Boyce/DiPrima/Meade 11 th ed, Ch 7.1: Introduction to Systems of First Order Linear Equations

Transcription

1 Boy/DiPrim/Md h d Ch 7.: Iroduio o Sysms of Firs Ordr Lir Equios Elmry Diffril Equios d Boudry Vlu Problms h diio by Willim E. Boy Rihrd C. DiPrim d Doug Md 7 by Joh Wily & Sos I. A sysm of simulous firs ordr ordiry diffril quios hs h grl form whr h k is fuio of. If h F k is lir fuio of h h sysm of quios is sid o b lir ohrwis i is olir. Sysms of highr ordr diffril quios similrly b dfid. F F F

2 Empl Th moio of ri sprig-mss sysm from Sio.7 ws dsribd by h diffril quio u + u 8 + u = This sod ordr quio b ovrd io sysm of firs ordr quios by lig = u d = u'. Thus = or = = = - - 8

3 Nh Ordr ODEs d Lir s Ordr Sysms Th mhod illusrd i h prvious mpl b usd o rsform rbirry h ordr quio io sysm of firs ordr quios firs by dfiig Th y y y y F y y y y y F

4 Soluios of Firs Ordr Sysms A sysm of simulous firs ordr ordiry diffril quios hs h grl form I hs soluio o if hr iss fuios h r diffribl o I d sisfy h sysm of quios ll pois i I. Iiil odiios my lso b prsribd o giv IVP:. F F I : < < b

5 Thorm 7.. Suppos F F d r oiuous i h rgio R of -sp dfid by d l h poi b oid i R. Th i som irvl h + h hr iss uiqu soluio h sisfis h IVP. F F F F /... F /... F /... F / < < b < < b... < < b

6 Lir Sysms If h F k is lir fuio of h h sysm of quios hs h grl form If h of h g k is zro o I h h sysm is homogous ohrwis i is ohomogous. g p p p g p p p g p p p

7 Thorm 7.. Suppos p p p g g r oiuous o irvl wih i I d l prsrib h iiil odiios. Th hr iss uiqu soluio h sisfis h IVP d iss hroughou I. g p p p g p p p g p p p I : < < b

8 Boy/DiPrim/Md h d Ch 7.: Rviw of Mris Elmry Diffril Equios d Boudry Vlu Problms h diio by Willim E. Boy Rihrd C. DiPrim d Doug Md 7 by Joh Wily & Sos I. For horil d ompuiol rsos w rviw rsuls of mri hory i his sio d h. A mri A is m rgulr rry of lms rrgd i m rows d olums dod Som mpls of mris r giv blow: m m m j i A i i i C B A

9 Trspos Th rspos of A = ij is A T = ji. For mpl m m m T m m m A A T T B B A A

10 Cojug Th ojug of A = ij is A = ij. For mpl m m m m m m A A i i i i A A

11 Adjoi Th djoi of A is A T d is dod by A * For mpl m m m m m m * A A * i i i i A A

12 Squr Mris A squr mri A hs h sm umbr of rows d olums. Th is A is. I his s A is sid o hv ordr. For mpl A B A

13 Vors A olum vor is mri. For mpl A row vor is mri. For mpl y No hr h y = T d h i grl if is olum vor h T is row vor.

14 Th Zro Mri Th zro mri is dfid o b = whos dimsios dpd o h o. For mpl

15 Mri Equliy Two mris A = ij d B = b ij r qul if ij = b ij for ll i d j. For mpl B A B A

16 Mri Slr Mulipliio Th produ of mri A = ij d os k is dfid o b ka = k ij. For mpl A 5 5A

17 Mri Addiio d Subrio Th sum of wo m mris A = ij d B = b ij is dfid o b A + B = ij + b ij. For mpl Th diffr of wo m mris A = ij d B = b ij is dfid o b A - B = ij - b ij. For mpl B A B A B A B A

18 Mri Mulipliio Th produ of m mri A = ij d r mri B = b ij is dfid o b h mri C = ij whr Empls o AB dos o ssrily qul BA: k kj ik ij b CD D C BA AB B A

19 Empl : Mri Mulipliio To illusr mri mulipliio d show h i is o ommuiv osidr h followig mris: From h dfiiio of mri mulipliio w hv: B A AB = æ è ç ç ö ø = æ è ç ç ö ø BA = æ è ç ç ö ø = æ è ç ç ö ø ¹ AB

20 Vor Mulipliio Th do produ of wo vors & y is dfid s Th ir produ of wo vors & y is dfid s Empl: k j i T y y k j i T y y y i i i i i i i i i i i T T y y y y

21 Vor Lgh Th lgh of vor is dfid s No hr h w hv usd h f h if = + bi h Empl: / / / k k k k k 6 9 / i i i b bi bi

22 Orhogoliy Two vors & y r orhogol if y =. Empl: y y

23 Idiy Mri Th mulipliiv idiy mri I is mri giv by For y squr mri A i follows h AI = IA = A. Th dimsios of I dpd o h o. For mpl I IB AI

24 Ivrs Mri A squr mri A is osigulr or ivribl if hr iss mri B suh h h AB = BA = I. Ohrwis A is sigulr. Th mri B if i iss is uiqu d is dod by A d is lld h ivrs of A. I urs ou h A iss iff da d A b foud usig row rduio lso lld Gussi limiio o h ugmd mri A I s mpl o slid. Th hr lmry row oprios: Irhg wo rows. Muliply row by ozro slr. Add mulipl of o row o ohr row.

25 Empl : Fidig h Ivrs of Mri of Us row rduio o fid h ivrs of h mri A blow if i iss. Soluio: If possibl us lmry row oprios o rdu A I suh h h lf sid is h idiy mri for h h righ sid will b A. S slid. A A I

26 Empl : Fidig h Ivrs of Mri of Thus 5 / 5 / 5 / / / / / / 7 / 5 / / 5/ / / / 5 / / 5/ / / / 5 / / 5/ 5 5 A I 5 / 5 / 5 / / / / / / 7 / A

27 Mri Fuios Th lms of mri b fuios of rl vribl. I his s w wri Suh mri is oiuous poi or o irvl b if h lm is oiuous hr. Similrly wih diffriio d igrio: m m m m A b ij b ij d d d d d d A A

28 Empl & Diffriio Ruls Empl: My of h ruls from lulus pply i his sig. For mpl: d d d d d d d d d d d d d d d d B A B A AB B A B A C A C CA os mri is whr si os 6 os si d d d A A A

29 Boy/DiPrim/Md h d Ch 7.: Sysms of Lir Equios Lir Idpd Eigvlus Elmry Diffril Equios d Boudry Vlu Problms h diio by Willim E. Boy Rihrd C. DiPrim d Doug Md 7 by Joh Wily & Sos I. A sysm of lir quios i vribls b prssd s mri quio A = b: If b = h sysm is homogous; ohrwis i is ohomogous. b b b b b b

30 Nosigulr Cs If h offii mri A is osigulr h i is ivribl d w solv A = b s follows: A b A A A b I This soluio is hrfor uiqu. Also if b = i follows h h uiqu soluio o A = is = A =. Thus if A is osigulr h h oly soluio o A = is h rivil soluio =. A b A b

