CH01 - u '7‘ Autunumnus Systems and e; stems 5m...

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Unformatted text preview: u '7‘ Autunumnus Systems and e; stems 5m IntTOdllCthl'l “.4 (trumpeting Species 515 *Jfi PtedstnrJ’rey Equalinns 523 |_,ittptinttt"s Secttnd Methnd 536 it? Perindic Sulutiuns end Limit Cycles 547 its (“linens and Strange Attrticturs: The Lnrenz Equations 553 Partial Differential Equations and Fourier Series 559 Htl 'l'tt-n—i’nirtl l'h'tunthtrjt- Value Prnhlents 569 ML? Ftturit+rHeI'ies 5% NH I‘ltei-tuuriel'{'nneereeneeThctit‘ent 537 tit—1 liten and field i‘llt'IIIIit'tFIH 594 “this 1"'rL'|"+.tl'LtllHFrHi ‘t’uriuhies: Heat ("tinduetittn in a Red 603 lth {Illitei iii'itl {I 'ttl‘rtitJL'tittn Plul'tiet't'lfi III? Ih- We. - r Lititi'itTli' ‘t ihiernuus u! an Elastic String 623 iii "i I I-1' I-'-' i_1'liri'.| "" {IJH . * _ In this eha ter we I in v ' I ' - - me. i f.~ Heiv .tllttti u; the Hex-tit ._utLtuetien Equatlen 649 (THE t. r , 1? 5E Era] different “HI-*3 “3' 2-1“? PefiPC-Em'e In 301.1! stuet e-t _ 1 . r . , H I- I i H. m H . iqmlinn fl,“ 1 TE“ 13 Equallflns First.we use two: problems teillustrate scene of the haste-idem rJI_r i_!.- '1 I L i“: ~t [:1' , '._ _ L _ __._ .. _ _ that we mu W’th m and Eiabflmw “P0” [TEQUEflliF lht'eugheut the remainder at the banks Latenwe indicate several ways ef classifying cqustiens in etder tn preside urgentzattnnal structure fer the beek. Finali}: tte eutline senme cit" the major trends Chapter 11 III-L IJHd r1; 1'.i-I'itt Fr-.i_tiett.-s [i537 mg. 1 I I “in I “I! I m- l“ 1 Pum- Hmmdm hm: thlemfi 657 in the htstertcsl devetepment of the subject and mentic-n a tee eut the etutstsndtne .. I I] , _ _ _ I _ _ ' mathflmfltlfllflflfi Whfl have eentrihuted In it. The studs ef differential equsttem this it sun .11, r .41: tile mun-thus Hill-'6 FINIJIL'IW fifi-‘J attracted the attentien et' mans ef the world‘s greatest mdthimdlifiiinh dunner the HA t.1t..,t..,.,s..ld,_,m” l';l,L”-H_Lm. mu“; [ll-Hmcme (,79 past three centuries Nevertheiesse it remains :1 estimate fielel ett' iantrt ladder-s Eh ILL! hmeului HHII m- | inm-ille i’ruhieitts (395 many interesting GPEH Clue-slicing 11.5 Further I‘let.'il1ttl'i't*-ttll the stetlmtt nl' Septu‘stien nt‘Varit-tbles: A Bessel Serieslzsl‘stl'ishut 7H2 [1h Series tit th‘tlmennni i’unelinus: Mean (femeergcnce 709 1 1 Same Basic Mathematical Mfld8|5 D-TECt' t ; l IUD it? 3 Answers tn Pruhlernte 7'19 . . . . . Befnre entbarktngen a serious study ell dtlletettlml .-e.,_=.t-iti=tt:- t‘: -r e t. . e ing this heel-t er majer portions nt' it]. you sitt'tulJ heme sans. iete. :‘ee . ~ benefits In he gained [1}; doing set. Ft‘tt' sum-e students the emit-I» -. ._..e.:. subject itself is eneugh ntetit'atitin. but hit most tt is the tit-“him- . applieatiens In other fields ll'tttl makes the untlettgtsitte 'I‘In1'iih‘thil‘i‘ti- Many ef the principles er huts. underttint: the hehnt itu‘ nut the :1 tutu. .i'. t- - _ statements er relatieins insulting rates Ltl tthtch lhll'ttlt- were: “ten -_ e. . : in mathematical terms. the relations are equtttmns and the Lites the tte ::-..t1--.._ Equalitiescentsiningdcrh'sth'esaredtflerentinlequstium. [hetetttteretest-t . e- and to itwesligate prnbletus il‘tt't‘ti‘t'ilttl the nttttien ert' fluids the item etutfi . electric circuits. the dissipatien e!" heat in sentid t‘tbieets. the prnpseetttt In .H'lti :e e1 we: I Indes ?31 FII -I' ._ I I . _ - '_ ' "' ._ .' I Li" II"- I I I I " 't- -- . -_l a. .I - - - ' . l“ I ._' u u I I. i..- .I I - I I _- . . _. _ I. l___ I -- ' ' . - _ -l . _ I I I. I I I‘" '- . J — I - u . 't-. a .I_ __I - I I I — .- J I I 't— I' 't. I: I '1 I Chapter I. Intradrrefimt _____________.__—a——"-_-—-"_——-—I—-'-‘ cit seismic waves: Ur “1t? infinite“ “r d“? _ I ‘ “aliflm is necessary in linear scin'iething aheut differeniiia I PIDCESS is fine“ {Ballad a math- _ r ' ' 1 ' r - arcrihes seine p ysii.“ . iifercntial equatien that dcs I i _ emigmi mfldal “f the precess. and many such medcls are discussed thrciughciut this begin with twe mndels leading te equalities that are easy in heist-i. In this scetinn we I i i I . selve it is netewnrthv that even the simplest differential enuattens previde useful I“ nindels elf impcrtant physical princesses. O in Happiise that an ehjeci is tailing in the atmusphere near sea level. Furn‘tltlfllfi a differential mg - Liatiiin that describes the incitiiin. I I f I_ I l I. I i E EA M P L E th'ite begin by inirnduciiia letters [a represent i arinus quelfltlltLh that may he elinterest tt‘l‘lhts FIGURE 1.1.1 Freeway diagram at the In an a failing flbjmi 1 [irnhlemh ThG-llltlllttll takes place during a certain time intervttl.t-ti} lift its use i te cleric-[e tin-m Ices i ilie veliieily at the falling abject. The vetecity will presumath charm, mm [mm .4” my [hint- ._-.i‘ r as a t'unciinn at i: in either wards. r is the independent Tin salve Eq. {4) we need In find a functicin t.! = all} that satisfies the equaticin- Il isiri'ihle 'lllLl i' l's the dependent variable. the chnice nt' units nf measurement is semewhat is Hm hard m dfl this, and w .u h h . h - F m t i i is nnthine in the statement til" the Weillth 1’3 WEE-55‘ apprflprifltfl “HHS, e m 5 Ow ya“ aw In} E “a?! Sflcufln- fir E FREE“ 1 - . hewever. let us see what we can learn abeut seluttnns vintheut actually findiri g any Eme- rflm-meihla. Te be specific. let us measure time . . . . . . t L m: will mum that v is Waive in the at them. Our task is'siniplified slightly if we assign numerical values In in and r. but A Filllnfl Alert let us. usi.‘ t' [H represen {thiect ttl‘i'lllltll'lL and tits. re si‘t we are tree in ni.il.e any el'ii_iice that se iii secunds ind telneilt i in rnclci‘s’seci‘iiid. Furthe :j‘m mm ._1 almmm _ Mm h “hm they use.“ it. tuning - the precedure is the same regardless at which values we cheese. Se. let us suppose lit .- i'nti-.-- -l tea. iii ll ginerns the n'nitieni cit lll'ljiSElh is Newton‘s secund law. Which states that m = 10 kg and y = 2 [cg/sec. If the units ffll.‘ 1/ seem peculiar. remember that ye iii ii Ii];— i.'..1,-- -ii the tiltieei iiine~. tis eeici'aiiiin is equal tn the net fiirce en the cibjeet. In must have the units ef farce. namely. kg-mt'seci. Then Eq. {4} can be reviTitten 35 iii -|iu.. !.l'.".--i t-_i;iis iiiis lav. i~ -. -.;‘-r'.. It. the equatit'in d“ v 31:21...“er {1) 3:9.3—5. ii"! --.:i_. ~ Fi'1._ ..i -..- l.l|-.' -i| Its .-..-.v.t iit31ttll. and t“ is the net force exerted en the .... : .. ~. Is .,-..:i i-.._-_i. Hid t'.-|L_':siti:.' Hi in hiltlgfilll'lSJI in meterstseccindiand _‘ ... :.. .. til. ...=-. ._ . is: he : . - ..- 2'. ..=";- 'riiL set we can rewrite Eq. {1] in the term Investigate the behavinr inf sclutiens nt'Eq. (5} witheut selving the differential equatien. i' -: win its: {2) ass M FLE We will preceed by leaking at‘Eq. {:5} treat a geeatetrical viewpoint- Suppose that e has . . ... . 2 a certain value. Then, by evaluating the tight side at Eq. tn}. we can had the ccirrespnndii'ig in" ‘i- " ""‘ il’" 'l ‘t' ' "" ii“: "hm" “i ” '“Hi' (“H-“"13" excl-i5 '3 fflrce “Illa! i” value [if dti/rlt. Fcir instance. if ii : 41]. then dvi‘dt = 1.3- This means that the sic-it‘s” el a Il‘lt i.‘.i_*i'..".li ii: '31- -ir|i_'I_ ' :-i tit. “.“i'l :l '. .1._' l} il‘ti: ._li..'t."i.'|t.'l'fli tLtl'l dtlt'.‘ lit—fl grayity. [11 [1'13 Uflilfi WE A Sfllutifln U = UH] hflfi. the value at an}: pflinl “.hflre t. = “Ir: {an ihia lnl'flm-IJut-in hit“ Li"'“-- i- '-T " *' “is t -" li'=='-'="' -! L?'~t"'-t'l“=1*~"rt'-51”1- l” I“? itPi-m'lilmfllflll' Etill-“itl “3 9-3 WSW: Objgct graphically in the tu-plane by drawing shnrt line segments with sinpe is at seseral points -.'-1‘i T1L'.tl llIIu' c-Ii"il-' '~r|i| i'_'=_' ill-:I' Is .z'isii .i iitl't.‘-.' i.li_'i..‘ ltt Liil' rthlfililnce- Dr drag-i [hfll is mare t‘ Had) the line U = Sll'l'lllal'l}". I? = ll'lE'l'l t'il‘fiii‘ : -r{l.:.1ii= 1tili'ir.‘ Lli‘fi‘i'i' line litigfl'lill’lli 1tr‘t'lll'l dttf‘t difficult in It‘li'-_ii.'i liii:- t- [ti-l the iilace liri' an i.”~.'Ici‘ided dist‘ussiian til‘ the drag farce: suffice {EDD 111 run: at several paints {m the line I. 2 en “It: “bu-1m Flam“: L! 2 hi. p'I'LHJLCClll'lg", =ti in: ..-:-e H 1” hill “it” i! i" “5”” i"-*'”"-t"'-i “it” '13“? little it l‘i't'llllr“lLlnfll 1“ the film”)? and WE “in way With ether values cif t'. Figure l.l.2 is an esarriple dt' “that is called a directiiiii held i'i'itil-tt: lhtll assiii‘i‘il‘itiiiii lieie. l'liiis the drag l'i'ii'ee has the magnitude F [in whgre y i5 3 EDI-[giant Sflmflfimeg a Elflpe field. " called the drag ciiciticicnt. Ilie numerical value iii the drag ccicl'licicnt varies widely from cine ThE impflrtflnce D" Figure L1 3 i5 Hm cacti. 1H”; egg—Emmi it. it tangent line its :" iih|i.‘t:i inaiinther:sniimili streamlined iihiccis have much smaller drag enefficientsthan rnugh a salutifln Bf Eqb [5L T5111; awn limugh we haw mu [mind an} mimwm .1! ._.I hmnl “TIFF- _ _ salutiflng appear in the; figure“, Unit.“ {3th [lt‘tt'lL'lllL'ltJE-‘a Lll'fi't‘s it‘il'lli.‘ L]l.ldlll-ilt‘- _' -i tl._lU:'.."' :- '- t I [n viTiiing an espressinn [er the net tnrcc F. we need in remember that gravity always acts the behwim DE Sfllmmm*Fm “mum; if I. E Ebb Hun HULL”; mum ._ JUL .1 n - in the dnwnward tpusitivel directicin.whereas drag acts in the upward {negative} directien,as 5E mania have pflfiimf ginpfil Lind the speed cit the tailing i_ihi..._i .nereas- :. ., i.~.t'_. _ [Shawn m Figure HI]. mus . ether hand.it It is greater than the critical value. then the line seernenis its - is“; v I; = mg — ytt [3} and the fallingflhjefl tilfltti's dust. [‘I 35 ll. l'Jllit. Eli-hat its thiscrttietl t .iltie - ~ii 'iiei ‘._!. 