CH04 - _|- .... . - . .'_. . '_'- _ ....

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Unformatted text preview: _|- .... . - . .'_. . '_'- _ . '_"-."'..'.'I' ' 'I .'.'...'. 'a-:- I. -. -'r-:'.".' ..I«. _.'.'I.- L'- '|i' I . .'I. ._ '. --.i-m ._ II. _ .: .._:_ i .l. - - __._E hl'glffif‘ WEE .= : I1 _-'.' _p .. I. 1. _I- I I 'I . #1.. .1 ' 4—1 General ;. men. ereer egeefien ie an equatien Hi the fem: - 'df'y- ' .. 'd""‘.‘y d1: y e... -._ --. ' _. .-—‘— Pi, '=U }. 1} _ a Ppfllflfl +318} mud + 'r n 10') {if +. {D} (I t _ We-eeeumithet Meflemfu. , Pfleend G are eenfinueue real-vetued funeliens u- e: r _{lfl,.end the: P13 is Hawker-e zero in llfie interval. Then. ' by Raft), we ehtflin fl -1 d1: . y .- ' :1:+---+p.+..:{n4*+pnu)y=3“! t3} d: linear differential eperater' L eferder fir-defined by Eq. (2) is similar be {he Seeeed ercler' .efrefe'ter-intradueed in: Chapter 3. The mathematical fhenry emailed web Eel! ._(g)._i}5 gig-511131533 ganglgggue tn mat. fer the second order lmeer equallflfl‘. fer the ffiflfigfi-w§ state the results fer the nth erder preblem. The preefs ef meet ef aim; gimflar'tatheee fee-the sewed urder equetien and are usually left as: exercises; ' ' ' ' ' 219 Theareai 4.1.1 "_,.._i. ...- .- 3"! Chapter 4. Higher Order Lineaer at I. _-—-_-—'—-—' - I _ _ _._._—--I-I— . ! Li.- _ ._. 'l .1 '1! T- __ I h derivative et_v with respect te r, it will, Eta-may.“- Each “f “135'” integral-mm : . tat.” . .'__ Since Eq. {2} invelves the at “m m 501% EL]. (3}. htain a unique seltt't'ici Hfi-ncfi' we can espect that. 10 U . I1...- require it integratt H initial cenditiens, arbitrary crmstant. necessary te specify 1-" I r r ,1. "_ H = 3 I! 11111} I Ft]. )3 “Iii :: ."IH‘ ‘ ' ' " -" If”) " u .- + .. . d i -.' vm'” is any set ef ream-{[3 Interval I an jut-Jun . i i: u P _,____fl_qt1~'f_- a selutien et'the initial value preblem (2)i.(3)2afla};fl istcnce and uniqueness theme-«1%); . “Elli: n-‘here a. may he any paint in the + there dees crust that it is unique are guaranteed by the lellewmg ea 1' ’11. .zi—t which is similar teTheerem .1. lithe functiens pi. pg. . . . .pa- flfld it } f i ' " ' 1 ' 1 I = f 0 there eststs esactl}. em. selutten _t ‘ ‘ _ . isties the initial cenditiens [3]. This selutten exists threugheut the tnterv Sttl the it ill net eit e a preef el‘ this theerem here. Hetvever. if the ceefficientspl ‘ _ i I 1pm a hen we can eenstruct the selutien et‘ the initial value preblem (EL-(333 see Set‘tiens 4.3 threugh 4.4. Even theugh we may net knee. that it is unique witheut the use flfmeflrflfii: an be [HUNG in [ace {Sectien 3.32) fll‘ Ceddingtfln. are cttttstnttts. I much as in thinner .1: attitliitlfi in this case. we de -i {.1 it. Itltltti- L'ii- lilt.‘ lilc'ttl'c‘ll'l L" ti_'ii;tlr_ttct M iii-.- ii’lTltifl-“r‘iti-lili- [gnu-tiara. .t‘ts in the cerrespending secend erder preblem, we first ' "' it-‘ts-I tier-its etitlnliun '.'[".L"‘ -". -= -—.w . -..'_ . . - "i—ferr—Iiri_i'r + [Quittv : 0‘ I I ._. . "I" I r' lilI I .t . are selutiens ef Ed. {-4}. then it ffliiDWS by direct cent—s iii l!l:"-. 1 _ "II l‘1.!lilt".-1'I litii the I.tle't'tt -_I_=ttli‘tlrt;'llltli'l 1' : iii-“1'4" {Lift-H "i' ' ' ' + EFLFI'IIIL it here r. .. . are ut'litli'arj. eenstants. is else a selutien et' Eq. (4). It is then natural [e ask ahether et'er} httiilllLtll el’ Ed- {4] can he expressed as a linear cembinatien- efrril. _ . . ._r.,. This will he true If. regardless el‘ the initial cenditiens {3) that are FIE)“ serihed. it is pessihle te cheese the censtants e1. . . . .cH se that the linear cernhinatien. til satisfies the initial eunditiens. That is. I'er any cheice ef the peint a] in i. and hit any eheiee el' ya. u... _..‘v.'1”_". vre must be able te determine e1. . . . .cn se that the equanens s'IJ't Uni + ‘ ' + Cnir'n'ifit} I J'tt ELY} Hui + ‘ ' ‘ “i” i‘n.‘t’i.”lt.i = Fl!) (6') tn—lt III—1| Uni = fun [El-1"}: [ii-"II He} + -- . + t‘flyfi are satisfied. Equatiens [6} can be selved uniquely [er the eenstants minute Prevtded that the determinant ef ceefficients is net acre. On the ether hand. 1i- ‘t " W;T 1" ' --- .- _|-“H‘!.- I‘ rla'l'i I Flirts" -' H-.-I1.I.'._"r1'1.-‘-_ .'. _ ___ rit- 'Ir ‘I-_ . . ' I id."_-F”Ii!"2'- “I'LL "F I -" _ -' - - ‘1' -_ . . .'. .' II ._ .' “"1" . , _'.‘_ -—_lll H I ' ' I 77! h‘ ' -. '1' -"L‘e- e-i' ..‘I.J-. . 13.5.“ ' 4.1 Gfiflm my . _ -._._-_i-_ ‘tu't‘htgg ‘ “mm—Eyaimfi _ _. I m _ _ I ._ ‘1 yflt'yilt '* it'll ' "mt-that'iEitE-iifil'edeiieeelteve-‘aaeluflta Heme alnemrjr. ' ' 5“ mm with“ i“: “It emanate a seleflen: (a) fer- value at. J" - - - y" . r’ y? ... WOM- - . .yn} = :1 .3 (T) Yin-" y'f'" yin—ti 15 “mi 351'“ a” = rtt- Since to can be any peint in the interval. 1’, it .is necessary and Enffifllem that “’(J’te'ai n Wyn) be nenaere at every peint in the interval. Just as her the secend erder linear equat'ien. it can "be shevvn that ifyi. v1. . . . . vs are selutiens 0f E'51- (4),. then Wtyti-je. . . . ,yn} either is acre fer everv r in ihe interval I er else is never zere there; see Preblem 20. Hence we have the fellewing theerem. Themem 4_1__2 If the functiens p; the, .