31 Empl : Nosigulr Cs of From prvious mpl w kow h h mri A blow is osigulr wih ivrs s giv. Usig h dfiiio of mri mulipliio i follows h h oly soluio of A = is = : / / / / 7 / 5/ / 5/ / A A / / / / 7 / 5/ / 5/ / A

32 Empl : Nosigulr Cs of Now l s solv h ohomogous lir sysm A = b blow usig A : This sysm of quios b wri s A = b whr Th / / / / 7 / 5/ / 5/ / A b 5 7 b A

33 Empl : Nosigulr Cs of Alrivly w ould solv h ohomogous lir sysm A = b blow usig row rduio. To do so form h ugmd mri A b d rdu usig lmry row oprios A b 5 7

34 Sigulr Cs If h offii mri A is sigulr h A - dos o is d ihr soluio o A = b dos o is or hr is mor h o soluio o uiqu. Furhr h homogous sysm A = hs mor h o soluio. Th is i ddiio o h rivil soluio = hr r ifiily my orivil soluios. Th ohomogous s A = b hs o soluio ulss b y = for ll vors y sisfyig A * y = whr A * is h djoi of A. I his s A = b hs soluios ifiily my h of h form = + whr is priulr soluio of A = b d is y soluio of A =.

35 Empl : Sigulr Cs of Solv h ohomogous lir sysm A = b blow usig row rduio. Obsrv h h offiis r rly h sm s i h prvious mpl W will form h ugmd mri A b d us som of h sps i Empl o rsform h mri mor quikly b b b b b b b b b b b b b b b b b b A b b b b

36 Empl : Sigulr Cs of From h prvious slid if hr is o soluio o h sysm of quios Rquirig h ssum for mpl h Th h rdud ugmd mri A b boms: I b show h h sod rm i is soluio of h ohomogous quio d h h firs rm is h mos grl soluio of h homogous quio lig whr α is rbirry b b b b b b b b b 5 b b b b b b b b b

37 Lir Dpd d Idpd A s of vors is lirly dpd if hr iss slrs o ll zro suh h If h oly soluio of is = = = = h is lirly idpd.

38 Empl : Lir Dpd of Drmi whhr h followig vors r lir dpd or lirly idpd. W d o solv or

39 Empl : Lir Dpd of W rdu h ugmd mri A b s bfor. So h vors r lirly dpd: Alrivly w ould show h h followig drmi is zro: b y umbr whr A b d ij if

40 Lir Idpd d Ivribiliy Cosidr h prvious wo mpls: Th firs mri ws kow o b osigulr d is olum vors wr lirly idpd. Th sod mri ws kow o b sigulr d is olum vors wr lirly dpd. This is ru i grl: h olums or rows of A r lirly idpd iff A is osigulr iff A - iss. Also A is osigulr iff da h olums or rows of A r lirly idpd iff da. Furhr if A = BC h dc = dadb. Thus if h olums or rows of A d B r lirly idpd h h olums or rows of C r lso.

41 Lir Dpd & Vor Fuios Now osidr vor fuios whr As bfor is lirly dpd o I if hr iss slrs o ll zro suh h Ohrwis is lirly idpd o I S for mor disussio o his. I k k m k k k I for ll

42 Eigvlus d Eigvors Th q. A = y b viwd s lir rsformio h mps or rsforms io w vor y. Nozro vors h rsform io mulipls of hmslvs r impor i my ppliios. Thus w solv A = or quivlly A I =. This quio hs ozro soluio if w hoos suh h da I =. l l l Suh vlus of r lld igvlus of A d h ozro soluios r lld igvors. l l

43 Empl : Eigvlus of Fid h igvlus d igvors of h mri A. Soluio: Choos suh h da I = s follows. A d d d I A l l

44 Empl : Firs Eigvor of To fid h igvors of h mri A w d o solv A I = for = d =. Eigvor for = : Solv d his implis h. So A I hoos rbirry l l l l

45 Empl : Sod Eigvor of Eigvor for = : Solv d his implis h. So A I hoos rbirry l

46 Normlizd Eigvors From h prvious mpl w s h igvors r drmid up o ozro mulipliiv os. If his os is spifid i som priulr wy h h igvor is sid o b ormlizd. For mpl igvors r somims ormlizd by hoosig h os so h = ½ =.

47 Algbri d Gomri Mulipliiy I fidig h igvlus of mri A w solv da I =. l Si his ivolvs fidig h drmi of mri h problm rdus o fidig roos of h dgr polyomil. Do hs roos or igvlus by. If igvlu is rpd m ims h is lgbri mulipliiy is m. l l l l Eh igvlu hs ls o igvor d igvlu of lgbri mulipliiy m my hv q lirly idpd igvors q m d q is lld h gomri mulipliiy of h igvlu.

48 Eigvors d Lir Idpd l If igvlu hs lgbri mulipliiy h i is sid o b simpl d h gomri mulipliiy is lso. If h igvlu of mri A is simpl h A hs disi igvlus. I b show h h igvors orrspodig o hs igvlus r lirly idpd. If igvlu hs o or mor rpd igvlus h hr my b fwr h lirly idpd igvors si for h rpd igvlu w my hv q < m. This my ld o ompliios i solvig sysms of diffril quios.

49 Empl 5: Eigvlus of 5 Fid h igvlus d igvors of h mri A. Soluio: Choos suh h da I = s follows. A d d I A l l

50 Empl 5: Firs Eigvor of 5 Eigvor for = : Solv A I = s follows. hoos rbirry l l

51 Empl 5: d d rd Eigvors of 5 Eigvor for = : Solv A I = s follows. hoos rbirry whr l l

52 Empl 5: Eigvors of A of 5 Thus hr igvors of A r whr orrspod o h doubl igvlu I b show h r lirly idpd. H A is symmri mri A = A T wih rl igvlus d lirly idpd igvors. A l = -.

53 Empl 5: Eigvors of A 5 of 5 No h w ould hv w hd hos Th h igvors r orhogol si Thus A is symmri mri wih rl igvlus d lirly idpd orhogol igvors.

54 Hrmii Mris A slf-djoi or Hrmii mri sisfis A = A * whr w rll h A * = A T. Thus for Hrmii mri ij = ji. No h if A hs rl ris d is symmri s ls mpl h A is Hrmii. A Hrmii mri A hs h followig propris: All igvlus of A r rl. Thr iss full s of lirly idpd igvors of A. If d r igvors h orrspod o diffr igvlus of A h d r orhogol. Corrspodig o igvlu of lgbri mulipliiy m i is possibl o hoos m muully orhogol igvors d h A hs full s of lirly idpd orhogol igvors.

