DIFFERENTIAL EQUATIONS

Transcription

1 DIFFERENTIAL EQUATIONS M.A. (Previous) Direcorae of Disace Educaio Maharshi Dayaad Uiversiy ROHTAK 4

2 Copyrigh 3, Maharshi Dayaad Uiversiy, ROHTAK All Righs Reserved. No par of his publicaio may be reproduced or sored i a rerieval sysem or rasmied i ay form or by ay meas; elecroic, mechaical, phoocopyig, recordig or oherwise, wihou he wrie permissio of he copyrigh holder. Maharshi Dayaad Uiversiy ROHTAK 4 Developed & Produced by EXCEL BOOKS PVT. LTD., A-45 Naraia, Phase, New Delhi-8

3 Coes 3 Chaper Differeial ad Iegral Equaios 5 Chaper Exisece Theorem Chaper 3 Uiqueess of Soluios Chaper 4 The -h Order Differeial Equaio 34 Chaper 5 Maximal Ierval of Exisece 43 Chaper 6 Depedece of Soluios o Iiial Codiios ad Parameers 47 Chaper 7 Differeial Iequaios 53 Chaper 8 Maximal & Miimal Soluios, Lyapuov Fucios 59 Chaper 9 Liear Sysems Ad Variaio of Cosas 63 Chaper Reducio of he Order of a Homogeeous Sysem, Liear Homogeeous Sysems Wih Cosa Coefficies, Adjoi Sysems 74 Chaper Floque Theory 8 Chaper Higher Order Liear Equaios 85 Chaper 3 Noliear Differeial Equaios, Plae Auoomous Sysems 96 Chaper 4 Classificaio of Criical Pois ad Their Sabiliy 5 Chaper 5 Criical Pois of Almos Liear Sysems, Depedece o a Parameer ad Liapuov s Direc Mehod For Noliear Sysems 3 Chaper 6 Periodic Soluios, Bedixso Theorem, Idex of a Criical Poi 3 Chaper 7 Prelimiaries abou Liear Secod Order Differeial Equaios 9 Chaper 8 Basic Facs of Liear Secod Order Differeial Equaios 37 Chaper 9 Theorems of Surm ad Zeros of Soluios 4 Chaper Surm-liouville Boudary Value Problems (SLBVP) 56

4 4 M.A. (Previous) DIFFERENTIAL EQUATIONS Paper-V M.Marks: Time: 3 Hrs. Noe: Quesio paper will cosis of hree secios. Secio-I cosisig of oe quesio wih e pars of marks each coverig whole of he syllabus shall be compulsory. From Secio-II, quesios o be se selecig wo quesios from each ui. The cadidae will be required o aemp ay seve quesios each of he five marks. Secio-III, five quesios o be se, oe from each ui. The cadidae will be required o aemp ay hree quesios each of fifee marks. Ui-I Prelimiaries: Iiial value problem ad he equivale iegral equaio, mh order equaio i d-dimesios as a firs order sysem, coceps of local exisece, exisece i he large ad uiquess of soluios wih examples. Basic Theorems: Ascoli-Arzela Theorem, A heorem o covergee of soluios of a family of iiial value problems. Picard-Lidel of heorem: Peao's exisece heorem ad corollary. Maximal iervals of exisece. Exesio heorem ad corollaries. Kamke's covergece heorem. Keser' heorem (saeme oly). Ui-II Depedece o iiial codiios ad parameers: Prelimiaries, Coiuiy, Differeiabiliy, Higher Order Differeiabiliy. Differeial Iequaliies ad Uiqueess: Growall's iequaliy, Maximal ad Miimal soluios. Differeial iequaliies. A heorem of Wier, Uiqueess Theorems, Nagumo's ad Osgood's crieria. Egres pois ad Lyapuov fucios. Successive approximaios. Ui-III Liear Differeial Equaios: Liear Sysems, Variaio of cosas, reducio o smaller sysems. Basic iequaliies, cosa coefficies, Floque heory, Adjoi sysems, Higher order equaios. Ui-IV Poicare-Bedixso Theory: Auoomous syems, Umlafsaz, Idex of saioary poi. Poicare-Bedixso heorem. Sabiliy of periodic soluios, roaio pois, foci, odes ad saddle pois. Use of Implici fucio ad fixed poi heorems: Period soluios, Liear equaios, No-liear problems. Secod order Boudary value problems Liear Problems, No-liear problems, Aprori bouds. Ui-V Liear secod order equaios: Prelimiaries, Basic facs, Theorems of Surm. Surm-Liouville Boudary Value Problems, Number of zeros, No-oscillaory equaios ad pricipal soluios. No-oscillaio heorems.