- i . , pi, are centinueth en the epen interval I , if the fut-retinas # y. ,Iygv . . ,yn aee'selutiens at E}. (4)._and if W091, ye. . . .. .ynttr} ge [I fer at least ene petnt in I, then every seltttien ef'Eq. (4) can he expressed as a linear cembinatien ef the” selutiens 32.1.. y}. . . . .yn. A set ef seiutiens y1, . . . , yfl ef Eq. {4) vvhese Wrenskian is nenzere is referred te as a fundamental set efselutinns. The existence ef a fundamental set at selutiens can be demenstrated in precisely the same way as fer the secend erder linear equatien {see Theerem 3.2.5). Since all selutiens ef Eq. (4} are ef the term t 5). we use the term genera] snlutien te refer te an arbitrary linear cembinatien ef any fundamental set ef selutiens ef Eq. (4}. The discussien ef linear dependence and independence given in Sectien 3.3 can alse be generalized. The functiens firth. . . - .fn are said te be linearly dependent en I if there exists a set ef censtants kt . kg. . . . . k”. net all zere. such that ilkLi": + ‘i‘ ‘ ‘ ' + kILIIi'r :1} {iii fer all r in I. The functiensfl . . . . ._t}l are said te he linearly independent net i it the-j. are net linearly dependent there. lf_t1.. . .._v,, are selutiens ef Eq. tat. then .t e..n i'~;' shetvn that a necessary and sufficient cenditien ier them te he linear'is ind-r pendent i5 that WU’], . . . “liniifl'li fer seme in in 1' {SEE Priihifm i. Hfilfi t.‘ ..'li-Lin'-Ji'“ni.i1li.ii set ef selutiens ef Eq. {4} is linearly independent. and a iincarly indeperdent st: n selutiens ef Eq. {4) terms a fundamental set el' selutiens. The Nenhemegenenas Equatian. New censider the nenltemegenenus equatiun i” ." 1. LL‘J] = Tm! +IJ1II1'F1II"ii + _ _ I +PHEHI‘. : LEN-L If 1"; and Y; are any twe selutinns ef Eq. {'2}. then it fellevrs immediateh Inen the linearity ef the eperater L that L[Y1— Yfltr] = Lll’lltti - Lll’fltrt :- gtn - gut : u. PROBLEMS - . - " megaheenep r et;:'~t:t;-:.-;s_-..t _ dif . twe selnttensef the nenhe l_ . ._ s . _ Henge ma fawn“ at any equattnn (e). 5in any fifllutlnnzzefthes at ~ binatinn m tmdententeleeh;_ _ i . r I d. I " ‘ I r 1" r. '. _. _ _l .. . 1:1 __ i;_4.§-F;r.__.ii _,__ ,, _ ._ r. a ,1, - _ ' =5? «ti -'. I_ E f .I d in the netsectifln- If the mffiflflflIS-arsnuts'1.. . 3‘; L» . ' - . _. “H. s.“ -4:___ I _ 1...: II ‘ _ 1' “:1: I “.2: I - zeta... meet - - .;. -. W .. _. a u . . -' I rfi__#rfl_s{:;l._ Lfirfijfjv 1".1’ - - ' " I. "" -t.l ' II .3 "" - 'II- I . .l. I'Il Jim-Ii. _ - _ . 3‘1- 1“?- ~ -. i {Abel‘s t” estimate-.. '7: " . inane.an ' .- - '2 eeeetuflemrera-tarso; Let r. (a) If. metheds similar tn these in Chapter 5. These tend te heeeme there J . . I q the nrder nf the eq natlen Increases. ' I ' _ I P _I:_ The methed of reduetien ef nrder (Seetten 3.5) else appltes he nth (mm. mil-Em... e uatinns I f u. is ene snlutien ef Eq. (4), then the substitutien y = q I ' r r .. _. a linear differential equatinn ei order it — 1 let u (see Prehlem 26 fer EEK! a : 3}. Hen-ever. if n 3 lit the reduced equatien is itself at least engendgxgfi WHEN I enltr l‘ttl'Efit' will it he signifi ea all}r simpler than the ertgtnal equatten. Thus, :7? _. tedhelien erl erder is seldem useful fer equatiens ef higher than seeend ' ' E‘- ' -I- - 1: .l' u.‘_-- "-‘.-— . -I- in t'LlL'i- .rl Phil-titans I threueh a determine intervals in whieh selutiens are sure-tn - l 1 e i, t - _.I. 1. t- 2. nr’” + taintjy” + 3y = nest _ Ir — ‘ a — — fit F H 4- PM + IT” +£2.13! +331? =1!” ,I I__ --Hint: The detitr'a‘litte .ef a 3-by-3 determinant is the suns at three Hay-3 determinants . . .1 l- .en t ,t : n e. [e _ the} + 12y” + By = a I -- antennae by segmental-lg the arse seennd. and third remreepectieely. (h) Substitute fet- _tr’".y§'. and y’f item the differential equatinn: multiply the first rew by i ; .- '1 I WI en s ‘ tin: ~ ah Ht 'l '-t=rtnine whether the ' iven set ef funetiette'i'e 'I' . . 1| . -1. t . at l r LIE LL L» g “sit- . pg,mtllt'1flly _the 5Wfld-fflW_prg,fifld add these te theiaslrew te oblate List‘utttiei'l r-r tan-uni rndepettdent If theyr are linearljtr dependent, find a linear fish; .iritltlltl ii'te'lt‘l _:‘ Iw. = —pllnw‘ t_~m=r3+t. _l';.ttt=3r3fi_r . 5“- flth -— 3t 3. tgtrt = 3:3 + ll [3“:- = 3:1“ +; i" ’1'“: 3‘ t 3- til.” :1“: +1. the = Eri — t. he] = :3 +r+1 1H. hm = 2r -— 3~ tL-trt =t-1 +1. hit] 2 2:3 — t. he) = r"- +t‘ +1 {e} Shew that Wiyttfa. slit} =fE‘P[—fptlndi]- It fell'nws that W is either always aere nr newhere aere en I . (d) Generalize this argument tn the nth nrder equatinn In eaeh ef Prebletns l l threugh In we fit}; that the given funetiens are selutiene et the diffe _- 1: equatinn. and determine their Wrensltian. .s‘“ + meet“ + - - - + est”? = U l l. + _tr' = H; i. eest. sintr a as: _. . . - . . . 1" i + i _ U‘ 1‘ 1" mi" I‘ 3”” with 'selntiens yl. - . . ._e,,. That 15. estahhsh Abel s fermnla. 13+ + 11:" —- ,t" —- 2y = U: 6" e". e'3 14. 19"“ + 2y” + y“ : U: 1, t. e". te“r way“ _ , , Jflttr'} = eexp [— felts} tit]. 15* sit-i” — _ " = l]: 1, ,e‘ I3 Hi. .eiy'” +13)“ - Zry’ + 23.: : t}; 1;, x3, 1;]; her thisjease. 1?. Shaw that Wtisin‘ Lens 2t] = [l fer all I. Can yen establish this result witheut 5' In each m- pmhlemfi 21 thruugh 24 use Abel‘s ferrntda (Prebient Zt‘tl tn find the Wrenshtan et‘ evaluatien ei the Wrenskian‘? Iii. "tr’eriftr that the differential eperater defined by a fundamental. set ef snlutiens ef the given differential equatinn. '31- W'+2Y""'}"—3}'=0 2'1. ,F”'+.t‘=t} ' . t at ,-r .r _q_ .2 My} = grim +p1(t}y‘”"” + . . . + put”). 23* by" + 2y" .. y. + 1): = 0 24. I‘ll." + [t +3“ 4} li' 4.2 Homogeneous Equations with Constant Coefficients solutions of yin! + F: mywu + _ _ * +3159”, _I_. fly”) is nowhere aero in I. Wt : .-. In I “III III .,y..)(el as 0,. and suppvfis that (a) Suppose that Wm,“ rtyitn+m+esysttl = D I U . I_ : ....._..,-:.,;i. I _ f It: in! By writing the equations corresponding to the firsts: _ _I a'iui'II shovv that e1 = - - - = efl = t}. Therefore, y“. . . ._v,. "are ltnearlgv,r magma-,5, 1: {is ' ‘- [b) Suppose that y1....._v,. are linearly independent solutions of'Eq, (i); _' i; - r If H1191“ ....