55 Boy/DiPrim/Md h d Ch 7.: Bsi Thory of Sysms of Firs Ordr Lir Equios Elmry Diffril Equios d Boudry Vlu Problms h diio by Willim E. Boy Rihrd C. DiPrim d Doug Md 7 by Joh Wily & Sos I. Th grl hory of sysm of firs ordr lir quios prllls h of sigl h ordr lir quio. This sysm b wri s ' = P + g whr g p p p g p p p g p p p p p p p p p p p p g g g P g

56 Vor Soluios of ODE Sysm A vor = is soluio of ' = P + g if h ompos of sisfy h sysm of quios o. For ompriso rll h ' = P + g rprss our sysm of quios Assumig P d g oiuous o I suh soluio iss by Thorm 7... g p p p g p p p g p p p f I : < < b

57 Homogous Cs; Vor Fuio Noio As i Chprs d w firs mi h grl homogous quio ' = P. Also h followig oio for h vor fuios k will b usd: k

58 Thorm 7.. If h vor fuios d r soluios of h sysm ' = P h h lir ombiio + is lso soluio for y oss d. No: By rpdly pplyig h rsul of his horm i b s h vry fii lir ombiio k k of soluios k is islf soluio o ' = P.

59 Thorm 7.. If r lirly idpd soluios of h sysm ' = P for h poi i I : < < b h h soluio = f b prssd uiquly i h form If soluios r lirly idpd for h poi i I : < < b h hy r fudml soluios o I d h grl soluio is giv by

60 Th Wroski d Lir Idpd Th proof of Thm 7.. uss h f h if r lirly idpd o I h dx o I whr X Th Wroski of is dfid s W[ ] = dx. I follows h W[ ] o I iff r lirly idpd for h poi i I.

61 Thorm 7.. If r soluios of h sysm ' = P o h I : h < Wroski < b W[ ] is ihr idilly zro o I or ls is vr zro o I. This rsul rlis o Abl s formul for h Wroski dw [ p p p p p W d whr is rbirry os Rfr o Sio. p This rsul bls us o drmi whhr giv s of soluios r fudml soluios by vluig W[ ] y poi i. < < b ] d

62 Thorm 7.. L L b soluios of h sysm ' = P h sisfy h iiil odiios rspivly whr is y poi i. Th r form fudml s of soluios of ' = P. < < b < < b

63 Thorm 7..5 Cosidr h sysm ' = P whr h lm of P is rl-vlud oiuous fuio. If = u + iv is ompl-vlud soluio of Eq. h is rl pr u d is imgiry pr v r lso soluios of his quio.

64 Boy/DiPrim/Md h d Ch 7.5: Homogous Lir Sysms wih Cos Coffiis Elmry Diffril Equios d Boudry Vlu Problms h diio by Willim E. Boy Rihrd C. DiPrim d Doug Md 7 by Joh Wily & Sos I. W osidr hr homogous sysm of firs ordr lir quios wih os rl offiis: This sysm b wri s ' = A whr m A

65 Equilibrium Soluios No h if = h h sysm rdus o Rll h = is h oly quilibrium soluio if. Furhr = is sympoilly sbl soluio if < si ohr soluios pproh = i his s. Also = is usbl soluio if > si ohr soluios dpr from = i his s. For > quilibrium soluios r similrly foud by solvig A =. W ssum da so h = is h oly soluio. Drmiig whhr = is sympoilly sbl or usbl is impor qusio hr s wll.

66 Phs Pl Wh = h h sysm rdus o This s b visulizd i h -pl whih is lld h phs pl. I h phs pl dirio fild b obid by vluig A my pois d ploig h rsulig vors whih will b g o soluio vors. A plo h shows rprsiv soluio rjoris is lld phs porri. Empls of phs pls dirios filds d phs porris will b giv lr i his sio.

67 Solvig Homogous Sysm To osru grl soluio o ' = A ssum soluio of h form = r whr h po r d h os vor r o b drmid. Subsiuig = r io ' = A w obi rξ r Aξ r rξ A riξ Thus o solv h homogous sysm of diffril quios ' = A w mus fid h igvlus d igvors of A. Aξ Thrfor = r is soluio of ' = A providd h r is igvlu d is igvor of h offii mri A.

68 Empl of Fid h grl soluio of h sysm Th mos impor fur of his sysm is h h offii mri is digol mri. Thus by wriig h sysm i slr form w obi ' = ' = - Eh of hs quios ivolvs oly o of h ukow vribls so w solv h wo quios sprly. I his wy w fid h = = - whr d r rbirry oss.

69 Empl of Th by wriig h soluio i vor form w hv = æ è ç - ö ø = æ ç è ö ø + æ ç è - ö ø = æ è ç Now w dfi wo soluios d so h æ ö æ = è ç ø = è ç Th Wroski of hs soluios is ö æ ø + è ç whih is vr zro. Thrfor d form fudml s of soluios. W[ ] = - = - ö - ø ö - ø

70 Empl : Dirio Fild of 9 Cosidr h homogous quio ' = A blow. A dirio fild for his sysm is giv blow. Subsiuig = r i for d rwriig sysm s A ri = w obi r r

71 Empl : Eigvlus of 9 Our soluio hs h form = r whr r d r foud by solvig Rllig h his is igvlu problm w drmi r by solvig da ri = : Thus r = d r =. r r r r r r r r r

72 Empl : Firs Eigvor of 9 Eigvor for r = : Solv by row rduig h ugmd mri: ξ A ri hoos rbirry / / / / / ξ ξ

73 Empl : Sod Eigvor of 9 Eigvor for r = -: Solv by row rduig h ugmd mri: ξ A ri hoos rbirry / / / / / ξ ξ

74 Empl : Grl Soluio 5 of 9 Th orrspodig soluios = r of ' = A r Th Wroski of hs wo soluios is Thus d r fudml soluios d h grl soluio of ' = A is W

75 Empl : Phs Pl for 6 of 9 To visuliz soluio osidr firs = : Now Thus lis log h srigh li = whih is h li hrough origi i dirio of firs igvor If soluio is rjory of pril wih posiio giv by h i is i Q wh > d i Q wh <. I ihr s pril movs wy from origi s irss.

76 Empl : Phs Pl for 7 of 9 N osidr = : Th lis log h srigh li = whih is h li hrough origi i dirio of d igvor If soluio is rjory of pril wih posiio giv by h i is i Q wh > d i Q wh <. I ihr s pril movs owrds origi s irss.

77 Empl : Phs Pl for Grl Soluio 8 of 9 Th grl soluio is = + : As is domi d boms gligibl. Thus for ll soluios sympoilly pproh h li = s. Similrly for ll soluios sympoilly pproh h li = s -. Th origi is sddl poi d is usbl. S grph.

78 Empl : Tim Plos for Grl Soluio 9 of 9 Th grl soluio is = + : As lriv o phs pl plos w grph or s fuio of. A fw plos of r giv blow. No h wh = = - s. Ohrwis = + - grows uboudd s. Grphs of r similrly obid.

79 Empl : Dirio Fild of 9 Cosidr h homogous quio ' = A blow. A dirio fild for his sysm is giv blow. Subsiuig = r i for d rwriig sysm s A ri = w obi -- r - - r æ è ç ç ö ø æ è ç ö ø = æ è ç ö ø

80 Empl : Eigvlus of 9 Our soluio hs h form = r whr r d r foud by solvig æ -- r ö æ ö æ ö ç = è ç - - r ø è ø è ç ø Rllig h his is igvlu problm w drmi r by solvig da ri = : r r r r r 5r r r Thus r = d r =.