11.:li. - i In e and Eli“ ‘2} mm hummfi whese speed is increasing [rent thiise whnse speed is decreasing.“ Reieiiiitg .-._-.nr -* - - - r “I. we ask what valufi. flf I. gauge; “lire-r“ [L1 l‘ic‘ £t.‘1't.‘LTlli. tlnhhlh‘ L‘t l'a-l. : t." it“:1 ir-I 4” t. m]; 2 ml" " W" (4] In fact, the ennsiant tunciiein it“ = 4'4 i5 ti ti—tifllis‘n ifl ELt- ti‘l- T” “i”? 17"” “"‘f " Equflmm {.4} i5 a mathematical mudfl m. {m m. h f, H. . substitute at” = 49 into Ell- (5} and nhsen-e that Ctldl‘l‘atdedfll ihe Cgijlaetilifiil Nu“: that m: mfldflt Enntlflnfi the threfi fiflaqlftmtjt.“ d ‘lng In the atmngphere [1331' 533 level. it does “[31 change} with llfl'lti'. lht': Sflltlllt‘tt‘l I'li'l :- 4'4 ll‘i-IL-Lllll tlt‘l Bill“ I flu-till I 1 2 I i 1* I ‘ SHLH'dnd i“ The “Unfilflnlfi "l and it depend seluticin that cerrespends ha a balance between Eras-it} altd dis—tit ‘ln Flaunt. “ _*“- directie-n held. l-inni inis tieiiie v-s ve rmuch tin th-i ' - "s * * . . - r} L particular ch}ch that is falling. and they are usually different far different the equilibrium mimic,“ 1.”; = 41) eupeniuia‘ised en the Chapter I. Introductien f Directien Fields. Dimmers-fields are “immaterial; in studying the admin Mm ential equatiens ef the-fern: - - ‘ . r recreate-35» cqmlilinum stilulten :1": l _f(£1?}j where f is a given functien ef the twe variables r and y. semetimes referred te as the rate functieu. A useful direclien field fer equatiens ef the term (6) can be constructed by evaluatingf at each peint ef a rectangular grid censisting ef at least a few hundred peints. Then, at each peint ef the grid. a shert line segment is drawn whese slepe is the; V311“: flff at that paint. Thus each line segment is tangent tn the graph ef the selutien passing threugh that point. A directien field drawn en a fairly fine grid aim a geed picture ef the everall behavier ef selutiens ef a differential equatien. The censtructien ef a directien field is eften a useful first step in the ineestigatien ef a differential. equatien. Twe ebsenratiens are werth particular mentien. First, in censtructing a directien field. we de net have te selve Eq. (6). but merely te evaluate the given functien fit. gel many times. Thus directien fields can be readily censtructed even fer equatiens that may be quite difficult te selve. Secend. repeated evaluatien ef a given functien is a 10 i task fer which a cemputer is well suited. and yen sheuld usually use a cemputer te draw a directien field. All the directien fields shewn in this heel-t. such as the ene in FIGURE l.|.1‘- :‘5 dim-1m” 13”“ ii” Eq- [51* Figure l.1.2,werc cemputer-gcnerated. itiilliilfxrzrxx shtiliiliiilllfffrrxx sstttttliillliffrxxx; \Htktllllliiffilffrxr \xtttttlllllllixrzr/x silllttlllllfffxrxxxx "."s‘ti'.L't\"1l|liiMilitia“!!! _asttll\llllllilfixxxr m HKKXKXllllIiliKKI/ffr E: Field Mice and Owls. New let us leek at anether, quite different esample. Cflfli‘idt‘l‘ " a pepulatien ef field mice whe inhabit a certain rural area- in the absence ef predaters we assume that the meuse pepulatien increases at a rate prepertiena] te the current pepulatien. This assumptien is net a well-established physical law (as Newten‘s law ef metien is in Example 1). but it is a cemmen initial htpethesisi in a study ef pepulatien grewth. If we deuete time b}; r and the meuse popula- tien by pm. then the assumptien abeut pepulatien greu-th can be expressed by __ the equatien _ . __ . '__ __ _ _.____'_.__:: rip I ' til _ ifffxxxxx illiffx/xxx ll lfii rp. where the prepertienalittr facter r is called the rate censtant er gruwth rate. - specificisuppese that time is measured in meutbs and that the rate eiutslui'e ." -. 2 ;. value Djfmenth. Then each term in Eq- lit has the units e't l'l'llCt.‘ menu New let us add te the preblem by suppesieg that siteral curls ii'lt: m 1:... ueighberheed andthattheykill iSI‘iele‘ mice per day. Te in'lt‘lt‘pi‘rTuL- l‘ s ll'ilt' e: -' . -‘| '. " 1.,l iute the medel. we must add auether term tn the differential L"-.1-.i...'ltil.i'-."l -. ~. . . \Kiiilill ttttttlll si- F'IGURE 1.1.3 Directien field and equilibrium selutien fer Eq. (5). becemes I LE :zlliri—-45H " if! The appreach illustrated in Example 2 can be applied equally well te the mere Obsflwfl that the predfltmn term i5 “:50 rather than _15 haunt um: i, 3.... Jim _ ~. general Eq. (4}. where the parameters in and y are unspecified pesilive numbers. The in nmnths and ma mflmhl}. pmdminn rah: i5 needed. results are essentially identical te these el‘ Example 2. The. equilibrium selutien Elf E9- (4} iii NH = "Le/1t. Selutiens helew the equilibrium selutien increase with time. these abuse ll decrease with time. and all ether selutiens appreach the equilibrium selutuin as r becemes large. 'A semewhat better medel eE pepulatien greuth is discussed in semen - e Investigate the solutions of Etl- till Erafihlillui} 1 l I 4 Fm. fiufficigntly large values ofp it can .45.. dii'ectitin field for Eq. [til ts shown In IEU L . . - fl is pushiin 5” that snlulinflfi . . . * 'tscll. that Elm" . . . lie seen from the hull”! “r till-“"5 “um Eq' {8} i is the case. Agami the critical r -:-- = iiptisite mum-M {in mu “m” mud! [in Elma“ lamina “(glitflmhi'llll‘l'ifl' that decrease is the value ofp for I . . _ 1, ' y i * ates solutions that Int. value not p tllat scpur . .- _ i ' torp we findthe _ . . . Eq {.5} and then solving . ._ - . _ .L Be-Hfinmorlpftlt uqualtuitcrfllfl mm [Tm inE ' 8- “hiilhliiiiiiiiiiiiliitriiin ,«im = sit“ for which the growth term and the prcda e q [ ) eqtll t "in . L ‘ l r f .-""" .e f r" 10m r r "i a: x I r“ "' I" If I f It" I‘- J - Hr"... -" if I". It! I r’ r’ ' a H .r’ r" “I _'__, Ha- I." ,p‘ a"; if f If... .a' _.--" _.--"" r“! If "Ff “I .u-I' __.--""' 3"" aid-IF. if Ill-FF.- Ilq-C .-r- I" I "F" _ J I I . _,..-' __,..-a-" ,F-d' _,.-" ..-"" -""" f, r; : .a—" ..-—-"' .-"" "'"- f...- FP-I- H :d- :..--' __.—--' ___--' _.__a- ___:—- ._—- --"" '-""- '-}i'_it:t __ _ __ H -—-— --..__I h___ -—.___ In... :H- : ""-n.. ""--. ""'--.- ""-- ""-s """-In. _ i— ._ .___‘_ a.____ -._I_ --...___ --..__ '--...._F till" :5 h: -_ '1.__ "-5.5 “h. “h... H a i '- ‘-.i_ "s. ""s. ‘H “H. Eh. H‘s. i i a. -.. s. *s. *s. ‘s. i s “s. “s. “s. ‘s. j “. ‘js. “s. “ls. “"s. ""s. I l I i d F "#'_" frflfiri‘F .3 a 5 i t it. t tti l t [Hr tort Iieltl and eqmlibrtum solution tor Eq. {is}. I ninp tl'lll'; l- --.-+anut..-~. .- out tie note that in both cases the equilibrium solution I=.Jt.|li:~ lllL'lI_'.I -t|i'_‘ ll'tll-ll -.lu_'~._':'t:.ltsitl:t tstjtlllllt.tl'l‘i. [n Eli'dl‘l'lpltfl 2 flll'lti'l' SfllUllflflE Cfll‘l- y eree lit. 11| site .t'. l.l.I~._lI._'|.,_l tit-1th.; L-quiltifirtttl‘tt stiluttt'irt. SL1 ll‘tt‘tl Etllti'l' the {lbletll fflllS far enouulr an oliserier u-Ill sec || mutiny; .tt iery nearly the equilibrium velocity. On the other hand. in l-Lsample 3 other solutions diverge from. or are repelled by, the equilibrium solution. Solutions helittie very Llil'l'erently depending on whether they start above or heluv. the equilibrium solution. As time passes. an observer might see populations either much larger or much smaller than the equilibrium population. but the equilibrium solution itself will not. in practice. he observed. In both problems, however. the equilibrium solution is very important in understanding how solutions of the given differential equation behave. A more general version of Eq. {it} is (a; where the growth rate r and the predation rate it are unspecified. Solutions of this nuire general equation behave very much like those of Eq. (8). The equilibrium “limit-l“ “l Est {‘9} i5- pltl = Mr. Solutions above the equilibrium solution increase. while those below it decrease. 'tfou should ltcep in mind that both of the models discussed in this section have “ll-i" “'“llililfln-‘i- The mndfll {5] of the falling object is valid only as long as the PROBLEMS shine?! in l"militia"fluent."trittlmiu.e.rasimmering ' um mg m - m (S) Eventually Prelim” negative-nitration ot-mie‘e (ipr e m). or anomaly my, numbers (if p- as 900}. Both these predictions are so this model tie-norm unacceptable after a fairly short-time interval. Constructing Mathematical Models. In applying differential equations to any ot the nu- merous-fields in which they are useful. it is necessary first to formulate the appropriate differential equation that describes, or models. the problem being investigated. In this section we have looked at two examples of this modeling procem. one drawn from physics and the other from ecology. In constructing future mathematical mod- els yourself. you should recognize that each problem is different. and that successful modeling. is not a skill that can be reduced to the observance of a set of prescribed rules. Indeed, constructing a satisfactory model is sometimes the most difficult part of the problem. Nevertheless, it may be helpful to list some steps that are often part of the process: 1. Identify the independent and dependent variables and assign letters to represent them. Often the independent variable is time. 2. Choose the units of measurement for each variable. In a sense the choice of units is arbitrary. but some choices may be much more convenient than others. For example. we chose to measure time in seconds in the falling-object problem and in months in the population problem. 