}'n_l{f”} = {I} for some en show that there is-a trousers: s'olu'titsrjififa fving the initial conditions I :1 II.‘:'I'I"'::I::-:‘I:-I:""I- -.I—' I It'- .4" THE-:11, -...‘ 'l'l'r-r.:_-'-.- _ l :71 D} an E '= E El :1 Cr I-un.II E M E L'.‘ ‘i d EL. I: fit "U pull CI E 1'1: .3 Ht- :7” fill I: E .El fl fit :1 a '5 fl H H.- g i; 1-. Sinee _v a . vields a eontradietion. Thus W us never zero. ..--I -- 3h. Show that ifs; is a solution of .- II I _ -% + puny” + pfltlyi + p3{t)y = n_ s, :- P .I I - 1...? in leads to the following seeend order -3j—:_-.;«§f._3-l then the substitution _v = _v. me '7. I '.- . .-. . .. ‘I.'| t'I' ~:~ 13v} +p._v.ltl"'+ t3_vi1'+ Ethin + pgyflu' = U, in tilt-J! Problems I? and is use the method of reduetion of order (Problem 26'}:;trs:l_~__: '{f'it-i'ifi; en's-5'- -itiIlet‘-.--Lut..+l e-.]u:'ttid|1- I? [II 'F-liFEI'lii- —-.'i'I-'rt'=|l. lei-31 _'I’tli4'i=l‘31r '. 3-“ m: 3n" — .i It — jit' —'-{1tl '+ £le — [iv = l}- I} 0: J31”) =f21| y2(§)-..=.__.__._.rl Consider the nth order linear homogeneous differential equation List 2 mow + mam” + - - - + essiy' + any = 01 Whfiffi Ethel. . . i .31: are real constants. From our knowledge of ascend order“-.;T..i_iist=" equations with eonstant eoeffieientsi it is natural to anticipate that y = e" is a sol of Eq. (1) for suitable values of r. Indeed, ..tltt!i_.'- Lle’I'] := e”{onr” + fl‘lf'II—l + - - - + tr,,_1r + on] = e"Z(r} for all rt where 213?: flflffl +fllrII-l + . .. +fln_lr+an' solution of Eq. (1). The polynomial Ztr) is ealled the characteristic the equation 22m = fl is the characteristic equation of the differential equfitifi-T-ii—‘ii ' 'a 1- | —I -. . .'_I I '__--. _,_'I I :r 2' I _i- :2. ‘ IIIIl_"E1-{l?.-"iII{I:Flrlllili-11llItllfjiI1I1!l| - I'I p' . - ' '- 1?; 1 . .' I‘l'I‘ I IIt.-II'-.II_ _' I I I- '1 T5.-I.I. :. _ -._ . -. I .I I. i'--_ I_q.. '_ I "-i' --'I- - -- -.. Ir IIhII-I II III I I "- - I?‘ —- ‘I- II I I IL "" 'i' -' - J 'II; 4“-'I-'Ih::ii . i. _. II' ' fill-- L .. I I y . I. I :l‘ _ _ . - fill-FEE? Phi-'1 pziiihLll-::.' .I r r «taunting-thee:enema-steam; Fibysolv‘ingflsg-pulymmial mean I II .I I ' e-s'I-l-‘fie—‘lh {a} - a "film's Waste-tee e F's. its-:2. r. = —s. - mature the general solution titties-{Em -' - _ ' ' =731If4‘sli'5tl-v'fisfi" + as” .+ area's (s) The initialfeonditionsgtfl-require that a. .h. . .a eatery the four equations Et+fi+ Es'l- Ct= L C1—E:+z03— 3121.: 0. 131-!th +463+ 9C4= *2. L'] - {'3 +3C3 *IEIh's = —1. By solvingrthis system of four linear algebraie equations we find that ._ It 3 E _l E] =iE-l', ('3': E. {'_t= —§.. f;— 3. Therefore the solution- of the initiai value problem is _ _ 1 a I I =%£+{se'—§fi—ae“ In! The graph of the solution is shown in Figure 42.1. than me years was 1stairwell-tor every poignant-ti at as; nation has at least one roots The-affirmative answer to this question the fundamental theorem an algebra: as given by Earl Frie'dri’eh Gauss “TN—1355) in his doctoral dififlflilllflfl to 1m. althtuigh his pmot Lots not meet modern standards of rigor. Several other proofs have been discovered stneei including three - - i- - ' damenlal theorem of signs-bra In a first flours-e on is Gauss-himselt. We}: students efisfl msfl 11“ m“ I _ . 1 h egmplexvariablestvvhete it eats be established as a corrch eenee oi some of the basic properties oi E11111]. h. s analysis functions IAn'-in1portantques_tion in mathemalies for more -1 FmURE 4.2.1 Selutien at the initial 1calue prehlem nt' Example 1. as Example 1 illustrates. the precedure fer sehting aninth erder linear differengjglg equatinn with censtant cecflicicnls depends en ending the rents-ef a cerresppndlgg nth deercc pehnnniial equatien. If initial cendttiens are prescribed. then a .--i- n tin—ear alnchraic equalities must he seleed te determine the preper values ef; h . Althnugh each cil'thesc tasks becenies much mere cemplicatedn'g H “mm. “Lg Hm it. n “p.311 he handled witheut difficulty with a calculater er cempulafg F.” that] and .I'nurth degree pelgnemials there are ferniulas.2 analegeus tn i... m u 5a it.» ~;L1.'tt.ir.i I ic etiua i inns hut in erc ceniplicated. that give exact espressiensffig Fit in 'i - I i | ' tliil: alerrrit hnis are readily available en calculaters and tempura-g; n:._ -. lei-leased in the differential equatien seleer.se that the precesg-nlfi hxremnal is hidden and the selutien ef the differential csinslantai'r. th-_- - Its .l':-._I fl. 1-. I'isl't i'it'! -. .t.=..attiattcall}; _ ' . i. . Illi; d 1c I'ictnr the characteristic pelyneniialh}.r handhefgf tin-n s helpi'ul. Suppesc that the pelynemial ' !. .-'. “l_ _‘._ ll._:J,..,:..-. _ _. _=F. fl”_1r + H” = I] he liltL‘fl._| Ii.t_."._'il‘.*.~-- ll r' I F. l-ilthJl"-_.I then it ninsi l‘1._ a lacttn' til it... and t; tT‘Itlst be a facter til en. Fer exemplain Ed. in". the incite. t'i a. are ii and the liacters til n.I are ii. i2. i3. and i6. the nnij. l'n.'t's*~ll"*lc ratinnal rents cl this ecluatit‘in are :|:l. i2. :le3, and :lzfi. By testing: these pessihle tents. we lint] that l. #1. 