81 Empl : Firs Eigvor of 9 Eigvor for r = : Solv by row rduig h ugmd mri: ξ A ri hoos / / / ξ ξ

82 Empl : Sod Eigvor of 9 Eigvor for r = : Solv by row rduig h ugmd mri: ξ A ri hoos ξ ξ

83 Empl : Grl Soluio 5 of 9 Th orrspodig soluios = r of ' = A r Th Wroski of hs wo soluios is Thus d r fudml soluios d h grl soluio of ' = A is 5 W

84 Empl : Phs Pl for 6 of 9 To visuliz soluio osidr firs = : Now Thus lis log h srigh li = ½ whih is h li hrough origi i dirio of firs igvor If soluio is rjory of pril wih posiio giv by h i is i Q wh > d i Q wh <. I ihr s pril movs owrds origi s irss.

85 Empl : Phs Pl for 7 of 9 N osidr = : Th lis log h srigh li = ½ whih is h li hrough origi i dirio of d igvor If soluio is rjory of pril wih posiio giv by h i is i Q wh > d i Q wh <. I ihr s pril movs owrds origi s irss.

86 Empl : Phs Pl for Grl Soluio 8 of 9 Th grl soluio is = + : As is domi d boms gligibl. Thus for ll soluios sympoilly pproh origi log h li = s. Similrly ll soluios r uboudd s -. Th origi is od d is sympoilly sbl.

87 Empl : Tim Plos for Grl Soluio 9 of 9 Th grl soluio is = + : As lriv o phs pl plos w grph or s fuio of. A fw plos of r giv blow. Grphs of r similrly obid.

88 Cs: Rl Eigvlus Sddl Pois d Nods Th prvious wo mpls dmosr h wo mi ss for rl sysm wih rl d diffr igvlus: Boh igvlus hv opposi sigs i whih s origi is sddl poi d is usbl. Boh igvlus hv h sm sig i whih s origi is od d is sympoilly sbl if h igvlus r giv d usbl if h igvlus r posiiv.

89 Eigvlus Eigvors d Fudml Soluios I grl for rl lir sysm ' = A: All igvlus r rl d diffr from h ohr. Som igvlus our i ompl ojug pirs. Som igvlus r rpd. If igvlus r r r rl & diffr h hr r orrspodig lirly idpd igvors. Th ssoid soluios of ' = A r r ξ ξ Usig Wroski i b show h hs soluios r lirly idpd d h form fudml s of soluios. Thus grl soluio is r r ξ ξ r

90 Hrmii Cs: Eigvlus Eigvors & Fudml Soluios If A is Hrmii mri rl d symmri h ll igvlus r r r rl lhough som my rp. I y s hr r orrspodig lirly idpd d orhogol igvors. Th ssoid soluios of ' = A r r ξ ξ d form fudml s of soluios. r

91 Empl : Hrmii Mri of Cosidr h homogous quio ' = A blow. Th igvlus wr foud prviously i Ch 7. d wr: r = r = d r =. Corrspodig igvors: ξ ξ ξ

92 Empl : Grl Soluio of Th fudml soluios r wih grl soluio

93 Empl : Grl Soluio Bhvior of Th grl soluio is = + + : As is domi d bom gligibl. Thus for ll sols bom uboudd s whil for = ll sols s. Th iiil pois h us = r hos h li i pl drmid by d. Thus soluios h sr i his pl pproh origi s.

94 Compl Eigvlus d Fudml Sols If som of h igvlus r r our i ompl ojug pirs bu ohrwis r diffr h hr r sill orrspodig lirly idpd soluios ξ whih form fudml s of soluios. Som my b ompl-vlud bu rl-vlud soluios my b drivd from hm. This siuio will b mid i Ch 7.6. If h offii mri A is ompl h ompl igvlus d o our i ojug pirs bu soluios will sill hv h bov form if h igvlus r disi d hs soluios my b ompl-vlud. ξ r r

95 Rpd Eigvlus d Fudml Sols If som of h igvlus r r r rpd h hr my o b orrspodig lirly idpd soluios of h form r r ξ ξ I ordr o obi fudml s of soluios i my b ssry o sk ddiiol soluios of ohr form. This siuio is logous o h for h ordr lir quio wih os offiis i whih s rpd roo gv ris soluios of h form r r r This s of rpd igvlus is mid i Sio 7.8.

96 Boy/DiPrim/Md h d Ch 7.6: Compl Eigvlus Elmry Diffril Equios d Boudry Vlu Problms h diio by Willim E. Boy Rihrd C. DiPrim d Doug Md 7 by Joh Wily & Sos I. W osidr gi homogous sysm of firs ordr lir quios wih os rl offiis d hus h sysm b wri s ' = A whr A

97 Cojug Eigvlus d Eigvors W kow h = r is soluio of ' = A providd r is igvlu d is igvor of A. Th igvlus r r r h roos of da ri = d h orrspodig igvors sisfy A ri =. If A is rl h h offiis i h polyomil quio da ri = r rl d h y ompl igvlus mus our i ojug pirs. Thus if r = l + im is igvlu h so is r = l - im. Th orrspodig igvors r ojugs lso. To s his rll A d I hv rl ris d h A riξ A riξ A Iξ r

98 Cojug Soluios I follows from h prvious slid h h soluios orrspodig o hs igvlus d igvors r ojugs ojugs s wll si ξ ξ r r ξ ξ r r

99 Empl : Dirio Fild of 7 Cosidr h homogous quio ' = A blow. / / A dirio fild for his sysm is giv blow. Subsiuig = r i for d rwriig sysm s æ ç ç ç è ç A ri = w obi - - r r ö æ ç è ø ö æ = ø è ç ö ø

100 Empl : Compl Eigvlus of 7 W drmi r by solvig da ri =. Now Thus Thrfor h igvlus r r = / + i d r = / i. 5 / / / r r r r r i i r 5/

101 Empl : Firs Eigvor of 7 Eigvor for r = / + i: Solv by row rduig h ugmd mri: Thus / / i i i i r r r ξ I A i i i i i hoos ξ ξ i ξ

102 Empl : Sod Eigvor of 7 Eigvor for r = / i: Solv by row rduig h ugmd mri: Thus / / i i i i r r r ξ I A i i i i i hoos ξ ξ i ξ

103 Empl : Grl Soluio 5 of 7 Th orrspodig soluios = r of ' = A r Th Wroski of hs wo soluios is Thus u d v r rl-vlud fudml soluios of ' = A wih grl soluio = u + v. os si os si si os si os / / / / v u os si si os / / / / W

104 Empl : Phs Pl 6 of 7 Giv blow is h phs pl plo for soluios wih / / os si si os Eh soluio rjory pprohs origi log spirl ph s si oordis r produs of dyig poil d si or osi fors. Th grph of u psss hrough si u =. Similrly h grph of v psss hrough. Th origi is spirl poi d is sympoilly sbl. / /

105 Empl : Tim Plos 7 of 7 Th grl soluio is = u + v: As lriv o phs pl plos w grph or s fuio of. A fw plos of r giv blow h o dyig osillio s. os si si os / / / /

106 Grl Soluio To summriz suppos r = r = d h r r r ll rl d disi igvlus of A. L h orrspodig igvors b Th h grl soluio of ' = A is whr r r ξ ξ v u i i ξ ξ ξ b ξ b ξ os si si os b v b u l + im l - im

107 Rl-Vlud Soluios Thus for ompl ojug igvlus r d r h orrspodig soluios d r ojugs lso. To obi rl-vlud soluios us rl d imgiry prs of ihr or. To s his l = + i b. Th whr ξ i ib os isi os bsi i si bos u i v os bsi v si bos u r rl vlud soluios of ' = A d b show o b lirly idpd.

