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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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