2. and —3 are actual reets. In this case there are an ether l'tll.ll.h. since the prilt'nnn‘iial is int" feurth degree. If seine ef the rents are irratienal er cnniples. as is usually the case. then this precess will net find [heat but at least the degree cf the pelt-henna] can he reduced by dividing eut the lactate- cnrrespending tn the ratienal rents. :T'hc ntethtid ltir snlt'ing the cubic equatttin was apparently disct'n’ered by Scipiene dal Fefifl life} ahtiut lilill. allhuugh it was first published in 1545 by Girelan‘te Cardane [[50145th in hiaill'l‘ .l-linnnr. This heel»: alse centains a nicthed fer seli-ing quartic cquatiens that Cardane attrihutes It} lll‘i-I Pull” Ludei'lce Ferrari tlfilfl—lfihil. The questien at whether analegeus ferrnulas esist fer the reels fll' higher degree equatinns remained epen fer mere than twe- centun'es. until in 1321: Niels Abel sheared lllll nu general selutien l'erniulas can esist fur pelyneniial equatiens ef degree live er higher. A mere 'gellilfil. thenrjr was deselnped 1w Ei-ariste I«Cialeis t 151 1—13.32] in 183 l . hut unfertunately it did net becenie width" ltnnwn I'er sci-eral decades. (a! n is a ratinnal rncit. where p and q have an cemm' fig EXAMPLE If the film We have the general-selutien- (El-lamp]? sealant exponentialfnmalm the selutien within rent. If this reet whereas if it. is. negative. then sala'tiens will tend exponentially In Berti. finally. if largest rent is as rather: selutiens-will appreach-anenaere censtant as r 'heeemes large- Of ceurse. fer certaininitial eenditipn; the eeefficient ef the etherwise deminant term Will be Zfil'fl; then‘the centre ef the selutien fer large I is determined by the next largest r'eet. bytes teen aa'nespanatng te-tIie-alg'ehraiflll‘y hen antennas will name expeaenfially the Complex Renta- If the characteristic equatien has cemplert reels. they must mm in centugate pairs, .1. i in, since the ceefficients an. . . . .n.I are real numbers. Presided that nene ef the rents isre'peated, the general selutien ef Eq. (1} is still ef the term (5). HeweverJust as fer the secend elder equatien (Sectien 3.4). we can replace the cernplex-valued selutiens elH'“ 1" and eli'll‘" by the real-valued selutiens 9“ sin lit {13 ebtained as the real and imaginary parts cf eli+it‘”- Thus. even theugh seine ef the rents cf the characteristic equatiert are ceinples. it is still possible te express the general selutien ef Eq. (1} as a linear cernbinatien ef realwalued selutiens. Find the general selutien ef e“ eesut. Alse find the selutien that satisfies the initial cenditiens it" till = —4. still = m. and draw its graph. Substituting e” fer y. we find that the characteristic cquatien is Therefere the reels are r = l.— Li. at. and the gc neral selutien cit Eq. t' 1-1] is .151: L'it‘: 'i' [31" If we impese the initial cenditiens (15}. we dad that thus the selutien ef the gts-cn initial l'.llllt.‘ prehleni is The graph ef this selutien is shew-n in Figure 1.3 2. Observe that the initial cenditiens [ 1:? l c. term in the general selutien in be acre. 'l‘heret I which describes an espenential decay he a steads esctll if the initial cenditiens are changed slightly. then r, is the selutien changes enernieusly. Fer es same, but the value etymtill is changed frem —2 te prehlem becenies l -I. yeti _ J; = {L [14] }"'t{ll = if. j." till = -—I [15! r4 — l =tr3 - l'it'r:+ li=ti_ " + i'_—.cest +- ssint. {1:23. Ii:-:l_l; l'.'.=—'l' - 3t" —' icesr - sitti- '-' ease the c..:-ei'li-.'tcnt 4': iii the car-tee- ntia1ltil gr: is “new. "' arc-pic. ii" the first three initial centlitnnisr — [ti-ti. tltcn the selutien tit tnc llllll.-..- -. .. Here this term rs .il‘setzt in the. st ' ' - 1' H I - II: 1 alien. .ts Figure -l._ _ she-m r» A liltcl's lt't l'Ic l'lllllr'tl'l't“. .tntl li‘a. n-..‘_tI.'-. -'.i.‘ sin till “It. I.|_. '5. lr' _l '. 'I _. I ' e?“ ' _.|i- _ .1”: __.i I..-__i . I_ J .I' i v " II”- ? '1 'E.i'i'i'1i'i'i"t't-fj [ii i it? t' H'ifr'i'r'l' 1; t'i'r- i t' .3 -' '- '-" i Lad-r -=I':_-f.=s:~eeeies'-:::5ere:- -"-' -' '— —'—-" 1— __ 2., .a‘ _I:__ '1._ ail-i“. . new- II F I_II_I:H 5-. I. ,. .-. .p _. _'.'-r —l-.'-1- -'- I II ' Ii. . ' '. r .- ii.tei':=-ii.vii I "e i i e i '- '- .-- -._' a it'll“ the” in 331- [lfil-riM-ifiei. l - . 7 difffif uni}? slightly * -_ . - -' .I.._l.-r;.__ H.____ m Eqi‘ (1 it relatively' small-nestlii-‘i’lflfli 0‘ 12" '33s Le%i.-Ei=lr.u. greWiflg iEm' “an mm t E er 5. This is clearly 535“ --ife.~.;l_.__i_i_._ii 1 I 4 ' =..'-li."..:.':. .._'_'1 __ ,I. fl ' - - iislat'get'thfifla -* - . selutien hit the “ma ‘ {16) and (17). -- 7. u._ | .‘._._:. . I ,I . the we millilin - -. it :. _ - . . '_ . ‘-=-i.r‘-I .. v H - .. ‘-‘-i_ - " use -. .. (pm plea) Hemline - event ‘ itie ease-tees? - i'_' sit .—_- net-5;: __ .- thfl - - 35ml I 1115 .. - .. i; Fine thegeeeiai entities: er? EXAMPLE y“i+.yeiii (20} .i' 4 The charaetetistieeqtta'tittn is.- fi+1=fl Te selve the-equatien.=w_eainust neinpute the. fettrth reels er -—I.- New —1. theth ef as a eer'rtpleit number. is -1 .-'l--Ui. It has-magnitude taint pelar an‘gle n. Thus -1=eesn.+isinrr =6". t -i ; i t Fit-:s lit the selutiens. [file] [light curve} and (1.?) (heavy Curve)_-_ "—4 Mereever..the' angle is.,dete_t'n'1inec_l enlv up te a multiple ef 2n. Thus I _.1 =nnsfir + Emir} + isintir + Emir) = elfimii it ~_.' _ I... it "t '_ I where in is seen er any. pesiti‘ve er negative integer. Thus [_l)_.iii = EiiFII-Himnfl} __. ms + + ism + _ -1 .‘.‘" r _ ._ l- -I Ir:r_-__ I. I q. 1 I + . _ I : El: _ I Repeated than. it til: E'-.‘:.t[.~_-L er the tIl'lflTtiLlLl'lSIlC equatien are net disttnet—ih._ .. Iii].-. it r- "i ii i‘ th 1 re "it" it“ 1': {.