108 Spirl Pois Crs Eigvlus d Trjoris I prvious mpl grl soluio ws / / os si si os Th origi ws spirl poi d ws sympoilly sbl. If rl pr of ompl igvlus is posiiv h rjoris spirl wy uboudd from origi d h origi would b usbl spirl poi. If rl pr of ompl igvlus is zro h rjoris irl origi ihr pprohig or dprig. Th origi is lld r d is sbl bu o sympoilly sbl. Trjoris priodi i im. Th dirio of rjory moio dpds o ris i A. / /

109 Empl : Sod Ordr Sysm wih Prmr of Th sysm ' = A blow ois prmr. Subsiuig = r i for d rwriig sysm s A ri = w obi N solv for r i rms of : - r - -r æ è ç ö ø æ è ç ö ø = æ è ç ö ø 6 r r r r r r r

110 Empl : r Eigvlu Alysis of 6 Th igvlus r giv by h qudri formul bov. For < boh igvlus r rl d giv d h origi is sympoilly sbl od. For > boh igvlus r rl d posiiv d h h origi is usbl od. For < < igvlus r ompl wih giv rl pr d h origi is sympoilly sbl spirl poi. For < < igvlus r ompl wih posiiv rl pr d h origi is usbl spirl poi. For = igvlus r purly imgiry origi is r. Trjoris losd urvs bou origi & priodi. For = ± igvlus rl & qul origi is od Ch 7.8

111 Sod Ordr Soluio Bhvior d Eigvlus: Thr Mi Css For sod ordr sysms h hr mi ss r: Eigvlus r rl d hv opposi sigs; = is sddl poi. Eigvlus r rl disi d hv sm sig; = is od. Eigvlus r ompl wih ozro rl pr; = spirl poi. Ohr possibiliis is d our s rsiios bw wo of h ss lisd bov: A zro igvlu ours durig rsiio bw sddl poi d od. Rl d qul igvlus our durig rsiio bw ods d spirl pois. Purly imgiry igvlus our durig rsiio bw sympoilly sbl d usbl spirl pois. b r b

112 Empl : Mulipl Sprig-Mss Sysm of 6 Th quios for h sysm of wo msss d hr sprigs disussd i Sio 7. ssumig o rl fors b prssd s: Giv h quios bom ' d ' whr ' d ' or d y y y y y k k y k y m y k y k k m y k k k d d m k k k d d m 5/ d 9/ k k k m m / ' d / ' ' ' y y y y y y y y y y

113 y ' y y' y y' y / y d y' / y Empl : Mulipl Sprig-Mss Sysm of 6 Wriig h sysm of quios i mri form: y Assumig soluio of h form y = r whr r mus b igvlu of h mri A d is h orrspodig igvor h hrrisi polyomil of A is r 5r r r yildig h igvlus: r i r i r i d r i

114 Empl : Mulipl Sprig-Mss Sysm of 6 For h igvlus h orrspodig igvors r Th produs yild h ompl-vlud soluios: Ay y y' / / i r i r i r i r d i i i i i i i i 8 6 d 8 6 i i d ξ ξ 8os 6os si si 8si 6si os os si os 8 6 os os si si si si os os si os i i i i i i i i i i i i v u v u

115 Empl : Mulipl Sprig-Mss Sysm of 6 Afr vlidig h u v u v r lirly idpd h grl soluio of h sysm of quios b wri s æ ç y = ç è ç os os -si -si ö æ si ç si + ç os ø è ç os ö æ os ç - os + ç -6si ø è ç 8si whr r rbirry oss. y ' y y' y y' y / y d y' / y ö æ si ç -si + ç 6os ø è ç -8os Eh soluio will b priodi wih priod π so h rjory is losd urv. Th firs wo rms of h soluio dsrib moios wih frquy d priod π whil h sod wo rms dsrib moios wih frquy d priod π. Th moios of h wo msss will b diffr rliv o o ohr for soluios ivolvig oly h firs wo rms or h sod wo rms. ö ø y

116 y d y rprs h moio of h msss d y y' y ' y Empl : Mulipl Sprig-Mss Sysm 5 of 6 To obi h fudml mod of vibrio wih frquy ours wh y y d y y y To obi h fudml mod of vibrio wih frquy ours wh y y d y y Plos of y d y d prmri plos y y r show for sld soluio wih frquy

117 y d y rprs h moio of h msss d y y' y ' y Empl : Mulipl Sprig-Mss Sysm 6 of 6 Plos of y d y d prmri plos y y r show for sld soluio wih frquy

118 Boy/DiPrim/Md h d Ch 7.7: Fudml Mris Elmry Diffril Equios d Boudry Vlu Problms h diio by Willim E. Boy Rihrd C. DiPrim d Doug Md 7 by Joh Wily & Sos I. Suppos h form fudml s of soluios for ' = P o < < b. Th mri Ψ whos olums r is fudml mri for h sysm ' = P. This mri is osigulr si is olums r lirly idpd d h d Y. No lso h si r soluios of ' = P Y sisfis h mri diffril quio Y' = P Y.

119 Empl : Cosidr h homogous quio ' = A blow. I Empl of Chpr 7.5 w foud h followig fudml soluios for his sysm: Thus fudml mri for his sysm is Ψ

120 Fudml Mris d Grl Soluio Th grl soluio of ' = P b prssd = whr is os vor wih ompos : Ψ Y

121 Fudml Mri & Iiil Vlu Problm Cosidr iiil vlu problm ' = P = whr < d < b is giv iiil vor. Now h soluio hs h form = Y h w hoos so s o sisfy =. Rllig Y is osigulr i follows h Ψ Ψ Thus our soluio = Y b prssd s Ψ Ψ

122 Rll: Thorm 7.. L L b soluios of ' = P o I: h sisfy h iiil odiios Th r fudml soluios of ' = P. < < b

123 Fudml Mri & Thorm 7.. Suppos form h fudml soluios giv by Thm 7... Do h orrspodig fudml mri by. Th olums of r d h Thus = I d h h grl soluio o h orrspodig iiil vlu problm is I follows h for y fudml mri I Φ Φ Φ Ψ Ψ Φ Φ Ψ Ψ F F F F