3. Articulate the basic principle that underlies or governs the problem you are investigation This may be a widely recognized physical law.such as Newton's Ian of motion. or it may he a more speculative assumption that may be based on your own experience or ohse ri itions. In any case. this step is likely not to be a purely mathematical one. but oil] require you to be familiar with the field in which the problem originates. 4. Express the principle or law in step 3 in terms of the variables you chose in step 1. This may be easier said than done. it may require the introduction of physical constants in parameters (such as the drag coefficient in Example 1} and the determination ot' appro- priate values for them. Or it may involve the use of auxiliary or intermediate variables that must then be related to the primary variables. 5. Make sure that each term in your equation has the same physical units. [1 this it'- rtoi 1h..- case. then your equation is wrong and you should seek to repair it. It the Lilli-ls .iei. .- :ti. your equation at least is dimensionally consistent. although it may has e other sh. u: tiles that this test does not reveal. ti. In the problems considered herethe result of step 4 is a single dit'r'eresiiaii .1': in. :- constitutes the desired mathematical model. islet-p in mind. though. in ll r. i_'. i problems the resulting mathematical model may its much more e'i-ti"':'ll- *' involving a system of several differential equations. for -‘..“~..rlt'. r-le In each of Problems l through ti drain a direction held for the insert thinner-at Based on the direction tield.deterniinc the heliaiior ot t. .tst m- s... it not iteit..i m.- on the initial value of v at r : ll. describe the dependency. l. l." =3H2l' g“) 2 1" : :1 _ I: 3._‘l"=3+2_l' it '1 l :_l_'l i i"=l+2v n "Il': mlutinns have the required behavior as r -.+ ets I I. I "hm 3-. All selutit'ins apnreaslu' = 2d?“ 7. Allselutiensa reaehyzli. ‘ I _ _ II in up; PF 10. All other EDIHIIDHE-dIVEng'fiflm-y-g; 1.13;. ._ it. All ether solutiens diverge frem y --.= 2. _ 1'3 rm "7'3 git-"EH differential-_qu [its a- In each at Prehlenis ll threugh 14 draw a direetinn fie _ _ _ _ _ .3 a; I It. Based en the direclifll'l fifiidtdflflmmfl the banana: a” a“ 4 m' H Ems bfihaflm niiti‘ijiiiiii-i': '1 i - :5 _' - ' . " -- - - . i: I- eat-51:: an the initial value aty at r =fl.deserihe this dependeflflt- “a” "13”" these tit-“blame - 35;; ._ it _ =2 .. . ' ' _- i. _-: - . ...:_' Ivy Tr. are a- equatiens are net at the farm y“ = try + b. and the hehavier nf'their sniutinhs a all... .- equatiens. same at tvhieh predueed the direetififl. _ . | a! mere eempiieated than fer the equatiens in the test. - is; _I I I I ’2 11‘ y: = — ’2 12- y, = '— -_‘-: I I I l j. I ;-_. : :‘1- $.11 _ ’2, y’ =y1 ‘2, Jr" =yfit-2} - l l 5'" -. é xii-hf,”- II - ' .- - -_ Cnnsider the [allowing list at differential I ‘ fields sheivn in Figures l.l.5 threugh till}. In each of Problems 15 threugh 2t] Identify-magi: differential equatien that enrrespnnds tn the given direetien- field. I I. J I I a. l I. . "I1 {at v' = 2y — l {h} i" = 2 + 5'" __ h;- is] _v' = _v — 1 (d) r“ = set + 31 . flit. tel y 2‘ yiy — 3] {f} _t" = l + 2y tie—:1; [gt _v' = #2 v— _v {h} j." =yi3 -,t'l .: til _v‘ = 1 - as {j} y“ = E - _v ..---:;.’3: 15. The direetinn field at Figure [.15. - In The tilreelnin field at Figure l. | -fi. l't'. The Lliret'tinn field {it figure it]. IN. The direetien tieid L'l'i Figure LL55: '21:;- |‘-t. The 1..ill'l.’l.'ilt"li"l held I.‘Ii Figure tit]. 3|} the tlireetirin heir: el Figure i.|.'ii.i_ ‘ . K... _I A term! “mull; eminent |.I.tiiiJ.:H_l’lgttl til water and an UI'Ikl'ltJlF-‘Il amnunt [if an undesirable _.j.- ' I " ' 1" m f- . ' ' ii I sirenntal e. .tlL'i' auteur-mg ILLii gram at this ehemieal per gailen flaws intn the pend at a .9 '- '1 5' i 'I ' 2 rate at _tttlignIthr. The misture finivseut at the same rate.se the amount nftvaterinthe pend I ‘T'i .1 fig; "are e at it remains eenstarn. Assume that the ehemieal is unifermly distributed threugheut the pend. . ::-- a? ' ' ta] Wr'tt' d'Ti '- :- v ‘ ' - ' - jE§€EEH$~HHHHH l t. a ti trential equatttm fur the autumn at chemical in the pend at any time. "i -- "it." $1M} E E a a {tit himv much at the chemical ivtll he in the pend after a very leng time? Dries this E ii limiting amnunt depend an the ameunt that was present initially? . Qt: .. _._ _ .. .- .. _ I: 31:: :1 :1 3: 1'1 1‘ r H '- l - I. ‘- l i I I I -. r '1 L _IP- _I- _ -- Iii-1. :- I -I.— H I. - . i.- _ I... a... A spheres] raindrtip evaptirales at a rate preperttenal in its surfaee area. 1lithrite a "differ— . '“Iie. - '5‘ J ' " 2- - ~' ‘F if! I“willEliilltltmrn fer the vnlume til' the rainan 35 n fanetien tJf time. I ' v - stilt . --"I-a.llt=- 5- f r r r r r r '.r i 7 " . 1. * . ,. . I 1 v ' h Fe ..3. hiewtnn s law at sealing states that the temperature at an nhjeet changes at a rate prepar— P by - 19 man fie: fur FIGURELLIB Dlmnfln new i“: [Trial tn the difference hetsveen the temperature at the abject itself and the temperature ' ' In em ' habit“: *u E “i fufiflurldmss (the ambient air temperature in mast eases]. Sappese that the am- ten em erature is EFF ' - — - - + Equalifln PM [he lempflmljfd 1:1: thilrate eanstant ts 0.05 (mm) '. Write a differential i2, 25 Fur small slnwly falling nhjeets. the assumpt' n d ' th 1 t i h d i ' . an an lectatflny “In: - . I -. as me e tn e es tiat t e rag .erse .s 14. A eerlain drug is being administered intravennusly In a hespital alient Fl 'd t ' ‘ I I Pmpuniunal m the mac“? ii a gm Una Fm lawn mu": rapid“ Mimi: Dhiflflfi n “1 ii myemi at the drug enters the patient's hiendstream at a ratepef lflfl‘emilfiirciihjlt‘iiiig I mare accurate in assume that the drag rum is Pmpflrfiunal m mt bqugf m- mfl WHEN ts nbserbed by bad? tissues er etherwise leaves the hiendstream at a rate-prepertienal IE (a) write a differential Equation fur the film“? M a falling nhjfl' “E mail!" m Hi L Jr" ' ' fnree is prnpertienal tn the square at the veleeity. the ameunt present. with a rate ennstant at {1.4 (hr}“. (a) Assuming that the drug is always nnifennly distributed threughnut the hiendstream't write a differential equatinn fer th E H 1 ' r . at any time. mflum “i "1'- drug “131 15 Present tn the hinedstreatn- (b) Determine the limiting velocity after a long time. lb} Haw mueh at t ' - he drug Is present In the blendstrearn after a lung time? - 2See Ler N. Long and Edward Weisst “The 1't’eltant'tr Dependents: Int items-name Drag. a. Primer tar . - Mathematiflannfdmerieen Mathematical Monthly we {19%}. 1. pp. lELtl'i. to .. Incas]: at wehlsmm'th‘ ,;:_.',,_f. - .' Hamil on the direction fields-MW” ' - __ _ - - - -' this _ __ _,. . _. h- . , : "" ‘ at l = '- .. -. h- - - .- i - :- tr'Ilil I '1i 1:: ‘-:.-.i an {as weflIflI-II: mfirflfflmlhflr mluimnsfirflfl EMEmfl-[E .. . I I I I flee.y=—a+r—y I I I 28.Iv'=e-"+y I — III_ ‘I g 30. y*=3sinr+l+_v ’2, 3i. f—Zt .1 fl 3:. Iv'=-*{2t+y1!2y (2, . _I_-_-_—_l-—-l-l—-'-'_ _-__——'I—-""-—-_— ' .._.._—-—-'-'"-'-'- L2 Solutions of Some Differential Equations M In the preceding section we derived the differential equations do rrr—I- = mg — Y“ I and If {I .3 | ---— = rip] ""‘ A: El rt ' tr; 1: Eclttnlfl In I' | t models a lizriiing {Ili‘IjL'Ci 21nd Ettt i2) 3 Pflpumtifln flf'fiflld “.11 he twglrm [Ltih these ULJUI-tiifllifi ii“: {If the gE‘flEffli form I If '51:" II r v -'— = rrIv — ii, iii." {h j:_ -. r-I- 1-. a. -|. where n and t: are given constants. We were able to draw some important-qualitat conclusions about the behavior of solutions of Eqs. (1) and (2) by consideringzrfi' ,_ I associated direction fields. To answer questions of a quantitative nature, however; I1 .' . we need to find the solutions themselves and we now investigate how to do that! I.“ ' [. - _ -':'Ls,'-; of Eq. (I 1} for seve values of r. - I 'J I . . li.___ I r f .4- -- - - 3*. t ' Graohs Consider the equation --,I .j . examas tr_pI =fl5III_IISfl I:-I I _ . 1 H" .I _ ' . -- NQIEmEtstHE-y maths character inferred from the deem ficid in Figure Lt For F. I . which describes the interaction of certain populations of field mice and owls [see fit]. (3)01?" iii 3 i i ': infinite. solutions-tries flfl' Emmi Sidfi' Bf “1E equilibrium Wuhan P = “1' "find “1 dltflt-"f told MICE . . , , - - - - . __ . I I n'ndflwis 5mm" Lu“ find sululmnfi “m” equalmn' i: - from that stilettos. {In} 1'. III To solve Eq. (4)I we need to find functions pm that. when substituted into the-equating; s'I__III*' . n in“ } reduce it to an obvious identity. Here is one way,' to proceed. First, rewrite Eq. (4.) inthe rennet . s ' fl 1*?1l . rs ' - J” p ".900 if I " ' Int-Example '1 wefound infinitely-many solutions of the differential equflltfl-‘I'I t4}. r m 2 i corresponding to the infinite}: man}! values that the arbitrary constant c in Eq. t" i 1] entry 51h 9m, -'usve. rypieal of'whnt happens when you solve a differential equa- dp‘f‘fl a. .1. -'tio'n-i Tire'so'ltttiontpnoeess involves an integration. which brings with it an arbitrary p — 9m _ 2 constant, whose-possible values genot'ale- e‘n infinite family of solutions Chapter I. Introduction _-_-d______________—————'_-'——_-_—‘——-fl-'-"'— on a single member of the infinite faring}-r . t t n. we - . - b t t, staining the value of the arbitraryr constant. bios; gefisfllufifldno m hummus .‘i ‘3va instead a point that must lie on the grap h I Ir lfldll’tfitlll} by“ Bill?