‘IE'tlfi‘Cl [hr-‘11 “‘16 Wit-“it'll” {5) is Email? flat the The feur feur'th reels ef —1 are ehtained by settingm = 0.1.2. and 3: they are 3L 1" C r I- _i_; .l_ _~t 1.: I. t J": ,___l__ g ' _ selutien ei' Eq. Hi. Recall that if r; is a repeated reet fer the secend erder- iii-ii- 1 +I. 4+1. _.1 “I. i __1 J2“ J5” fi' if? it is easy te verify that. fer any ether value ef in. we ehtain ene ei these fetir reets. Fer example. ‘ ' ' i' : 3 I equatien iii-.i' 4 e; i' + H: i- = t}. then twe linearlj»,r Independent selutiens unit. ___I. “1"”. Fer an equatieri et' erder ii. if a reel ef Z tr] = [L say r = r1, has multipli;__ E _- [Where i E “l' thfln J . J, [j 1 n, i cerrespendiflg In in. :4. we ehtain [1 + rib/Z. The general selutien 0f 34- [3“) it n: .Irl re. [1 i _ i _ 1 _ E till-{I ;. , are cerrespeiiding selutiens ef Eq. ill; see Prehlem 41 fer a preef ef thJS-StfltfllJJEiif " y _ 1 ff * v5. . ' J5 «52 If a cempleii reet i. + in is repeated 3 times. the cemplex cenjugate A . alse repeated 5 times. Cerrespending te these 25 [temples-valued selutiens. l-il|i_.i__. _ find 25 real-valued selutiens by neting that the real and imaginary parts ef el" . Eli-“ti”. . . . . tHe‘H‘” ” are alse linearly independent selutiens: " '~'+ " In cenelusi'en. we nete that the prehlern ef finding all the reets ef a pelt-nemial equatien may net he entirer straightferward. even with cempeter assistance. Fer ii "if ' f g“ ' instance, it may be difficult te determine whether twe reets are equal er_inerelv 6} ms Ht 8’ m “I! mi EDS Mi I 51“ pm very c-lese tegether. Recall that the term ef the general selutien is different iii these {ti—lei” ces tit. (“i—1e“ sin pit. '1 '5 - twn fiases‘ * E (1) l embers. the selutien ef Eq (ll _ _ _ .. .. _. tams finial“ _ “an in q. are cemp eit n_ i * -_ Hence the generalselittten ef Eq. ( 1) can always be espressed as a linear Cir-III! . _, iii.“ -_ If the cum ' . h h 1 ate: 1 Hanna . _ , i, -- ' ' ' . i 5 case, hewever. the rents eft e c are: ert cfl ef it real-valued selutiens. Censider the fellewnig example. ‘ ; I; 1-5 5n” {if the form (4) In a 1 F4 n.” In each of *Pt’flhiEMERefi M Rteosfl+tstnfli- - 2. _1I+J3-l_ I—r-III-u- -_— —...-. _ __ _ _ .- 'Ir -.p. r-"'—.'r_.__ __-' _..".. I. I I ' J h‘t I Far-l ‘t --t -"'-,_ 1'5} I'HI h ‘ ' “mt-mlwnumm I 3:33: is also-a root. The no. - siting-salesmeer I . "7.1.: ' ‘ lets another ul'thfii r- it 1st". 1 through ti express the gfl'flfl comp . HI: I .. "i _ -'.' through 23 find the general solution of the given dififorgg m I.” I y’”—3y"+3y‘ -—'y=fl I4. yi‘“ - 4y” +4)“; = {I I 16. y‘“ — 5.3:” + 4}: = I} In eaeh of Problems I l 1]. t"" r _t"' — _t" + _t' : it I}. it” — 4y" ‘ 2y” 1— 4;: = {l 15. It'”” :— r'i' : ii I? t"“' r— 31”? + 3}" F _t' = U 13 Jim — V” = I”. 1"" - .'n"“' 4- fit” a— it" + I)" = H 20. yi'“ — Sy’ = ‘1. ~ ' I “It'!':' .:—1a-.-:tt 22 y”‘+2y”+y=fl at . ~71 -- F-t- t = H 14. _t""+5}'"+fi}"+2}'=fl .Ilt 5*") .' :n ' : I-n' -—ty :1; 2e. yi‘“ —?y”’+6y"+3{]y sage-gt}; i" - . - = .i 17h- +.~a-:u '73. 2s. rt” +6r"+17y'+22y+-tsr:% “1.14;. Fe ilrt-_.rug.ii it: find the solution of the given initial fljri;;-.-;_-.ti.r._ In t..n.'i' 'l I": . ii-w. tli.}L"'~1.-iii:’ solution behave asir —+ on? - Inst -.:~ amply. LU. -.- :— :r H. mm =11. you : 1. J'uffii =2 iii} 1”“ 4; _‘.' -_ ii; :iiii : H. flit} : i}, y”{ii} = —-~]L.I ymfli} =0 3t. yi' - 4y ' + -tt-' : it. til: : —t. _t"t1}= 2. y*’(1}=0. y’"(1}=0 .1: f3. 32. -- r" + fr t' = it; til“ i 1 I'm} = ‘L W01 = ‘2 . 5- ‘1. 33. It”: — — U)" + 4*.“ 1" 4_t‘ = [i1 HUI = r—E. y’tfl] = U... y"(U} = —-2, i3. 34. 4pm + t“ + :ii- r. {1; ya}: = 2. flit} = I. y”t_fl} = —1 35. hit.” -i- is” + it" = H: It'itlt = #2.. y'ifl} 2' 2. y”{[l} = I} iii: 3h. y‘“ + tie" + lit!” + 21)" + 14y = U; ytfl] = "Lr y'tfl} = _2_ fan = {)1 .1: 3?. Show that the general solution of _t""" - y = i} can be written as i;._ if = Et eost + e; sin i + e; eosht + ea sinht. r! .-. -r I Determine the solution satisfying the initial conditions ytfl) =0, yt-{fl} =0; ;_. ,ttwtflt = 1. Why is it convenient to use the solutions eosh t and. sinh I“ lit!!! in“! L' 4? F‘ 33. Consider the equation y”? — y = i]. . (a) Use Abel‘s formula [Problem aim) of Seetion.4.1] to find the - mental set of solutions of the given equation. - ' .-.' -' . r5” _ as r .- _ I Fats-Summit. ns+ht=2ut tit (ti) sans the" (i) for-u: no substitute into the second equflitm. thereby " “. j at: "‘= set an a1“+?n:'+tiul=0. t'eit general solution of Eq. (ii). (It) that“ the initial conditions are “1(0): L == 0.. Hair.“ ='- I. Iii-“31 =13 ‘. '1} Use the first of Eqs {i} and the initial mentions-{oi} to obtain statues for u: nit. .tntt u'jttn. Thea Show that the solution oi Eq. (ii) that satisfies the four tom's} cent-onions. on sr_ is tam .= cost. Show that the meeting solution a: is on” = .‘t ens .r (d) - New espouse that the initial conditions are asin part-(e) to show that the corresponding solutions are min -= a .. ms t. in and tuft) = menial. (e) Observe. that the solutions oblam in parts to} and (d) describe no distinct tit-mines ofstihratiott'. In thefirst. the frequency of the motion is I. and the two masses mote tfl I- “as I III' 'I. i e phase. bath mnving up nr rinwn tngetflpgéfijizem masses mnve nut nf pha. up. and vice versa. tit these twn mercies. nutline nn ependent nn -—ei: a' J sat in shew that if n. . - . . n, are all real and .dtfffll'enh.lfim::. e i ‘ I- . . - .. I ' -_—_ I 1—! l 411 In this prnhlem we r: r e m. Tn dn this. we ennstder {hangar LU] any + my I! J + . . . +flflwly + an}, = g“) (le e’“ are linearly ind II.I.I _ I can be detained by the methnd nf undetermined enemeieats. prnvided that gtt't is of relatlnn I I II _IIIII: III I _I: m an apprnpnatefnrm. Mlh'flllgh the merlhad fif‘ undetermined eneficients is not as “er s . - - + in e ~ - general as the methnd nf variatinn nf parameters described in the nest sectinn, it is usually much easier tn use when it is applicable. fl [I thflrehp flhtaining - Just as fnr the secnnd nrder linear equatinn. when