124 Th Fudml Mri F. d Vryig Iiil Codiios Thus wh usig h fudml mri h grl soluio o IVP is Φ Φ Φ This rprsio is usful if sm sysm is o b solvd for my diffr iiil odiios suh s physil sysm h b srd from my diffr iiil ss. Also o F hs b drmid h soluio o h s of iiil odiios b foud by mri mulipliio s idid by h quio bov. F Thus rprss lir rsformio of h iiil odiios io h soluio im. F

125 Empl : Fid for Sysm of 5 Fid suh h = I for h sysm blow. Soluio: Firs w mus obi d suh h W kow from prvious rsuls h h grl soluio is Evry soluio b prssd i rms of h grl soluio d w us his f o fid d. F F F

126 Empl : Us Grl Soluio of 5 Thus o fid prss i rms of h grl soluio d h fid h offiis d. To do so us h iiil odiios o obi or quivlly

127 Empl : Solv for of 5 To fid w hrfor solv by row rduig h ugmd mri: Thus / / / / /

128 Empl : Solv for of 5 To fid w similrly solv by row rduig h ugmd mri: Thus / / / / /

129 Empl : Obi 5 of 5 Th olums of r giv by d d hus from h prvious slid w hv No is mor omplid h foud i E. Howvr i is ow muh sir o drmi h soluio o y s of iiil odiios. Φ Ψ F F Y F

130 Mri Epoil Fuios Cosidr h followig wo ss: Th soluio o ' = = is = whr =. Th soluio o ' = A = is = whr = I. Comprig h form d soluio for boh of hs ss w migh p o hv poil hrr. Idd i b show h = A whr A F A I is wll dfid mri fuio h hs ll h usul propris of poil fuio. S for dils. Thus h soluio o ' = A = is = A. F A!! F F

131 Coupld Sysms of Equios Rll h our os offii homogous sysm wri s ' = A wih is sysm of oupld quios h mus b solvd simulously o fid ll h ukow vribls. A

132 Uoupld Sysms & Digol Mris I ors if h quio hd oly o vribl solvd for idpdly of ohr quios h sk would b sir. I his s our sysm would hv h form or ' = D whr D is digol mri: d d d d d d D

133 Uouplig: Trsform Mri T I ordr o plor rsformig our giv sysm ' = A of oupld quios io uoupld sysm ' = D whr D is digol mri w will us h igvors of A. Suppos A is wih lirly idpd igvors d orrspodig igvlus. Dfi mris T d D usig h igvlus & igvors of A: No h T is osigulr d h T - iss. D T... l...l

134 Uouplig: T - AT = D Rll hr h dfiiios of A T d D: Th h olums of AT r A A d h I follows h T AT = D. D T A AT TD

135 Similriy Trsformios Thus if h igvlus d igvors of A r kow h A b rsformd io digol mri D wih T AT = D This pross is kow s similriy rsformio d A is sid o b similr o D. Alrivly w ould sy h A is digolizbl. D T A

136 Similriy Trsformios: Hrmii Cs Rll: Our similriy rsformio of A hs h form T AT = D whr D is digol d olums of T r igvors of A. If A is Hrmii h A hs lirly idpd orhogol igvors ormlizd so h... i i = i k = for i = d for i k. Wih his slio of igvors i b show h T = T *. I his s w wri our similriy rsform s T * AT = D

137 Nodigolizbl A Filly if A is wih fwr h lirly idpd igvors h hr is o mri T suh h T AT = D. I his s A is o similr o digol mri d A is o digolizbl. D T A

138 Empl : Fid Trsformio Mri T of For h mri A blow fid h similriy rsformio mri T d show h A b digolizd. W lrdy kow h h igvlus r = = wih orrspodig igvors Thus A ξ ξ D T l l

139 Empl : Similriy Trsformio of To fid T ugm h idiy o T d row rdu: Th Thus A is similr o D d h A is digolizbl. / / / / / / / / / / T D AT T 6 / / / / / / / /

140 Fudml Mris for Similr Sysms of Rll our origil sysm of diffril quios ' = A. If A is wih lirly idpd igvors h A is digolizbl. Th igvors form h olums of h osigulr rsform mri T d h igvlus r h orrspodig ozro ris i h digol mri D. Suppos sisfis ' = A l y b h vor suh h = Ty. Th is l y b dfid by y = T. Si ' = A d T is os mri w hv Ty' = ATy d h y' = T ATy = Dy. Thrfor y sisfis y' = Dy h sysm similr o ' = A. Boh of hs sysms hv fudml mris whih w mi.

141 Fudml Mri for Digol Sysm of A fudml mri for y' = Dy is giv by Q = D. Rllig h dfiiio of D w hv!!!! D Q

142 Fudml Mri for Origil Sysm of To obi fudml mri for ' = A rll h h olums of osis of fudml soluios sisfyig ' = A. W lso kow = Ty d h i follows h Th olums of giv h pd fudml soluios of ' = A. TQ Ψ Y Y Y

143 Empl : Fudml Mris for Similr Sysms W ow us h lysis d rsuls of h ls fw slids. Applyig h rsformio = Ty o ' = A blow his sysm boms y' = T ATy = Dy: A fudml mri for y' = Dy is giv by Q = D : Thus fudml mri for ' = A is y y Q TQ Ψ Y

144 Boy/DiPrim/Md h d Ch 7.8: Rpd Eigvlus Elmry Diffril Equios d Boudry Vlu Problms h diio by Willim E. Boy Rihrd C. DiPrim d Doug Md 7 by Joh Wily & Sos I. W osidr gi homogous sysm of firs ordr lir quios wih os rl offiis ' = A. If h igvlus r r of A r rl d diffr h hr r lirly idpd igvors... d lirly idpd soluios of h form r ξ ξ If som of h igvlus r r r rpd h hr my o b orrspodig lirly idpd soluios of h bov form. I his s w will sk ddiiol soluios h r produs of polyomils d poil fuios. r

145 Empl : Eigvlus of W d o fid h igvors for h mri: Th igvlus r d igvors sisfy h quio A ri = or To drmi r solv da ri = : Thus r = d r =. r r r r r r r r r A = - æ è ç ö ø

146 Empl : Eigvors of To fid h igvors w solv by row rduig h ugmd mri: Thus hr is oly o lirly idpd igvor for h rpd igvlu r =. ξ A ri hoos ξ ξ

147 Empl : Dirio Fild of Cosidr h homogous quio ' = A blow. A dirio fild for his sysm is giv blow. Subsiuig = r i for whr r is A s igvlu d is is orrspodig igvor h prvious mpl showd h is of oly o igvlu r = wih o igvor: ξ

148 Empl : Firs Soluio; d Sod Soluio Firs Amp of Th orrspodig soluio = r of ' = A is Si hr is o sod soluio of h form = r w d o ry diffr form. Bsd o mhods for sod ordr lir quios i Ch.5 w firs ry =. Subsiuig = io ' = A w obi or ξ ξ ξ Aξ ξ Aξ