“ hU-mnsmm F in EL]. {.1 1L we muld require that t_e{popu_ align. esaniple. to dc tetrnthlt 1 firm” mm: Such as the value list] at time t‘ —- . rt other hm: a til-ii“: iiitflhuiit‘ilni solution must P355 [Waugh “‘3 Pmni [U‘SSOJ' Symbflhcflnyi wot s. 13.] i ‘ we can espress this condition as . a _ . _ 1 Frequenllv. we want to locus our attenttot pill} = 35L]. (12) Then. substituting: : tl andp : ‘lfitl into Eq. t II). we obtain Sill : dill] + c. 1.1 [mg mine in E4, {1 l }_ we obtain the desired solution. Hence c = —_‘it 1. and by insertit - n1 "it. H _,_ I “J L I p : Llilll —- I‘llt‘r '. ltttl we used to determine c is an example of an initial "I ‘._tilillillliillELlIlElllll'lil l lf'll ‘ I I _ I i 1 1L 1 on {4] together with the initial condition (12) form condition. The differential equati .in initial value problem. \- W “Hamil-.1.“h.[HHI.C._iL-n€[;1| Prrfil‘ilcm flint-timing till. lhE‘ diffflI-E'l'ltlfli equation I I n i. ' T r. rl't -— : rll' — 1h ti'lltr I'MJ' 'i -!| ll |_tt"llil|r'llt1 ..' 1II:I._ . .-.. t . . . moat .'.:._- -. .i n anti e this problem by the same method t l l- - i ti 'o'. i - tnen we can rewrite Eq. [3) as vii-gttt l—— : it- (15) i' — ihl'tt’l Eu". integrating both cities we find that in ~i — iii 'n'il = or + {I (16) where t' is arbitrary- Then- talting the exponential of both sides of Eq. (16) and solving for _t'. we obtain _t' : {hint} + tie'". (l?) where t' = its" is also arbitrary. Observe that c = ll corresponds to the equilibrium solution _v = bra. Finally. the initial condition (14} requires that c : _vn — (bfnl. so the solution Di the initial value problem l3}.(14] is .' = [life] + Lit. - tbfaile‘”. {13) The expression {ll} contains all possible solutions of Eq. {3) and is called the general solution. The geometrical representation of the general solution (1?) i5 311 infinite family of curves called integral curves. Each integral curve is associated With a particular value of c and is the graph of the solution corresponding to that value of c. Satist‘ving an initial condition amounts to identifying the integral curve that passes through the given initial point. EXAMPLE 2 A Falling Object (continued) T” [film "he lawman-(13} tfi-'Ea+.(3)iwltinh mneetsthe field mouse" pm ' ' ' “ WE Hilfid fl r b thfl Predlafiflfl ma #— fl SUIUt-lflfl . . . _ _ P = (H?) + [Po - (kirllfi'fl. (19) where pa is the initial population-of field mice. The solution (19) confirms the con- clusions reached on the basis of the direction field and Example 1. Hpo = It I r. then fromiEq. (19) it follows that p '= ll: / r for all I; this is the constant. or equilibrium. solution. If pa 9E kf r. then the behavior of the solution depends on the sign of the coefficient Po * {it/r} of the exponential term in Eq. (19). If pa :- kfr. then p grows exponentially with time t; it pi} e kfr. then p decreases and eventually becomes zero, corresponding to extinction of the field mouse population. Negative values of p. while possible for the eitpression (19). make no sense in the contest of this particular problem. To put the falling—object equation (1) in the form (3). we must identify a with —r for and b with —g. Making these substitutions in the solution [13). we obtain in = (weir) + [L‘n — tinefvile‘tllm. (10) where an is the initial velocity. Again. this solution confirms the conclusions reached in Section 1.1 on the basis of a direction field. There is an equilibrium. or constant. solution it = org/y .and all other solutions tend to approach this equilibrium solution. The speed of convergence to the equilibrium solution is determined by the exponent —y/m. Thus. for a given mass m. the velocity approaches the equilibrium value more rapidly as the drag coefficient it increases. Suppose that. as in Example 2 of Section 1.1. we consider a falling object of mass or = l” kg and drag coefficient 1.! = 2 kgfsee. Then the equation of motion t 1 l becomes E = 9.3 — tit Suppose this object is dropped from a height of 3th or. Find its velocity at any time t. How long will it take to fall to the ground. and how fast will it be moi irtg at the time of impact ' The first step is to state an appropriate initial condition for Ea- ill }. The word “drape to the statement ofthe problem suggests that the initial 1~e.'ocit_t is rare. so we will e-.— rli- o .:- 1 condition 1| :— llll :1 -— l’lli‘l : ll. The solution of En. {Ellcan be found h_v substituting ti e t-‘tiiucs oi the c .-._-t=-'t--.._tn=~ solution {20).but we will proceed instead to solve Eq. til i ditectl}. Fen, rev -' h- tit- . ..a .~. - .i as ilt' Hill] 1 t‘ * 4"? _ 5' By integrating both sides we obtain init'—-lql=—-+li__. t.__-' and then the general solution of Eq. [Ell is _ 4. .-l r1 :~J:“--l".-t‘It‘r . |._"' 1.“; '1'" I . _ . -'.s . . . _ - ___ __ l__‘_ _ __ MM 1-" .. . lines: I- “h- 1 '_: EU ' we substitute t = [l- and u s: i} from the initialm .r' i c c. t -- .'_ - _- n=4llll -f‘ ' - ' ' at an sitivc time (before Ir. Equation [26} gives the eeloeilv of the falling object 1} p0 I}? TEE - are shown in Figure 1.2-2.with the 5am- . : vident that all solutions tend to approach the equifibfl :11.- ; ached in Section 1.1 on the basis an; . ground. of course}. {iraphs oi thesolution {2 I i ' '- ‘ 'urvc. Itse [It’ll shown by thi. heavvt. I I r i nilution e = 4v. This confirms the contlusions we 11. direction liele in Figures ill and 1.1.3. 5) for several values oft: .- ._w—*7_ e---.e-—er-"if,_...er--r“ {10.51. 43201-1. IIIJ-f/ f. :- Jit...’ 1 _ i:I_Irlitji,] -. _..'_ . _ ..- . - Mme._._|_.._L___I__.-. l _I: -'] i. 8 1'3 tlt . tJttl-I 1.2.2 i_rt.Ii."lI'-. n; the solution £25] for several values of e. '|'._. Imil the telnent “I the IlhleL'l v.'hen ll hits the errjiuntl. we need to know the time ill which irnrun't rit't'lll's. In other curds. we need to determine how long it [sites the object to fall Hill] [11. in do this, we lltllL' that the tlistnnce .'t the Ul'tlL'Cl has lullen is related to its velocity ti h}; the eqttnttrin i- ~— rltrrlt.trt it's: = is l - , ill I E T t'rrnsetluentlv, by integrating both sides of Eq. {2?}. we have I = 4t1r+ 245s“ + e. {23) where r is an nrhitrarv constant ot' integration. The object starts to fall when t = D. so we know that .t 2 ll when r : ll. Frtint liq. {2H} it lulltlws that t' = —E¢l§. so the distance the object has l'nllen at time t is given h}: .l.‘ = 49: + 245st" — 34%. [29) Let T he the time atwhieh the object hitsthe grtiunditltenr = 301'] whenir = T. B)! subfilllUllflE these values In El]. {Ell}. we obtain the equation tar + 245i:- M — 545 = o. {30) -49. Then the solution of the-initial value p: _ jg}; E. _“i'j _ :IL lei: I: ' I '1' , - '."r:.I as s.— at; - e- - _. ._ . . ._. . . Willi flbseflflfiflflfitflr'expcfimmfil melts. Wis-have no actual damnation; or capa- .~ ~. fittest-sienna * imflmal result-'5 “3 “3-3 fer-WNW beret-hut there are seven} more at possible In the case eta-the object, the finderljing physical mp1s (Hemline law or mill-fin“) WEHJ'ilSEflblifihfid and widely applicable. However. the motion that the drag force is proportional to the veloufi‘ty'is t certain. Even if this assumption is correct... the determination of'thc drag-confident y by direct mmt presents difficulties. [camisoles-lineman finchthedragmfim‘ ' at indirectly—for crampk. by measuring the time of fall from a given height and then calculating the value oi in that predicts this observed time. The model. of the field mouse population is subject to var-ions uncertainties The determination of the growth rate r and the predation rate it depends on obs-en} tions- of actual populations, which may be subject to considerable variation- The assumption that r and k are constants trial.1r also be questionable. For example. a constant predation rate becomes harder to sustain as the field mouse population hes comes smaller. Further, the model predicts that a poptuation above the equilibrium value will grow exponentially larger and larger. This seems at variance with the be— havior of actual populations; see the further discussion of plantation dynamics in Section 2.5. If the differences between actual observations and a mathematical model‘s pre- dictions are too great. then you need to consider refining the model. matting niere careful observations, or perhaps both. There is ttlttttiltil she sits a tri-stlet'it‘tr been een .it- curacy and simplicity. Both are desirable. but a gain in one usually. int-filters a less .n the other. However. even if a mathematical male! is inetnttplete or sunset-shat nu..- curate. it may nevertheless he useful in explaining qualitative features in the er a tee:- under investigation. It trial.I also give satisfactory results under sore. etrettiitst. but not others. 111115 you should always Lise gt'l'fld _itttlerrieet .enti coalition s-.- us. i“: constructing mathematical models and in using their prettietions. fl 1. Solve each of the following initial value problems and plot its. solutions Mt s. i . :.-~ a of vii. Then describe in a few words how the solutions resemble. ti.ntl tirtier . .‘ flll‘lfif. ta) {1'}?de = —y + 5. villi = _‘l'o (it!) tillde = "*1? + 5. _t‘ifll 1* ‘l'n {c} tl‘t'ftll' = -2;v + 10. villi : _l‘o TE fl 2. Follow the instruetions: 4. Consider the differential equation alt-Mt = try - h. i . ._,_ it. Use the method ol' Problem 5 to solve the equation d't'frlt = —tf}" + b. T. The field mouse population in Example 1 satisfies the differential equation _i_. emit: = usp — 45o. ' - {ill Flnd lhe time at Which the population becomes extinct if pill] = 35ft - -- ; {h} Find the time of extinction if tht : punvhere t} e on e. 90!]