the cnnstant eneffieient linear differential nperatnr L is applied tn a pnlynnniial Aer“ + Adm" + - i - + Am. an E1- pnnential functinn e‘". a sine functinn sin fit. nr a cnsine functinn cns fit. the result is a pniynnrnial. an espnnential funetinn. er a linear cnnihinat'inn nt' sine and cnsine functinns, respectively. Hence. if gtt} is a sum nf pnlynnmiats. espnnentials. sines. and ts are tern. d differentiate with respeet t rim—1"” : UI and shut that all the ennstan tut Multiply Eq-{ili1§.'e"“ an t-ttr: —- Hiya—H” 'i' +i'nirfl '— “if . m. —H H and differentiate with respect tn 1* tn elitism I II I I I I II III fl IIIIIIIIII _ rI mun—qt: = DI cnsines, nr prnducts nf such functinna we can espect that it is pnssihle In find Yin . .r — r71“ — r It + I =1 “ by chnnsing a suitable cnnthinatinn nf pnlvnnmials, expnnentials and sn fnrth. mul- I II I_ IIIII IIIIIIIII [III and (MI Eventually. ghlaining tiplied by a number nf undetermined cnnstants. The cnnstants are then determined m [entities the Prim-dim ” " by substituting the assumed enpressinn intn Eq. (1). IIIIIII I {I I I l , , _ “II _ n tyre-""- *" = U; The main difference in using this methnd fnr higher nrder equatinns sterns frnrn the fact that rnnts nf the characteristic pnlvnnmial equatinn may have multiplicity II II I III IIIIIII HIIITII‘t-IIWIH greater than 2. Cnnsequentlv. terms prnpnsed fer the nnnhnrnngenenus part til the ' ' ' II I I I I IrII_I[ = “I snlutinn may need tn be multiplied by higher pntvers nfr tn make them different frnm i'i " ' T I” “i terms in the snlutinn nf the enrrespnnding hnmngenenus equatinn. The fntlnwing I I IIIIIImIIm m film. that (HF, :11 in a similar way it inllnws that examples illustrate this. In these examples we have emitted numernus straightfnr— ' “- l” WILI 1:]: mum.” _ _ _ are linearly independent. ward algebraic steps. because nur main gnal is tn shnw hnw tn arrive at the cnrreet ‘ ~ - i- t- 'in - w .i. it?- shew that ill r = 1‘1 l5 H [Um {if mUItipliCilY-‘inf-‘ihii. farm far the assumEd Sfllmmn' "’ H" ' 'II 'l ...‘.-. _ . .. an: snlutiuns at Eq. ('1). "as I .III'. i'tLI-tltllletl’lti the metth given in Prnhlem 22 nl Sectinn 35 I I . I - I! . - tan till‘i'i'h- the start t'rtnn Ett. [it in “1'2 lfl'iii- Find the general snlutinn nt II I 1'” '- - . EXAMPLE. i'r'~3.i"'+3i"e.i =43 '3‘ up“; : filed. (it) '_ 1 I I I _ _ I The characteristic pnltrnntntal fer the hnningenenus CttUfllltll'l enrrespnnding tn En. t-t :-.- rl after each differentiatinn. -.1l-.‘tllt. Willi t'tL'sPe'Cl [Li Litjiling r z I II F - i 1 ‘ r..5'r !|I|_' "iT '1 " ' q. I ' I ._lTIt_i _llllsl- 3-1 . .- e _. I!” _._1r_:_:__Irf-_ .1_ _ - - __ _ .5“ ._ , r _ . .. . tut tir~~..er-.e that 11. r'._ l' s t't'ttil nt multiplicity .i'. “if” El“ _- lr rli ‘i'iri-Iffifl'e in”; t Ptrltnnnu «-I Hi tit-ere; n e i and WW I “- Slim" mi” 2‘“ i' Z i" l” ' ' “z- ”‘i me an the general snlutinn nt’ the hnfinngenenesetiuatmn .s Eurii'hu" EMU” in Kit} :t'_r' rt- fjte':+-'_‘.l:r" -" [[1] Br ditl'erentiutit'te E-ti- iii "EP'fi'diCdil—t' With mil-1'3“ m r‘ 5"th that I I h . . I I L. I __ ' ' Tn find a particular snlutinn i muffin. titres" flail-13.- “iiiiiimil int” 1‘ "-" "' -*' *i' "‘" H ” a ” uteri} since e’. as". and tie" are all snlutinnr at the Inninngerteainm entrain-n u: i':‘.'t-7 "itil‘rf‘._ i ELIE l : E: ‘ initialehniee bvri. Thus nur finatassunintinnrs that 1 ..-'.~ : .4: :".i ate-re -t 'T :t.i=..-t.~t.-~.z... .. t .. ‘-__ _‘I _-. 'a . 1tl_l_:l 1._ enefficient. Tn find the enrreet valut' let .»i. we tiliii'tt'fltiJL'... '1 it! Hit“. tints L l'_ .t t. i and its derivatives in Eq. {IL-and enlleet terms in the resultingequattnrr in ililr- we; w. [ill—Ii _‘II nt- : J". —-L[e”] : Lilli t.’ i '.ir“1 I I I II r Thus A = and the particular snlutinn ts te} Shnw that ewJe'i'. . - . .t"'ie"‘ are seluliflnfi flf Eq. {1 l- H“ ___ IgIF.-II__.I -. —- -—-- ' —-' -' ' ___—— —-—-r --' ——"—'”———' — "‘"mwflJ—rfi; The general snlutinn gf Eq {2} is the sum nt't', ltl item ELI. {3} Mid it” “it!” Est l'ii 234 Efi-AHP-LI EXAMPLE - ‘i __s lilll‘ifi I F‘gtB- :t find a particular salutien a! the “1113159” I I u I _ f" +2)” -'I- y = 3smr. — See-st. The general salutiea ef the hemageneeus equatien was tuned In Example-31 a ._II H u yr“) = e. ensr-t- rs sin: + etreasr + etrsin i, I I ' —-i- nf the eharaete-ris'tie aquarium we: 1M” enrrespandmg tu . _ _ assumption far a p by t3 In make it diflerent [rem all so: assumptiun is all '.I Yin -— Arisini + 31* east. III- s substitute intn the differential equating {gm .: Nest. we differentiate Yin feur time terrns. uhtaining finally _ —H,4 sint - 33 east = lenr — Sensr. . and the partieular snlutinn at Eq. (4} is 11 err: is a sum nt‘ several terms+ it may!' be easier in practice tn enmpum-gg- . ' ' . '1." eqnatiatt; spending te each term In 31!}. As far the €13.“er . 1_. f-f—f-l‘rag‘ftei: the partieular sniuliatt etirre h] _ h f h - _ . - . a ' ~ t esurun t'e-.""'a. - . -- _ meannnshe partnerilarsnlutinnnfthe Lamplele pre _e_rnrts ‘ I _: 1 f+f+ + =_fl_ a“ _ ‘ sultllittns at the individual enmpenent prehlerns. This ts Illustrated tn air-{rag ' . 5 H. _4y,,ftz 1" #- —. 4Ler - Ill IIWHNL Jim ' + E. f+2f+yaii+ens2at ' -r-+r"-_'= . fif+f=si21 In each e'f Problem mind a! the w w ' ' "I: m plate-graph a! the - i m '9. 1'” '=. f; ' lfl.y"’+2r+y=3:+4; 11-!”*3.V’+2f=r+e': flfll=ftfllr=ts flan; wi=ymi=fl -""l‘”1=.v"tm=t lib-.1. r. r..'.til=_nl..- -nl~;|ti-;H Ill,- r'" -— «ir' : t + I‘lensir are—3'. .1 li I. tilt hutlttagt:nettith uqtjillifln. The eharaeteristie equ'atien i5 T3 tlitjE-l2':t'.i'.L_-_: $533?) I31: sl tt-u‘ lltt. I'- H ll‘. :tlL' ll. ! :1. TIL'IILTL' 1 1 I 12. 