examl of Also W W W

35 Examl of Ths a aila solio i igal fom is Y = å m= é ê ë ò gsw m s W s ù dsú û m gs -s - = ò ds + gs s ò ds + - gs s s ò s = ò éë -s -+ - s + --s ù û gsds ds

82A Engineering Mathematics

82A Engineering Mathematics Class Nos 5: Sod Ordr Diffrial Eqaio No Homoos 8A Eiri Mahmais Sod Ordr Liar Diffrial Eqaios Homoos & No Homoos v q Homoos No-homoos q ar iv oios fios o h o irval I Sod Ordr Liar Diffrial Eqaios Homoos