. I ‘ {c} Find the initial population pit if the population is to become extinct in 1 year. it Consider a population p of held mice that ‘ ‘ I grows at a rate Pupulfllmmsfl that {1pm, 1 mi proportional to the current 4' in} Find the rate constant r if the population doubles in 30 days ii- lb} Find it if the population doubles in N days i 3. Consider the differential equation for Problem 1 for the fellutvittgitiiti' ' (in dyftfr = y - 5-. still =' so is) try/e: = 2y —- s. yell} = u. {c} dvfdt = 2); - lfl. ylfll = J’s dyfdr = —-oy + bi where hath o and f: are positive numbers. {a} Solve the differential equation. d ' I 1 solution for several different initial conditions. i. it increases. ii. it increases. iii. Beth o and it: increase. but the ratio bfo remains the same. in} Find the equilibrium solution v... I H I '- tht Let i’iii : i- — is; thus You is the deviation from the equilibrium solution. mails; differential equatitjttl satisfied by l’itl. :31,” I - l- Llndetermined Coenieieots. Here is an alternative way to solve the equation '5 -: a -' sir-'o’i 2: div m b. I iitl hr'il'ri. 'ht -llTlPlL'i I:L:L't'ilt.'I'l til-ll” '—: fl'jF. ‘ .1 IL'_ .ill Iii-r ‘t'ii-lilttl'. _i'- li'l dis-.1“ - ‘ ' . L5 J ' _ ‘h' ' m " i' “‘1” “‘5 Uri": diffs rcric l‘-.'l".t.i.'1t:fl Etta. ii i and in) is the constant hf;- refills-[m TL. -' I 'l'herel‘iui ii roof. set-tn resistant-title tei assume that the sitlutiens of these two EqUflfifll-Jfig. " .ilmdiili r trill} h}. a constant. Test this assumption by trying to. find a cflngtafltk Such 12:."- _t' 2 t':tft s l. l‘s u Stilulitin til Eq. [ll _ i in ("ampute- your soluliun 1mm Part lbl With 111*: Sfllutiflfl given in the text in EC], (NJ-set?!” Note: This method can also he used in some cases in which the constant it is replaced'bllihir .i- tuncmm W L It dfipw'di‘ “‘1 “him” TU" ‘33” gut-‘55 [he general form that the solution-iii?! *-" .l liker to take. TI'Ii's method is described in detail in Section 313 in connection with aeeoti 1‘11??- order equations. - '. t. if "r‘t j- M. (h) It stat. -= rue iii-iii ei to meet 1 " i it (25) em ample a on the nine (ii-fl Fessidouyowfilttté the efiect of! quflIIIiEdfl-e W *iih “Ml eeaitiueer-deagforee. (at Fifi-fie slistsnne'ett}.-flist'-ths ration i. (f) Fea.--trie=esae the sum Infill sen meters. 12:. Etudiouefittehtttterittl, such-as thehmope thoriuunmfiisintegrates at a rate propmfioml to the sitteiifltet‘i‘iieiifli' fifieetitilfflftfiath’e amount present attituet.thett do he = —rQ. where re {1 lie the decay rate. ' (a) If 100- mg of thorium-234 deeays to 3104 mg in 1 week. determine the deeagir rate r. (h) Pied-n11 expression for- the-atnount of thorium-2.34 present at any time t. 0:) Find the time. required for the thorium—234 to decay to oneshnlf its original amount. 13. The of a radioactive material is the time required for an amount of this material to decay to one-half its original value. Show that for any radioactive material that deeass fleeording to the equation Q’ = — rQ. the helfslit‘e r and the decay“ rate r satisfi' the equation rs = to 2-. 14. Radium-226 has a half-life of 1616 years. Find the time period during which a give it amount of this material is reduced hy-one-quarter. 15.. Aeeording, to Newton‘s law of eooting (see Problem 13 of Section 1.1}. the iernreraturi aft) of an object satisfies the differential equation dujdf == —klu — T}. where T is the constant ambient temperature and k is a pmitive mast-ant. Supiaise that the initial temperature of the object is utflt = n.3,. (a) Find the temperature of the object at any time. {13) Let r be the time at-whieh the initial temperature difference to. — T has been reduced by half. Find the relation between t and r . ' on with Nee-“tone Inst" of eta-tiling tsee P‘rohr- tom 15) and that the rate nonstant it has the value 1115 hr". Assume that the interior Chapter I. In '. __'_i '- -v...i.- -- -r I-PF'_.I I . _ _ I 1..-: J. -_ __ __.I_ . _ I. ‘- when the heating intent Me if "1“ “mil 'fmp‘immm Hes-1s . Mum! tempcrfllum i5 flrF for the interior temperature to fall to 351T? - 7" '. 4 I‘m Yflilihaw am 11‘ ii ill m- huw tiling “1”!” circuit containing a capacitor. resistor. and battery: see Figure 31:7 enable: ' i if 3m. "— m d!‘ l?. ti'onst or an e in.“ ‘_ H .H . _ 1 1 “minus a; _' - u . _ . ._-\vill1m=. : '. Thu chargc Qt't'i on the capacitor satisfies the to .I (d) Find the time T at which mfllmmmmfim ,3de mm “In “In: mm Reg + Q :1; e‘ (s) Findthe-finw-ratsthatissurfinem machine the mneennnnmr attegrgat wttaiarax. dt C" I . -'-"' . ivltere H is the resistance. t" is the CHPflCiianc’i'i and V iii the Eflnflflnl vflhflge supplifid'fiili the but large. in] it Lilli! = ‘ lhi Find the limiting value (it that Qttt approaches after a long tune. [r] Suppose tltat Qlt'll = Q, and that at timet = r] the batter}; is removed and the airtillit. ' closed again. Find (gm fort t, and sketch its graph. n, and (lot at anv time t. and sketch the graph of Q versus r. 1.3 Classification of Differential Equations W The main purpose of this book. is to discuss some of the properties of solutions of differential equations, and to present some of the methods that have proved effective in finding solutions or, in some-cases approximating them. To provide a framework for our presentation, we describe here several useful ways of classifying differential gt equations * ) l.” t J. air—.H Ordinary and Partial Differential Equations. One of the more obvious classifications is [7. based on whether the unknown function depends on a single independent variable Ht; ti Rt: 1.1.1 on. --I.-ett-ie circuit of Pnzthtent to. or on several independent variables In the first ease.on11tr ordinary derivatives appear in the differential equation. and it is said to be an ordinary differential equation. In the second case. the derivatives are partial derivatives and the equation is called a -~i I. . in wt; : !t l tile-t 1% ti itialli- fr“ it a certain un e' ‘ . . . . I ' ii if is” is I, ,_ 'l. __ __ . .. .iata sitttiLtiu-tl otthe chetiiic'iiiittge Chen“?! partial dll'fErentIal equation. 1 "(I i fl If i" 'I W i .' , '1 '_| i! _' ‘l'if' _' ' '{jm'm ‘Pflnd at Ihc same ragsiflifiz All the differential equations discussed in the preceding two sections are ordinary- Ht -. - .. .etL't -- '.-_‘.rt.'.t. tilt. '.- :t~t"'-"-‘-~- “ ' . . . . _ - . ... __ - . * ‘_ tit .: at.- ._-'-tL:-..i. _a1-.m-..t..;t..:_., .i; i. its-amt {influgiigm mu [mug differential equations. Another esample of an ordinary differential equation is t.tl l -.I [h w the aitttttuti or iii. rl-ettttttti m the [toilet tll time I. Write down an initial dEQU} dQU} I 1 H "..'Ii'lt_'l'Iit'lti-_lltltll' Lyliil ‘i" E = fl'l‘t i‘ttli‘t't' tilt; FtTtIl‘tlLi'll !rI ["rttlt {til ltlr {hill}. Hi'itt [Hugh Chi-initial i5 in [he pflnd flfiflr] tent" for the charge go} on a capacitor in a circuit vo'th capacitance C. resistance R. and inductance L; this equation is derived in Section 3-1%. Typical csan‘tplcs ot partial ie} .--"tl the L'lltl to l tear the source of the eltetttiettl in the pond is removed; thereafter difffirenthl Equatiflnfi are the heat punduction cou'ttiot‘t I I a . PU” “Hist “HM “Hittite pond. and the mixture lions out at the same rate as before. Write tlttvtn the initial value pitihlertt Ihttt tlcsei'tites this new situation. 1 H-MI.“ Fiifi_}__l'_ " is . ' ' . . l l r _ . _ , '* —-———-—q = n _ — '. till hoist. tltt. initial ‘nlltlt. prohlent in part ict. lion- much chemical remains tn the pond U ill" “7' after I additional ji'e'al' l2 vents truth the beginning of the problem}? tel HHW ltllltl does it [til-it: lt.tt' Gill in he: TL‘ijUL't'd {Li I“ E'} 1"! i H alt H I] . I ‘- t“l.t'. “'I'i-t ~ Ill Platelet versusttte s tears. ei——— : -. ‘ “' "I it t' - "i t' ' and the wave equation ti ' . - _ 1_ 1 . . t - ‘tnur sutntnttng pool contatntng titl'.tlil'il gal of water ltas been contaminated by 5 he of .I nontoste die that leaves a stvirnmer‘s skin on unattractien green The pangs [sharing system can take water from the pool. remove the dye. and return the water to the pool iii a lloiv rate of Elli] gallntin. Here. or2 and a3 are certain physical constants. the heat contlu. tan-c taunts-a scribes the conduction of heat in a solid body. and the state entatter. .t: e. t * variety of problems involving wave motion in soith or flutsz Mac the e a - Eqs. (2) and (3) the dependent variable it depends on the too ntdcpettttee: t ~ I and r. {at Write down the initial value problem for the oldve tn the pool at any time I. lit] Solve the prohlem in part {a}. tiltering process; let not he the amount Systems of Differential Equations. Another classification of differential mutualiIt-ri‘: pends on the number of unknown functions that are insulted. it theft. is .t sauce function to be determined. then one equation is sufficient. Hones erltt the re .tte tn ti or more unknown functions. then a system ofequations ts required. i't‘ll csaniple. lite —————_——— .I . - - - 1- r 11 I. I . I l .L I h' I I f1. 5. I r .I -. 1 I 1 ' - a. I. 1. a 1 re:- - 2-. II I-IiI I I LII—lif- I _l'r. .1-1- I}. 1.-.._I _ . . ' .- +‘t‘-J..I |.'. '“ ‘i Effie-H 3i . '__.- . I II .I" _ it ' iII.E-:.I_.'—L-I_rI'-:'_ " -"': ..."'._- - It _. . I I I I flilj'l'I-fltsFIFIQ-li.l:fI.I;Ia ' rlr -- ' ' -, r I .- 1 a - . - _' - _. _rLglul-L‘L-Iflnl_5_ “I _ ....II_. _.__-- -__ - --._- —I , _ _ 'l _ I "In: 'J M:%=-:{F_Ft ____ _._- I -_' '_' " a. _._— - u.