149 Empl : Sod Soluio Sod Amp of From h prvious slid w hv ξ ξ Aξ I ordr for his quio o b sisfid for ll i is ssry for h offiis of d o boh b zro. From h rm w s h = d h hr is o ozro soluio of h form =. Si d ppr i h bov quio w osidr soluio of h form ξ η

150 Empl : Sod Soluio d is Dfiig Mri Equios of Subsiuig = + io ' = A w obi or ξ ξ η A ξ η Equig offiis yilds A = d A = + or A I ξ d A I η Th firs quio is sisfid if is igvor of A orrspodig o h igvlu r =. Thus ξ η Aξ ξ Aη ξ ξ h h h

151 Empl : Solvig for Sod Soluio 5 of Rll h Thus o solv A I = for w row rdu h orrspodig ugmd mri: k η η ξ A h h

152 Empl : Sod Soluio 6 of Our sod soluio = + is ow Rllig h h firs soluio ws w s h our sod soluio is simply si h ls rm of hird rm of is mulipl of. k h

153 Empl : Grl Soluio 7 of Th wo soluios of ' = A r Th Wroski of hs wo soluios is Thus d r fudml soluios d h grl soluio of ' = A is W

154 Empl : Phs Pl 8 of Th grl soluio is Thus is uboudd s d s. Furhr i b show h s sympoi o h li = drmid by h firs igvor. Similrly s is sympoi o li prlll o =. - -

155 Empl : Phs Pl 9 of Th origi is impropr od d is usbl. S grph. Th pr of rjoris is ypil for wo rpd igvlus wih oly o igvor. If h igvlus r giv h h rjoris r similr bu r rvrsd i h iwrd dirio. I his s h origi is sympoilly sbl impropr od.

156 Empl : Tim Plos for Grl Soluio of Tim plos for r giv blow whr w o h h grl soluio b wri s follows.

157 Grl Cs for Doubl Eigvlus Suppos h sysm ' = A hs doubl igvlu r = d sigl orrspodig igvor. Th firs soluio is = r whr sisfis A r I =. As i Empl h sod soluio hs h form ξ η whr is s bov d h sisfis A - rih =. r r Ev hough da I = i b show h A - rih = lwys b solvd for h. Th vor h is lld grlizd igvor.

158 Empl Esio: Fudml Mri of Rll h fudml mri for ' = A hs lirly idpd soluio for is olums. I Empl our sysm ' = A ws d h wo soluios w foud wr Thus h orrspodig fudml mri is Ψ = - æ è ç ö ø = - æ è ç ö ø + - æ è ç ö ø Y

159 Empl Esio: Fudml Mri of Th fudml mri h sisfis = I b foud usig whr whr is foud s follows: Thus Ψ Ψ Φ F F = YY - F Y

160 Jord Forms If A is wih lirly idpd igvors h A b digolizd usig similriy rsform T AT = D. Th rsform mri T osisd of igvors of A d h digol ris of D osisd of h igvlus of A. I h s of rpd igvlus d fwr h lirly idpd igvors A b rsformd io rly digol mri J lld h Jord form of A wih T AT = J.

161 Empl Esio: Trsform Mri of I Empl our sysm ' = A ws wih igvlus r = d r = d igvors Choosig k = h rsform mri T formd from h wo igvors d is k η ξ T h

162 Empl Esio: Jord Form of Th Jord form J of A is dfid by T AT = J. Now d h No h h igvlus of A r = d r = r o h mi digol of J d h hr is dirly bov h sod igvlu. This pr is ypil of Jord forms. T T T AT J

163 Boy/DiPrim/Md h d Ch 7.9: Nohomogous Lir Sysms Elmry Diffril Equios d Boudry Vlu Problms h diio by Willim E. Boy Rihrd C. DiPrim d Doug Md 7 by Joh Wily & Sos I. Th grl hory of ohomogous sysm of quios prllls h of sigl h ordr lir quio. This sysm b wri s ' = P + g whr g p p p g p p p g p p p p p p p p p p p p g g g P g

164 Grl Soluio Th grl soluio of ' = P + g o I: < < b hs h form v whr is h grl soluio of h homogous sysm ' = P d v is priulr soluio of h ohomogous sysm ' = P + g.

165 Digolizio Suppos ' = A + g whr A is digolizbl os mri. L T b h osigulr rsform mri whos olums r h igvors of A d D h digol mri whos digol ris r h orrspodig igvlus of A. Suppos sisfis ' = A l y b dfid by = Ty. Subsiuig = Ty io ' = A w obi or or Ty' = ATy + g. y' = T ATy + T g y' = Dy + h whr h = T g. No h if w solv digol sysm y' = Dy + h for y h = Ty is soluio o h origil sysm.

166 Solvig Digol Sysm Now y' = Dy + h is digol sysm of h form whr r r r h igvlus of A. Thus y' = Dy + h is uoupld sysm of lir firs ordr quios i h ukows y k whih b isold d solvd sprly usig mhods of Sio.: h h h y y y r r r y y y h y r y y y h y y r y y h y y y r y k h y r y k k k k ds s h y r k k s r r k k k k

167 Solvig Origil Sysm Th soluio y o y' = Dy + h hs ompos y k r k rk s rk h s ds k For his soluio vor y h soluio o h origil sysm ' = A + g is h = Ty. k Rll h T is h osigulr rsform mri whos olums r h igvors of A. k Thus wh muliplid by T h sod rm o righ sid of y k produs grl soluio of homogous quio whil h igrl rm of y k produs priulr soluio of ohomogous sysm.

168 Empl : Grl Soluio of Homogous Cs of 5 Cosidr h ohomogous sysm ' = A + g blow. = æ è ç - - ö ø + æ - ç è No: A is Hrmii mri si i is rl d symmri. Th igvlus of A r r = - d r = - wih orrspodig igvors ξ ξ Th grl soluio of h homogous sysm is h æ = è ç - ö æ ø - + è ç ö ø ö ø = A + g -

169 Empl : Trsformio Mri of 5 Cosidr h rsformio mri T of igvors. Usig Sio 7.7 omm d A Hrmii w hv T = T * = T T providd w ormliz d so h = d =. Thus ormliz s follows: Th for his hoi of igvors ξ ξ T T

170 Empl : Digol Sysm d is Soluio of 5 Udr h rsformio = Ty w obi h digol sysm y' = Dy + T g: Th usig mhods of Sio. y y y y y y y y y y y y 9

171 Empl : Trsform Bk o Origil Sysm of 5 W us h rsformio = Ty o obi h soluio o h origil sysm ' = A + g: 5 6 k k k k k k k k y y

172 Empl : Soluio of Origil Sysm 5 of 5 Simplifyig furhr h soluio b wri s No h h firs wo rms o righ sid form h grl soluio o homogous sysm whil h rmiig rms r priulr soluio o ohomogous sysm. 5 5 k k k k k k

173 Nodigol Cs If A o b digolizd rpd igvlus d shorg of igvors h i b rsformd o is Jord form J whih is rly digol. I his s h diffril quios r o olly uoupld bus som rows of J hv wo ozro ris: igvlu i digol posiio d i dj posiio o h righ of digol posiio. Howvr h quios for y y sill b solvd osuivly srig wih y. Th h soluio o origil sysm b foud usig = Ty.