1".“ +2)?” +y"+fi_tr' -- 12}? = 135ml — #4:. Hill 2 3- still = [3 r. m 2 r. + rte-i + ea" J. - :Wtfl} = ——1. fun =2 - . . . ."l We can write a Puttieular M tlutitut ttl' Elia {H} 115 “1": 5m“ flrpflfliflular salutinnfi fifthfldi I I i In each of Problfims 1-3 My: 13m '1 5mith fan“ i“! 3 '-” fl “*3 mm“ “1' “Tr-3C- _-'| 7w . ; - termut' ed eeeffieienls is In be used. Do net Hm: the enastants. 13.f-—2y'+y=r’+2f Il.t"*_r'=tr"-Iefist 15.y""=-'-Zy"+y=c‘+sint lt‘i.jr'*'-=L-lr'=1aa:rsse “t .l I" 1?. y‘“—-y”—_v'+y'=r1+4+rsint titulajluflhizif 1.1::- 41’ iii-.111: 19. Consider the nenhomogeneeus nth. order linear difiereattst equath I, i:.‘_'- - L‘Lttlalintts H! I] _. __ y — 4} _- e . .__. .II: fi'm — 41" = I'. Fm -- 4}: z 3 Egg [1 {Jet initial ehniee fur a partieular sultltinn FIN} {11' the first equatien is Ag: + iii-:1 ' ' ' ennstant is a snlutinn at the hnmngeneaus equatinn. we multiply by t. Thus ' ' iii"- I. 1 Van = HA“: + 21]}. I will +fllyslrtit + = gull II; Far the seennd etlualinn we ehnnse where no. . . . .a. areeaastsnts Verify that it eat is of the [was Y1“): Bensr+fsina .' |'__ and there is an need tn ntndiiy this initial eheiee sinee east and sin! are net 1:91}! ' the Itnmngeaenus equatintL Finally. [er the third equatien. since if” is a selutiflti. -’- '- hnmngeaenus equatinn. we assume that :1 i i fiM+---+b_ti then the sutflitutiutt 'y = fun} reduees Eq. {i} In the rim Vain == Err-‘1‘. hum + t.a"'“" + «- - + Ln = ha” + --- + in... an In 'I I greenest. I . ., _ t . “ha” R0” "*"demminmga particular-selutien at 'EhFflflgi-nal‘fir -- r - ._ . .. merrier- seltttt'en et ..I:.;.'_i."I_ " -..'--:.-'.-st';t._I...t elem at“ determining a pa he ain't ler PW -' _ II ireefficignts and a petynertual fer the nenhe.trieetfi'lflii'1.15 Um“ I t.I.. . I I II I I _ .I .I I_ _IIII_III_II-_II_I II III II I Method el' Annihilator-s. la Prehlerns 2.0 threueh 2'3 we “amid? "ii‘i-i5.iii:ii .. . III III IIIIIIIIII IIIII. “If; fer use in the methed ef undetermtne _.I.I-eteI_I_III..tnIttrIIIIItt.tI.I.. II _ II _ I III II II II I I II II I it“ this: Pb P IIIIIII the (Ibsenflfifln that; Expunfiflflflit Pfijfflflmlfilt UT __-=,I.I“I.I__ “a I -' I II ..ii'jr-*.I-ZI__,.“.i. .-l:_;‘I-_"-'. . l - r" iIE-I' I I .I I dud: lsfldaliffiifi ef such terms) can be viewed as selutiene ef est-tea 1—159” ' ' 3" I " L53? ' ' t 3'. : ‘ TE"- ‘fl. 773-- ;3 it ";'-.:'='=' I r I . III I II II I IIII II IIIIIIIIIII I II I I ._ .t t t... It . I I t- t I_| E. III Ii . I IIII II I :3 Fiittl equatiens with censtant ceeffictents. It is centrentent WItlfifif-I t I . IIIIIIIItIIII I ~I.I.£II_II III II . I I. _I 1:“ f sample e“ is a selutien of [D + I}? = D: “"3 differential “WrflmE-Efii’ i‘t; . . T'"_' ': ' '= :f“‘r-. - '- ' ‘ '~ '_ ‘ it“- “Hi' ‘ . - - - " 3 ‘ a ihila'tertii . . “Mitt—ti.— .- ' ' ' -- . '_'.' Humbert,pr m “It be an tinntt‘tthtrer ef. e '. StmtIIlarly,D + 415 an II I. I {D it: - 3 — {it} + 9 is an annihilater efe' er tefltand se fertht " “' ' * "-.~ - t '3 . iii-Er" - -- *- _ - . Lt '-' ' . ‘- I ..r'T-...-'- ' ear differential eperaters with censtant. ceeffietents flhflyu-thfillj - ~ . II.I_I III.I.I._ _I.~: ta — int!) r as = {D - but) — aif -'-. :' a" I .I_ I.I.I ._ ,. its; her are: twice differentiable t'uttctien f and an!" “(instants fl and b" Tim is” The I encc it"! attft' finite number Ltf t‘acters. - l' i --; -.._.£’..' ,. ' ‘ " L' -. = ' - - ' E I ' .. the.;m.ethetiet='saaahsastatements-terse" " ' T. element the” ‘ a the term ef a particular selutien 1’“) cf 2L [I'tttsid‘rthc prehlcnt uffindmII _ _— .. __ I __ _I L L naaheraageaeatts a. i311.a‘fllflfllfi'IIfifiar-Idiffgrgnfi'fl.aquatic“ 4 .- _ - Ehrl= 1'3""+-.at--(t)y‘"““ +- - - - wast-creams» =gtr) (I) {L titre er the equatittn is written in a term cerrespending te the. faglI'tIifrrn-I-Igjh _I_- I 1' _l in s zr‘te + 11'!" = 3s" - it- is a- direct eatenaien tif"2t11e .Lntethe'd'fhr the ascend erder differential equatien (see Seciien- 33):.- A-s' heteret these the mathed .ef variatien ef parameters, it is first nec- essary ite- Is'eltteIIthe e’erresp'entfltrg.hemegeneeus differential equatien. In general. this may he diffith unless the-eeeffici'ents are censtaats. Hewever. the method at satiatiett e‘f parameters i-is..s_t-i_li mere general than the methed ef undetemtined ceef- ficients in "thtiIt._i_t' leads tti an espressien fer the particular selutien fer any centinueus functien 3. whereas. the methad tif undetermined cecfiicients is restricted in practice test limited class ef functie'ns g. Suppese then that We knew a fundamental set ef selutiens In, Ire. . . . , y” at the hemegeheeus-equatien. Then the general seintien ef the herttegenceus equatien is " " " - "- [Ejr- ' spectiseltt arc annihilaters ef the terms'ent urge-great .tht'tt'. niart H r I and ID + I]:. W I I _ _.II_ I' 3H!) + l} annihilates iItI_ItI'tI.-_I:_t_.-;i-.= it-‘lc t" L}. !. 1 t. :ee that the cettthinetl eperater {D — .rtt- rice: title ct it; til stntultaaceuslt; I II II ' “i“t'-‘1‘Lt'+tii' it? e int"! + 1i1 te Eq- {i} and use the result cf -_t_i_"-_J._i-.'-.:tt= ' Z- [i'tl '_III-I-Iit- tIIl-rIII: tn — :c'te +1131! = s. Th "i te - '.3t_"i?_ti'.'I'-IT| at the httmegenceus cquatien {ii}. By seltring Eq. ('ii),ahew;III - 1. '1 ‘1 1 'l _ __ tart = .- _. + t-tte“ '-. tar—e" + t'tt's"r + rte ' +ctte '+c-tt2e , Iii-(T) = time-1- Carri!) + - -- + email}. (3) .c~ arc censtants. as set undetermined. [ct Ohserte that E; rte”. Fei'. and e"’ are selutiens ef the hemegeneeusequaI '.I;_I I. tthcrc t'l ..