— botlta—Volterra. or predator-prey. 1- form They have the III/III = III I_ IIIIIIIII divfdr = -c_v -'t- yxy. } are the respective populations of the preyand predators; I and e are based on empirical observations and diagrams High; died. Systems of equations are diseusseel m ell-fi- where .rtr] and _vtr “I The constants ran. c. articular species being stu I I I II __ End 9* in particular, the Lotka—Volterra equations are examined in “TaIiIIiIig. I some areas of application it is not unusual to encounter very large systenisenndffl' -I I I I I II I I I I I I I hundreds. or even many thousands. of equations. '_ I tIIIIIIIIIIIIII _";'i'-_~I I - - - . ' - L. 1 ' : - " " - Order. The order of a differentialequation is the ordorIof the highest derivahugfl _I . I I I . . I II I I appears in the equation. The equations in the preceding sections are all era}. . I . .I the equations. whereas Eq (l) is a second order equation. Equations (2) and _ {if Lmnkfi a I iI III I I .. ‘.IIII':-'IIIII:-II. WII II - IIIGIII I .I E II-__' II I I I III It I . - F .. . II - II .. - second order partial differential equations. More generally. the equation . - -. . - - fi- - -- .. equation F L: tt _ti rt.....tr“”{t 2 U II__I _._ I=I. I isan ordiniirjvdiffcrciitialequation ofthe nth order. Equation (5) expressesarel ff ' between the independent variable t and the values of the function it and its " ..n“”. It i: cont-'enient and customary in differential equatin"""sl whose diarittationi'is outlined in 29 through iii. The presenoe of the term - - - involviitgs-ainfl- Eqi (12) nonlinear. derivatives H .ti . _ to virite v for “ME. with _i '._i'. ..._i'”“ standing for tr’trl.tt"{t}. . . . ,irimfi). Thu_s_i'if_II‘ tfii is written or " ~- quI ' _.— i‘-'It _i . . . ._i'””_} = U. g: I l3: Fill L'ttit'tifilt‘. J_ 1 I .I I. I. II I I. I_ 4 .-I..._ii. i wear i . 3.). ..r t is a third oi'del oiiierentiiil equatit'rn for v 2 Hit]. Occasionally. other letters Will l: i used instead til i and _r lot the independent and dependent variables; the meaning i should he clear from the. contest. if? _i..-e .. .. _I c ' ' _ _ a - _ v . . , _ I. E i I I Wt. asaLttnt. that it is always possible to solve a given ordinar}.F differential equation nig ti” 1h“ h‘gh‘ii‘i GHWHHVK “mi-lining I i FIGURE 1.3.1 An oscillation pendulum. yin} = IfIIrIIII.IIIIItIIII_oI I I I I Irtrt— ] l-jI Ti f1ilUdi' “Int? “quallnnfi _flf the form {3). This is mainly to avoid the ambiguity- The mathematical theoryr and methods for solving linear equations are hie-tin. -.i..- t at may arise hcca use a single equation of the form (6) may correspond to several : . veloped. In contrast. for nonlinear equations the theory is more complicated and istillill'ftfls 0* it“? {Wm {bl- Fm" Mil"‘lPIE- “15 Equatifln Fl. ' methods. of solution are less satisfactory. In view of this it is fortiiaaie iiiai t'i'h'lrt'n Significant problems lead to linear ordinary differential equations or tan b: .irrie imated by linear equations For example. for the pendulum. if the anete e is small. then sine ‘i—_‘= ti and Eq. (12) can be approximated bv the linear equate-n -— Z_, . yr III-_- l+ l I _.f _. “'2 __ 16- I III-{H III I III _v"2 + tdv' + 4y = U (9) leads to the two equations 2 or y = (Ill)- IIIIII + In 2 n, .-~. Linear a if N l' ' - This process of approximating a nonlinear equation by a linear one is called linearirm Whethe: the?" :15"; Equmm‘ A- crucial Classificalifln “f differential equations is ' - tion; it is an extremer valuable way to deal with nonlinear equations. Neieriiieti T. y rt. inear or nonlinear. The ordinary differential equation ' there are many physical phenomena that simply cannot be represented .ioequiIaiL-a ' ' - ' ' - ii '~ i' i eat FILM”! - i of“) = [i by linear equations To study these phenomena it is essential to at al uith ii iii in equafions 22 "P; __'_'.-_._-_—-.- _-l-—-—'_— _l :3 . ..__ rfifi; Ii In an elementary text it is natural to emphasize the SIHFELth _ forward parts of the subject. Therefore the greater PM}: :5 Ch I: e i_.-.__.yE.-:._I_ linear equations and various methods for solving them. owe up I as well as parts of Chapter 2. are concerned with nonlinear equa ions. an.“ is appropriate. we point out why nonlin and why many of the techniques that are he applied to nonlinear equations. . fl “I I ear equations are. in general. more I; I. useful in solving linear equat_ion5._ __ Salatians. A solution of the ordinary differential equation (8) on the" n e i e a is a function it: such that o'.d”.... .o‘m exist and satisfy _ 113.? - own} =_i’ti.airt.o'ti_t. . . . .o“"“ir}] for everv r in or e. i e it. Unless stated otherwise. we assume that the faucet-fluff; ‘1'? J ' ' ' r - . . _ :i; of Eq. {til is a realvvalued function. and we are interested in obtaining real~vglp_g&y.‘ solutions y = tutti. Rum” “1;” in youth-tn LE we found solutions of certain equations by a process direct integration- For instance. we found that the equation -' .I'l- . rip _ _ iii-T — = “.5 i — «tall . I” i (151. un- ll'llt'lt‘t 1“ tII'1 4-1 hiltttt't't-tiistant. it is tilten not so easy to find solutions of differential . '.'{.llt:tllt Ill“: H: i'.'. I."'.'L‘l ii itill llilJ It ilJl':t..‘l.lI_It] [hat ytiju think may [33 a salutifln Ufagiven _- .1: stilt-411'”? I' i'- n'ui'iliy rt Lillsely easy to -.leterinine whether the function is actuallya ' “th” ""‘T'i'l" "'1 “Ul‘illl'i'ma il'e- l'llnt'llfl“ in“? “it: fiduation. For example. in nay ii is easy to shiny thai the ttint-iion _t-.iit : cost is a solutjgn flf Ill.“ ‘i‘ _l' = lorall i. loci inlirni this. I ninety e that y'I tit : — sin t i—indy’l'ltl = — cos I: then it follows that yl'iii + t'ttlt : H. to the same way you can easily show that light) = sin! is also a solution at liq. [ lit. in course. this does not constitute a satisfactory way to solve most dit'terential equal ions. because there are far too many possible functions for you to have a good chance of finding the correct one by a random choice Nevertheless you should realize that you can verify whether any proposed solution is correct b J substituting it int-o the dillercntial equation. For any problem that is important to yetiP this can he a very useful check. It is one that you should make a habit of considering: has the solution. .II t” 2 “in” --.'- te'”. it‘lliEt'IJHItflorltanl questions. Although for the equations {15] and {1?} we are able to Emmi. it le‘llilIlt't simple functions are solutions. in general we do not have such _ ions readily available. Thus a fundamental question is the folio ' . D aflflalgipn of the. form (it) always have a solution? The answer is “No “ Sigriin :fiiiii; i' r n it r . 1 I y = m” ism": (ill tines not necessarily mean that there is a function u fluiuufln? Tim is tht:- tit1 tow can we tell whether some particular equation has theorems stating that urildleLrEtdEiltaih flf a smuuun’ and it is answemd bY . t - mas o s ' - Hmmm" “mail hi” “Wilmi- Hflwswr. this is nntna[Siriiilhiiiiitdmliiifiicharieeiiii itF . T . h __._,_..._' _ ' I_I_ ,. __ _-l i :1. i r. '-'....- .-‘-._. .s.1---_':_* of " sums; -. '- . .- ties "_ _ f'gfififlflltilli‘mfimtilfi . if}: - .- ..i one or mun - filiin at '_ _" '_'_ '-=..tna:-nieimfimnr‘ " ' .efponihte strum ' of the mu t e. As Wotan ia-gsetttieetat-ttntsf ' "i. _ i. this condition will determine a value-star have eaten no the on there may he other-salutinus-ot Eut--El-§).-tltat also have the value of p at the presto-thou limit” 1- 155w” flf'fifiiquflnofi also implications. If we are fortunate enough tot-find .a solution-at a; given problem. and if we know that the problem has a unique solution-.- then-"we be sure that we have crimpletety solved the problem. If there may be other solutions, then perhaps we should continue to search for them. A third important Question is: Given a differential equation of the form {3]. can we -'actuall_.y determine a solution. and if so. how? Note that if we find a solution of the given equation... we have at the same time answered the question of the existence of a solution. Howeve-r..-without knowledge of existence theory we might. for esarnple. use a computer to find a unmodcal approximation to a "solution" that does not exist- 0n the other hand. even though we may know that a solution exists. it may be that the solution is-not expressible in terms of the usual elementary functions—polynomial. trigonometric. exponential. logarithmic. and hyperbolic functions Unfortunately. this is the situation for most differential equations Thus. we discuss both elemen- tary methods that can he used to obtain exact solutions of certain relatively simple problems. and also methods of a more general nature that can he used to find ap- proximations to solutions of more difficult problems. 1. . Computer Use in Differential Equations. A con‘ipurer can he an extremely y aluahle tool in the study of differential equations. For many years computers has: been ate-Lt to execute numerical algorithms such as those described in Chapter s. to eon-area: numerical approximations to solutions of differential equations. these algorithm- have been refined to an extremely high Icy-cl of gene ralitv and" einriene y. .Jt ion i: it... ~- of computer code. written in a high—level programming language and en. cured i'lilt. r- within a few seconds) on a relatively inespensiye computersuinee til‘Pti-‘yltltt'tir. 1 ~ a high degree ofaccuracEir the. solutions of a v. it'lc range at diiteien ti.'il eqit aliens hint _ sophisticated routines are also readily ayailahie. these routines romaine the .ii‘li". to handle very large and complicated systems a itlt numerous diagnosi a.- lea! in t s in alert the user to possible problems as they are encountered _ _ The usual output from a numerical algorithm is a table of lttJt't‘ll‘t.‘ rs. listing it values of the independent variable and the corresponding tallies Hi the tie-'[H'l'ltie'l'tl variable. With appropriate software it is easy to display the soluuon tit .i etitte ie 1112.1: equation graphically. whether the solution has been obtained numerically oi .ts illi. result of an analytical procedure of sonte kind. Such a graphical display is olten l l iLclL. 