174 Udrmid Coffiis A sod wy of solvig ' = P + g is h mhod of udrmid offiis. Assum P is os mri d h h ompos of g r polyomil poil or siusoidl fuios or sums or produs of hs. Th produr for hoosig h form of soluio is usully dirly logous o h giv i Sio.6. Th mi diffr riss wh g hs h form u l whr l is simpl igvlu of P. I his s g mhs soluio form of homogous sysm ' = P d s rsul i is ssry o k ohomogous soluio o b of h form l + b l. This form diffrs from h Sio.6 log l.

175 Empl : Udrmid Coffiis of 5 Cosidr gi h ohomogous sysm ' = A + g: = æ è ç - - ö ø + æ - ç è ö ø = æ - è ç - Assum priulr soluio of h form v b ö ø + æ è ç d ö æ ø - + è ç whr h vor offis b d r o b drmid. Si r = - is igvlu of A i is ssry o ilud boh - d b - s miod o h prvious slid. ö ø

176 Empl : Mri Equios for Coffiis of 5 Subsiuig i for i our ohomogous sysm ' = A + g = æ è ç w obi v - - b ö ø + æ è ç d ö æ ø - + è ç Equig offiis w olud h æ A = - Ab = - b - è ç ö ø A = - æ è ç ö ø b - + = A - + Ab - + A + Ad + æ è ç ö ø Ad = ö æ ø - + è ç ö ø

177 Empl : Solvig Mri Equio for of 5 Our mri quios for h offiis r: A Ab b A Ad From h firs quio w s h is igvor of A orrspodig o igvlu r = - d h hs h form W will s o h slid h = d h

178 Empl : Solvig Mri Equio for b of 5 Our mri quios for h offiis r: Subsiuig = io h sod quio Thus = d solvig for b w obi Ad A b Ab A b b b b b b b b Ab hoos b b k k T

179 Empl : Priulr Soluio 5 of 5 Our mri quios for h offiis r: A Ab b A Ad Solvig hird quio for d h fourh quio for d i is srighforwrd o obi T = d T = / 5/. Thus our priulr soluio of ' = A + g is v = æ è ç ö æ ø - - è ç ö æ ø - + è ç Comprig his o h rsul obid i Empl w s h boh priulr soluios would b h sm if w hd hos k = ½ for b o prvious slid isd of k =. ö ø - æ è ç 5 ö ø

180 Vriio of Prmrs: Prlimiris A mor grl wy of solvig ' = P + g is h mhod of vriio of prmrs. < < b Assum P d g r oiuous o d l Y b fudml mri for h homogous sysm. Rll h h olums of Y r lirly idpd soluios of ' = P d h Y is ivribl o h irvl < < b d lso Y' = PY. N rll h h soluio of h homogous sysm b prssd s = Y. Alogous o Sio.7 ssum h priulr soluio of h ohomogous sysm hs h form = Yu whr u is vor o b foud.

181 Vriio of Prmrs: Soluio W ssum priulr soluio of h form = Yu. Subsiuig his io ' = P + g w obi Y'u + Yu' = P Yu + g Si Y' = P Y h bov quio simplifis o u' = Y g Thus u Ψ g d whr h vor is rbirry os of igrio. Th grl soluio o ' = P + g is hrfor Ψ Ψ Ψ s g s ds rbirry

182 Vriio of Prmrs: Iiil Vlu Problm For iiil vlu problm ' = P + g = h grl soluio o ' = P + g is Ψ Ψ Ψ Alrivly rll h h fudml mri sisfis = I d h h grl soluio is F Φ Φ I pri i my b sir o row rdu mris d solv ssry quios h o ompu Y d subsiu io quios. S mpl. Ψ Ψ s g s ds s g s ds F

183 Empl : Vriio of Prmrs of Cosidr gi h ohomogous sysm ' = A + g: W hv prviously foud grl soluio o homogous s wih orrspodig fudml mri: Usig vriio of prmrs mhod our soluio is giv by = u whr u sisfis u' = g or = - - æ è ç ö ø + - æ è ç ö ø = - - æ è ç ö ø + æ è ç ö ø - + æ è ç ö ø Ψ u u Y Y

184 Empl : Solvig for u of Solvig u' = g by row rduio I follows h / / / / / / / u u u = u u æ è ç ö ø = / - / + / / - / + æ è ç ç ö ø Y

185 Empl : Solvig for of Now = u d h w muliply o obi fr ollig rms d simplifyig No h his is h sm soluio s i Empl. / / 6 / / / = - æ è ç ö ø - + æ è ç ö ø - + æ è ç ö ø æ è ç ö ø - + æ è ç ö ø - 5 æ è ç ö ø Y

186 Lpl Trsforms Th Lpl rsform b usd o solv sysms of quios. Hr h rsform of vor is h vor of ompo rsforms dod by Xs: Th by dig Thorm 6.. w obi s L L L L X X s s L

187 Empl : Lpl Trsform of 5 Cosidr gi h ohomogous sysm ' = A + g: Tkig h Lpl rsform of h rm w obi whr Gs is h rsform of g d is giv by Gs = s + s æ è ç ç ç ç ö ø s s s s G AX X

188 Empl : Trsfr Mri of 5 Our rsformd quio is sx s AX s G s If w k = h h bov quio boms sx s AX s G s or si A X s G s Solvig for Xs w obi X s si A G s Th mri si A is lld h rsfr mri.

189 Empl : Fidig Trsfr Mri of 5 Th Solvig for si A w obi s s s A I A s s s s s A I

190 Empl : Trsfr Mri of 5 N Xs = si A Gs d h or s s s s s s s X s s s s s s s s s s s s s X

191 Empl : Trsfr Mri 5 of 5 Thus To solv for = L - {Xs} us pril frio psios of boh ompos of Xs d h Tbl 6.. o obi: Si w ssumd = his soluio diffrs slighly from h prvious priulr soluios. s s s s s s s s s s s s s X = - - æ è ç ö ø - + æ è ç ö ø - + æ è ç ö ø - + æ è ç ö ø - 5 æ è ç ö ø

192 Summry of Th mhod of udrmid offiis rquirs o igrio bu is limid i sop d my ivolv svrl ss of lgbri quios. Digolizio rquirs fidig ivrs of rsformio mri d solvig uoupld firs ordr lir quios. Wh offii mri is Hrmii h ivrs of rsformio mri b foud wihou lulio whih is vry hlpful for lrg sysms. Th Lpl rsform mhod ivolvs mri ivrsio mri mulipliio d ivrs rsforms. This mhod is priulrly usful for problms wih disoiuous or impulsiv forig fuios.

193 Summry of Vriio of prmrs is h mos grl mhod bu i ivolvs solvig lir lgbri quios wih vribl offiis igrio d mri mulipliio d h my b h mos ompuiolly omplid mhod. For my smll sysms wih os offiis ll of hs mhods work wll d hr my b lil rso o sl o ovr ohr.