- The methed ef-viiria‘tien of parameters fer determining a particular seiutien et‘ Eq. {'1} respendint; te Eq. ti}: hence these terms are net useful in selving the I I_ _ I I I I I I II I I I I equatien. Thercfetc. cheese t‘1.t‘3. cIt- and et It] be rem in Eq, {iii}I5g that 't '-f rests en the pessththty flf-dB-tB-l‘n'flfllng fl-"funt‘tl‘l'tlfis ts}. Hg. . . . .rtt" such that Ft!) 15 ed the lffl'l'im' ' - _ . 3 ,1: . -t .. -r l" in — Let t. + tale + title . Y0!) = “1(0)” (n + “Efflyflfi + - - - + “nutIttIIn-L {In This is the term at the particular selutien l’ ef Ee. (i). The values at the ceeificie" I - and t”: can he feund by substituting frem Eq. (it: in the differential equatiett (t). 311163 WE 113“ *1 fflflitmflfi t” dilemma we W1“ “EVE “3' SPEC'fi H Cflndflmfle 0H: ef- these is clearly that Y satisfy Eq. (1).. The ether n — 1 cenditiens are chesen se as te make the calculatiens as simple as pessihie. Since we can hardijr aspect :t Summary. Suppese that simplificatien in determining Y if we must selre high erder differential cqttatiens t'ttr LiDly = git}, I _ _ I I I _ at, . . . , It", it is natural te impese cenditiens te suppress the terms that lead te higher where MD} is a linear differential eperater with censtant eeefficients. and git}. 1.5 tI'tI'HFj-‘jI'i-if : derivatiyfls flf “III I I I I “III Fran-1 Eq_ (3) we flhtflin ifett'ii-I'f-‘I' i.- : preduct ef experiential. peiynential. er sinuseidal terms. Te find the fennel tit-ti. selutien ef Eq. {r}. yeu can precced as fellews: I" = (any: + tray; + r * - + unfit} + {liin + “is?! + * ' ‘ + “nJ’si- 1‘” -I I II-- _-ILI - «salts... ' - :- -I- r .- h.;u:IIII_ Er: - "II-I and the fellewing-n -1eend1ttens an the functians in, _ ,1 . t + ' _ I‘. 1 If .r {II-1} _ U” = (“IIyII'In'H + . . I + Hnyhn} + (“lyl + he ennditinn that 1’ must be a snl’utiun hf.EgI_I-,II;IIIII . illustre int dset _ I I IIIIIIIII wattle the deprivatives at Y tram Eqs. (6) and (3). Bflllflfitlflg lama-art? II I- WM'FIFI :7:- chg: 4II El ,_ :Li- __II III. I. I I ._ II '— we nbtairt use tifthe fact that Isl-yr] = DJ —- L2“ 1 - um I_ _II I _. .r Fll IF {Fl—1] = " II” Ill.” vi!” + ' + tiny" ' I f‘ [Immune [a]. enupled with the H a l equattIens (7}; gives it SIIHHIIHHEIJHSHIIILIII hIInn'I-ieneuus algebraie equatinns tar H] .Hlt - - - Iflnr I} J' nar’1 + _‘I'gh‘E + ~~+Ivnttfl — 0, r ,r I I II__ I?! “I + Iva“; + " ' ' + jinn" '— fl! I”! a r .r _ I II..IIII-.-.II I It1 I”! + Itl'j’tt: + -. . + Full" — 0.. III _II.I.I_ I I—lt J _ I111; + . - - + )TIIIII _ The system [ I”; is a linear algebraic system fer theIunltnnwn qIUflfltltIES-Higg Ba snlrina this aft-alert] and then integrating the resulting eitpressrens, yeti - II the enelfihients n1 . . . . . it”. A sufficient ennditidn far the exiateneeI at a saluttnII . ssstem at equatinns {mi is that the determinant Inf eneffietettts is mannere- .I 1Ir-alue at t. Hewever. the determinant nf eneffietents is precisely WI(yI.y1-, . and it is nnwhere :tere since 3:1,. . . .y“ are linearly independent salutiens I, I mngenenus equatinn. Hence it is pessible tn determine at"7t , . . . Using _IIIEE.'lI;...'I,~. -- rule. we can write the snlutinn at the system of equatinns (10) in the farm 'IIII g{”Wm(” : I i If = _—___II = llzillljfl' I til-I.- ”irt{” 5;. if .- j‘lll‘raraer‘s rule is ereditetl tn Gabriel Cramer [1?04—1T52). prefesser at the Aeadémie (it: I' 'IjI'I' whu published it in a general [arm {hut withent prenf] in l'l'St]. Far email systems lht‘r ‘-'r':!""'-'" ' - .. III . ~- 4: Itnnwn earlier. . -_]', I-'* 1 '7 EXAMPLE- -I I - . . 'II— . I . I . -I ..I._ "I' II" . I. "" ' - I - . .:_ I II . ":.' '11. . 'IZ' :- I. ...-1" 1... . ' h- _' . -' I. r " =" J! . I .". — :- - II III I ._ I I II I I. r 4 _ I ' I; r ‘ . t . I II . . . . I" |- ._ . _ I . I H ‘ ' 't- 1 -" ' . I __ _ l I - I I 1 I r. -I I" I I 1. . . -I___ I a. __' - —_ I III II , I 4 I : __' . I -_ - 1 IFI I p a I ... I I .- ' r- - = t ' ' - I 't 'l-I 1'. . I.. .II . ,_ I eafleapqndigg:m - a _- . . Wm“ f—‘f-f+a=atr}. my determine a Earti'e'iilai'aeljitieti etEq. {13) in terms at an iategI-at we use Est (iii- lia'w': ' I e' te‘ It" We): Wte'ne'te‘fitn = 6' tr + it! -e-' . 2' it + Eh" e‘tl Faetnring e' fruits each at the first use enlumns and e"' from the third enteritis. we obtain 11 r it Wei=eit r+1 allI- E1 r—t-F. I- Then, by subtracting the first raw from the see-anti and third [enlist ee hate [1 r l; Wit} = a in I 4; I] 2 n, Finally, evaluating, the latter determinant la}- rninnrs asrturiateti with the tirst winter. a : and that HI“) =: 4:“. Neat, {1 re" I'"[ may: a {l+1lt‘ --t""’i. 1 it + Eu" e"; I l manna“... Ian-sun: Him“! m 11;... H2} H m . 1 1' viii g”:- I--llfmryl Eifl#*r.i h f:}.. h. F I 'r" "L i 1" 3H HI * f " "iltfld'l- ~ +:,,-m.um1+d unqu-mdwflllflfi 1. .- i'- ~- 1-.I||-..-.Ju: £111.. uni rrqunmn -.:.. h r'“ i 1'?” +' F ‘1‘: rd .-L.. I 1| mluimn Hi "it Will ' .|. 41 -.l'- 1' I“. " ITHL'IIH'”? :_ .'i i-l. | _ 1-. 'r1ni'lll‘ 1' 1- ' I I f 'I' .. +1 . ” r 7;: .' In. 1 .1. I1. “1' I'LIIH'; up: '1' Hunting?“ I. {In-J til-I: HJI’HIHH‘I "I I“ I“ _. I'IIHI _|. FIJI'I'H-H'1hr- L'HIIHhH'I ' p I”) a- I r m: -- 1' +1 1. p, I “n, n H”: n I” r' I ‘r r r -- mu. um -- .‘ r m. — H. Flt]! u- ---I. I‘M." II I r I]: v-ml. Hill-+1. Hill... I. rfllihiI-I L' t' r ain't. rum-'1': —r- 1'. “till -— I. I'Hfl'r Hr -l h “mama: I.I",nml m .m mlumnunnhu hump-mm -- - 1- 1: n. In”: firth—.11. Iwfl. Ik‘tifl‘m'ifll I fillih‘ulfl mlulhm I _ H. I'm-d 1 human HI'm-lflflfl Ifllllhlh In! in 9mm: m 0! III _ L. . r" r‘ trivia-.1”- ...
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