38 P ROBLEMS ting and helpful in understanding-and interpreting; n than a. table ef numbers er a Z e-veral well-era [led an 'd- relativelxlflflxpenfifi all i .- investigatien-efal puters has hreugltt pewet'fitlielegy-tn much mere illumina _ at a differential equatten mule. There are enthe markets _ I purpese seftware packages fer the graphical ' v ‘lahilit ef rsenal cem _ _ _ _ E::::;Pgrf::hilc;leapabillily Itiltiithin the reach ef individual Students... censider. in the light ef yeur ewn circumstances. how best he take taillig- available cemputing reseurces Yea will surely find It enlightemng _te tin-5?; _- i _- Anether aspect el' eemputcr use that is very relevant te the. study el’ equatiens is the availability ef extremely pewvrful Enid gflflflifll Sflflwat‘e that can perferm a wide va riety ef mathematical eperattens. Ameng thesearejflg LEI-E: Mathematics. and MATLAB. each ef which can be used ea varteus lands at .3;qu - cemputers er werkstatiens. All three ef these packages can. execute esteem-1%.; merical cemputatiens and have versatile graphical facilities. Maple and Mathew: ' else have very estcnsive analytical capabilities. Fer example, they can perfume. _ analytical steps invelved in selving rnany differential equatiens. eften in TESfiDfiglll a single cemmand. Anyene whe expects te deal with differential equatiens than a superficial way sheuld bcceme familiar with at least ene ef these-meditate": caplerc the ways in which it can he used. Fur yea. the student. these eeruputing reseurces have an effect an hew yen-sham“ study dtfl'ercnltal equalities. Te hceume cunfidenl in using differential equatien'ni"-t- is essential In understand hew the selutien metheds werk. and this underst-andjngsfig-'r .- '_ tttll‘tlevetl. lll pa l l. by swirl-ting eat a sufficient number ulT examples in detail. “ll-i}: L't'crllllltllji' yeti sltuuld plan In delegate as many as pessible ef the reutine (nitihff. ' l’L‘pulllit'ul LlL‘iilllh ltl it t'tftl't'IFIIJIEF ts'llilt; _t'I':Ll lUCUS DI] il'lE': Pl'flpfll' ffll‘l‘lflllfltlflfl '- ...‘ -|. r" ' I r 1 .l 1—“ Ir always at} In us:- tlic hcst ructltculs Ftl'ltl reels available fer each task. In. partietilhie't it!“ hititlild “Irr- r 'u cumhuw; :iurriei‘icsl. graphical. and analytical metheds sci as iii-i5 attain uia sirutliu i.u'u.lcr~“.t.tlulirig iil'lhe hehavieir {ii the salutifln and uf the unmet-lying.- ' i ['tl‘tttJIJ'sh [hijl lltt.‘ j"il'l_lltlt.;ll| tt‘tt'tdels, TU“ ghguld men remember that same tasks-flan. ' best he dune with pencil and paper. while ethers require a caleulater er centputeii r.- (ieedjudgmcnt is alien needed in selecting ajudicieus centbinatien. I '- hit-i. I- Is. =3. '- _II. 1"}- ln each at Preblcms l threugh h determine the erder ef the given differential equatien' alse state whether the equattun is linear er nenlinear. I! I I-is'i, chi—fitted l. lifl+lfl+2v=sim e 1 “El” 113’ tie it: - ' ... tier-if”: +rE+y=g tl‘t' it”); flip it 3. —*- + —— __- fl _ rly 1 tll‘ ch" + ilti + ilt H u 4' E + If 2 U . 5- + sinll +yl = sin: d"? d? '1'" 6+ 333 H; + teesirty = t3 in each ef Preblems ? threugh lsl veri enfialequafien. 7. y“ -— y = l}: yiltl = e‘. yzlt} = eesht 3' yufly'i‘yfli Filth-2"”. ritu=e I -' fy that each given functien is a selutien ef the differ— in ~.'. uh. Elf-h ._ i. L. _ 17 i r.- .I: w. ' -. - ta. L1; ~ 2' '- an. ef.aay+sy'=i-it In eachetrerehtents ;_ _" - mm :' .. .. en. _ km mm. In partial diffflteflliflleattatien.“ ' '. 35- “a “We mill"- - "-"t-lIsJH-‘=tvvsavvvh'-n Hall!) = latr-I +331 as. see... = he ' - line-a = e*'3‘ sis-e. use; n = iii-rift an it. t a real eonstanl Ii. size” = _ttg; iii-{Lt} "= sin lxsin let. altar} = sintx - all. li. a real nonstant 23. etc... = is; ' _=--. {auntie-3W], l' s a 29. 'Fe'lllewthe steps indicated here te derive the equatien of metien at a pendulum Eq. ill} in the teat. Assume that the red is rigid and weightless. that the mass is a paint mean. and that there is ne'fi'ietien er drag anywhere in the system- (a) Assume "that the mass is in an arbitrary displaced pasitien. indicated by the angle H Draw a freesbedy diagram shewlng the ferees acting en the mass. (it) Apply Newten's law ef metien in the directien tangential tn the circular arc en he. a .eh the mass meves Then the tensile farce in the red dues. net enter the equauee Ui‘ .e rs..- I... that—yea need te find the cemponent ef the gravitatieaal farce in the tangential -.~ iie‘n. Observe else that the-linear acceleratien. as app-used te the anguiat melee. in n a. Ldiflfdti. where L is the length ef the red. (e) Simplify the result from part {bl te ehtain Eq- {tilt in the test. 30. Anether way te derive the pendulum equatien it'll is based en the principle of e- laser. -- tien ef energy. (a) Shaw that the kinetic energy 1" el the pendulum in meuen is "I i -. rlil ' T: -—fllL'(-—) . 2 ill (b) Shaw that the potential energy l" el the pendulum. relative in its rest rename is t" = mthl weasel. I .' "i._ 26 1.4 Histu Chapter 1.. ram '.' u—-—-'—-— _ __.-v1 .- - a! I11- 1’ I _N# I. ii iii .- r l‘ the total energy E = T + V is mm the resulting equation redum.fireg}; e of conservation of energy. -. r i rinci it it} B!- ”‘L P l . equal In Em and show that ":1. .‘-J ' III . - iion depends on the principle of angular mamm tuni about any point is equal to the net dammit“ t 1 Show [hit the angular momentum M. or moment of momentum. about the point of? .‘ '7' it - . t I- t is given by M = .tl'lf.:timfflf. i it In the moment of the gravitational force. and show that the resulting: nte that positive mnnrents are counterclockwise. supluir thl tier rill-trill equ equation reduces to Eq. {IE}. N -_-- l— —I —-——I—fl——I _ ——‘—-—u—Fu_—b_'_-I'_Ill_ rical Remarks -—_—-—-——-:.—""——_' Without lslltt‘i'illlil something ahout differential equations and methods of solving. them. it is difficult to appreciate the history el this important branch ofmathematins. l'urthcr. the deselopnienl of differential equations is intimately Interwoven with the gL-nufitl [lift eh ipnient of mathematics and cannot be separated from it. blevertlteless, in I‘tltti'lLlL' stittte historical perspective. we indicate here some of the major trends in the historic oi the '--tihic-ct and lilerlllli the most prcuninent early contributors. Othel- l'iisiiiriial irilm illitltllil is -._-u'1:.iii':c+J ti'i iiirititt‘ites scattered throughout the bflflk and Ill llts I'_l* l-..!"-.i. l:'-l~.'l-l :i'. ill-r i t‘ttl Hf rlli.‘ L'l'lttlllti’l‘. I'Iie .anijeei .. .i .iit'tei._-n.:..| _'i.i'..‘.-.tl:.ilt~. -..riginatecl in the study of calculus by Isaac roe-slum t Ins- FE": -.llt-.i - .r. lll"i=.l'_l ii' lt'F-Irltf'. Leibnisllti-lfi—l’ilb) in the seventeenth i sniuiv .‘--.e'-. . !| _. :in in ah..- lfnylisti countryside. was educated at Trinity Col- iL‘LE'. ‘5 anitirt- -. .-..i._l '.'n._i ..|.'|I. i u- ..-'-3"..i l’i'iilcsstir [Til Mathematics there in 1669. His I- lurch ll til'skti -i'—'.' . er ...i-_.iie-. .1Illi --i !llt.‘ t'urulaniental laws of mechanics date frnm into the. ._ i. --i’t.'||l._‘l-'_‘t__l Iii-ii .=tr_‘.lt' among his friends. but Newton was extremely sensitive lH- inlet-.ni -l|li.l .hel tit-t begin to publish his results until 168? with the ap— pe.ir.-in+.'e iii his finest tzn'nnirs tiiiiil; f‘iiiloiruu’riiri' i'liirrrr'alis Principle Mathematical. Although Nest Inn LllLl relatively little v. fill; in dilflerentialequatierts as such.hisdevel- npmcttt ill the calculus and clutridatiiiit ill the basic principles of mechanics provided a basis let their .il‘r—plicatii'iits til the eighteenth century. most notably by Euler. New- ton classilietl lirsl order dillcrential equations according to the forms sly/ctr. rafts}. ill-fits : l'tyi. and drills = ,t'i-r._-.'i. For the latter equation he developed a method nl solution using infinite series when _l't.i._i'i is a polynomial in .r and v. Newton‘s active research in mathematics ended in the early lbillls. except for the solution of occasional “challenge problems" and the revision and publication of results obtained much earlier. He was appointed Warden of the British Mint in 1696 and resigned his prnlcssorship a few years later. He was knighted in Ullfi and. Upon his death. was buried in Westminster Abbey. anftlglomplelicd his doctorate in philosophy at the age in several different liblcls. Heuiv'ts Litalltlhifi me EngagEd m‘ “lightly 5'”qu terest in this subject dcvclttpvd whenih :- finial-h" I‘lliight m inathfl-nfllim' Ell-"CE his I“. l'undamental results of calculus indc slid“ dill" lib lwemms: Lmhmz firtwad m the but was the lirst tn .ublisl 1h; * mi en fl-lhflu‘gh 3' “HIE lit“ than Newmn‘ P ~ 1 cm. to least. Leibniz was very conscious of the power (Se-tile“ 3-21 Emblem Stil- in-‘t-est. and the for solving an: order m saustivnfi't . .'nn 2.1) in less. as spe'ht'his life ah'ammder and rubber to sev- eral” Slim“ will“! families. Which tn herein-itier and to carryt on an extensive mflmnndenee wilh'nlher mathematicians. especially the Bernoulli bimhs era. In the course of this correspondence many problems in diflerential equations were solved during the latter part of the seventeenth century. The brothers -Jaltn.b-(.1654-—l?fl5) and Johann (16674743) Bernoulli of Basel did mum “3' dawn-ll) methods of solving differential equations and to extend the range of their applications. Jakob beaame professor of mathematics at Basel in 168?. and Johann was appointed-tn the same position upon his brother's death in ITGS. Both men were quarrelsome.jealnu5. and frequently embroiled in disputes. especially with each other. Nevertheless. both also made significant contributions to several areas of mathematics. With the aid of calculus. they solved a number of problems in mechanics by formulating them as differential equations For example. Jakob Bernoulli solved the differential equation y' = [Eithiy +- 511”: in 1690 and in the same paper first used the term “integral” in the modern sense. In 1694 Johann Bernoulli was able to solve the equation sly/sir = y/nr. One problem which both brothers solved. and which led to much friction between them. was the bruchistochrone problem lsee Problem 32 of Section 2.3). The brachistnchrnne problem was also solved by Leibniz- Newton and the Marquis de L‘Hdpital. It is said. perhaps apocryphally. that Ken ton learned of the problem late in the afternoon of a thing day at the Mint and sols ed it that evening after dinner. He published the solution anonymously. but HPGH seeing it. Johann Bernoulli exclaimed. t know the lion by his paw-“ Daniel Bernoulli ( 1700—1?82l.son of Johann. migrated to St. Pctersburg as a young man to join the newly established St. Petersburg Academy but returned to Basel in 1733 as professor of botany and. later. of physics His interests were prinianh in partial differential equations and their applications For instance. it is his name that is associated with the Bernoulli equation in fluid mechanics He was also the tirst to encounter the functions that a century later became known .is Bessel functions (Section 5.8). _ fl _ The greatest mathematician of the eighteenth centers. Leonhard hater : I :r USS). grew up near Base] and was a student ot Johann Bernoulli. H. iniiim . - - friend Daniel Bernoulli to St. Petershurg in I'll? For the remainder or -"'-=.- I was associated with the St. Petersburg Academy tl'iET—titl and fen—1's the Berlin Academy (till—libel. Euler ssas the l‘l'tt'l'st proiitie t'l'l.:ll‘.l.'.i'l..'l-‘ -:. all time: his collected works till more than it! large vi tomes. His ea. r. er: over all areas of mathematics and many fields or application as _ '1 its sign . blind during the last 1? years or his life. his vian continued ouch-nice l‘-'. at very day of his death. Of particular interest here is his ls'tt‘t'l'tttlerut .1; I I'll.“-I ‘1'1~"-I in mechanics in mathematical language and his deselopnient or aieiin . u these mathematical problems. Lagrange said of Euler s work in nu. cherish. t :-. i great work in which analysis is applied to the science ot t‘t‘u‘ucrticnt. -5ll‘..lt!li'___ . things. Euler identified the condition for esactncss till tirst order. Llllls‘li. n i . it up s (Section 2-6} in 1734—35. developed the theory ot integrating tactors the-cites _ on .n . . I - ,I_ - . 11‘ i! the same paper and gave the general solution ot homogeneous lines: Let-Halli l. iii. "l 2) in 1743. He calender! the latte": ' ' 4.5. and 4. constant coefficients {Sections 3.3;: fititl-‘il Beginning about 1?50.Enlgr mafia qua I . - . results to nonhomogeneouse - r ' ‘ 1 equations. He alan- . ~ 1 i '. - t r 5} In Willing d'ffer‘mna . t-rcouentd use of pfl'WLI’ scrres {Chap e j} and 8‘” in 1768491 made Impflrtflm propose a human ' lions .. cal procedure (Sec F l .5 Stematic "Eat ontrihutions in partial differential equations. and gave the first y mam E I . r of the calculus of variations. Joseph-Louis Lagrange {Hindi-€13} hecame professor mgtheriitégztligshig :1: - ' - h' age of 1*}. He succeeded Euler in the charro ma _ a IE “I” .Turm :1; .t L: lion and moved on to the Perla ACHdEmF 1“ 1:731 HE IS most fa- BerLitafllicfiifflinumental work Mecauione aualvrioue. published In 1788. an elegant comprehensive treatise of Newtonian metilflflfflfii Till? tipiifi Elilfiiegiary tllt'fertfnilitl equations. Lagrange showed in lihaeont til- I: e g ch. tiflfl 0f ' flan nth order linear homogeneous differenth equation as a linear cigar:l [:3 n m g pendent EL'tllllltll'It-i [Sections 3.2. 3.3. and 4.1]. Later. in till—'1.-. qu7ave 3 animate development of the method of trot-iation tifparitmeters‘fSecttons e. an . ). La, eranee is also known for fundamental work in partial dttferenttal equations and the .. . - '.'_- _t 1:4- Ldtiff:-ii‘iiitiilihidi'nl..ttplace {Ii—ttl—ldiiil lived in Normandy“ {15 [103' bull came “3 Parts. in [ins and quickly made his mark in seientilic circles. winning election to the rhgldpnm LIL-t. HL"|L'nL'i_'f-t in liii. l-le n as preeminent in the held ofcelesttal mechanics; . liaite tie ch’t nonun- t'efesre. was published in five volumes between union is fundamental in many branches ofmathematical phi ‘-it."~.. and I_.lp];1._-;_~ nadir-cl ll es tensi: elf. in connectionvn-‘ith gravitational attraction. The I =.'["|lii'i._- true-inwa- it h-IJ-ler at is also named [or 111m. although Its usefulness tn Huh H. l LtIJ'IL.1-l_-II1.E.1_ L-ihlntintltti‘lri '~.'.".i'-'t ntjii feetjglllflfd Uill'll lTlUCl'l later. '. .m- ._-|ent..~._-nth eentno Insan- elementary methods ofsolving ordinary M...“ glut .I5:_|=.1-.i'i_l.t'[lI.,'I;1‘tLil'-¥L't"'--. red. In the nineteenth century interest turned as He to the It“ --t._~_' a 'l't:,"tl'~'_‘l.it.'ill questions of existence and uniqueness .md IN the ties -l. -:"!'t'trh'iit-t'l less I.'lc'litcliil'tl'j.' methods such as those based on power “my. L-._J-_~-.n-._.;ma-~ t 'Iiaptet it these methods find their natural setting in the Ltittttftle't plane. Fons-Janeen}. they henehtted from. and to some extent stimulated. the more or less .ill—l'lLillilllLtlLl‘r- development it! the theory of complex analytic func- tions. Partial differential equations also hogan to he studied intensively. as their crucial role lf'l mathematical physics became clear. In this connection a number of functions. arising as solutions of certz-nn tn'dinary differential equations. occurred re- peatedly and we re studied eshaustively. Knott'n collectively as higher transcendental functions. many of them are associated with the names of mathematicians. including Bessel. Legendre. Hermite. I:C'hehyshev. and Hankel. among others. The numerous differential equations that resisted solution by analytical means led to the investigation of methods of numerical approximation (see Chapter 8). By latttl fairly effective numerical integration methods had been devised. but their im- plementation was severely restricted by the need to esecute the computations by hand or with very primitive computing equipment. In the last 50 years the devel— opment‘of increasingly powerful and versatile computers has vastly enlarged the range of problems that can he investigated effectively by numerical methods. Est tremer refined and robust numerical integrators were developed during the same pr. nod and are readily available. ‘v’erstons appropriate for personal computers have ltls great-est vet nri "|.' In?” mid iii-.-“ litpiilL'J-‘t L‘Ll Chapter 1. Inflation - REFERENCES brought The ability to solve-a significant problem within the mad! 01’ individual students. Another characteristie at differential equations in the twentieth oenmfi' “33 I!" creation of geometricalor topological methods.- for nonlinear equations. The goal is- it! understand utilise! the qualitative behavior of solutions from a BED“ metrical. as well". as from ananatjrtieat. point of view. If more detailed information is needed. it can Usually be obtained by using numerical approximations. An introduc— tion to geometrical methods appears in Chapter 9. Within the past few years these tuto trends have come together. Computers and especially computer graphics. have given a new inaperus to the study of systems of nuns linear differential equations.- Unertpected phenomena (Section 9.3). such as strange attractors. chaos. and fractals. have been discovered. are being intensively studied. and are leading to important new insights in a variety of applicatiflfli Allhflflgh i1 i5 an old subject about which much is known. differential equations at the dawn of the twenty—first century remains a fertile source of fascinating and important unsolved problems. Computer software for differential equations changes too fast for particulars to be given in a been such as this. A good source of information is the Software Review and Computer Corner sections of The t‘uiiege Mathematics Journal. published by the Mathematical Association of Ame rica. There are many bo-ol-ts on the use of computer algebra systems. some of which emphasize their use for differential equations. For further reading in the history of mathematics. see books such as those listed helon; Boyer. C. B..and Merchach. U. [1.1-1 History of Mathematics Lind edJ [Hen York: Wiley. l'ie‘s t. Kline. M. Mathematical Thought from Ancient to Modern THHES thien Torte; Deford L'nnerat} Press. 1912]. A useful historical appendix on the early development of differential equations appears I“ lnce.E. L. Ordinary Differential Equations {Londont Longmans. Green. 1927'. hen ‘i'orlt DL‘HL'L ldent An encyclopedic source of information about the lites and achiet past is ._ ... _ - _ -1 _. Le. Gillespie. C. C..ed.. Dictionary of .‘irrenrurr Biography t to H'dml tT's-ev. ‘tt rs atnhncr s. t t l t ' 1- ..u .a .— "' i! ‘ kl - Much historical information can he found on the it'llLrl'lci Dee .stt....nt ..lc is Elfin?-EiiP.d£S.il-.lilt_l.L1t2.Ul-L-"“'lltiit"fi Bio-gt one i. html created bv John J. O'Connor and Edmund F Robertson. lhpetment HI Klan-ea» -.-.i:.- . . ~- University of St. Andreas Scotland. ements of Ft.tll'tr:fl“-.slt.‘t.t"ti of the ...
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