5 DIFFERENTIAL AND INTEGRAL EQUATIONS 5 DIFFERENTIAL AND INTEGRAL EQUATIONS The subjec of differeial equaios is large, diverse, powerful, useful, ad full of surprises. Differeial equaios ca be sudied o heir ow-jus because hey are irisically ieresig. Or, hey may be sudied by a physicis, egieer, biologis, ecoomis, physicia, or poliical scieis because hey ca model (quaiaively explai) may physical or absrac sysems. Jus wha is a differeial equaio? A differeial equaio havig y as he depede variable (ukow fucio) ad as he idepede variable has he form dy d y d y F, y,,..., = for some posiive ieger. (If is, he equaio is a algebraic or rascedeal equaio, raher ha a differeial equaio). Here is he same idea i words : Defiiio. A differeial equaio is a equaio ha relaes i a o-rivial maer a ukow fucio ad oe or more of he derivaives or differeials of ha ukow fucio wih respec o oe or more idepede variables. The phrase i a orivial maer is added because some equaios ha appear o saisfy he above defiiio are really ideiies. Tha is, hey are always rue, o maer wha he ukow fucio migh be. A example of such a equaio is : si dy cos dy =. This equaio is saisfied by every differeial fucio of oe variable. Aoher example is : dy dy y = Classificaio of Differeial Equaios dy y + y. Differeial equaios are classified i several differe ways : ordiary or parial; liear or oliear. There are eve special subclassificaios: homogeeous or ohomogeeous; auoomous or oauoomous; firs-order, secod-order,.,h order. Mos of hese ames for he various ypes have bee iheried from oher areas of mahemaics, so here is some ambiguiy i he meaigs. Bu he coex of ay discussio will make clear wha a give ame meas i ha coex. There are reasos for hese classificaios, he primary oe beig o eable discussios abou differeial equaios o focus o he subjec maer i a clear ad uambiguous maer. Our aeio will be o ordiary differeial equaios. Some will be liear, some oliear. Some will be firs-order, some secod-order, ad some of higher order ha secod. Wha is he order of a differeial equaio? As a rule, oly hose differeial equaios are cosidered which are algebraic i he differeial coefficies. Defiiio. The order of a differeial equaio is he order of he highes derivaive ha appears (orivially) i he equaio.

6 6 DIFFERENTIAL EQUATIONS A his early sage i our sudies, we eed oly be able o disiguish ordiary from parial differeial equaios. This is easy: a differeial equaio is a ordiary differeial equaio if he oly derivaives of he ukow fucio(s) are ordiary derivaives, ad a differeial equaio is a parial differeial equaio if he oly derivaives of he ukow fucio (s) are parial derivaives. Example. Here are some ordiary differeial equaios : dy = +y (firs-order) [oliear] d y + y = 3 cos (secod-order) [liear, ohomogeeous] 3 d y d y + 3 5y = (hird-order) 3 [liear, homogeeous] Example. Here are some parial differeial equaios : u u = (firs-order i x ad y) x y u u = c (firs-order i ; secod-order i x) x u u + = (secod-order i x ad y) x y u = 3 (secod-order) x y Lieariy. We ow iroduce he impora cocep of lieariy applied o such equaios. This cocep will help us o classify hese equaios sill furher. Defiiio. A ordiary differeial equaio of order, i he depede variable y ad he idepede variable, is said o be a liear equaio which ca be expressed i he form d y d y dy a () + a () + + a () + a () y = Q() (*) where a () is o ideically zero o [a.b]. The righ haded member Q() of (*) is called he ohomogeeous erm. If Q() is ideically zero, equaio (*) reduces o d y d y a () + a () + a ()y = (**) ad is he called homogeeous. Thus a liear homogeeous differeial equaio of order does o coai a erm ivolvig he idepede variable aloe. Examples. The ordiary differeial equaios d y dy (i) y =, (ii) 7 3 d y dy y = si, are boh liear.

7 DIFFERENTIAL AND INTEGRAL EQUATIONS 7 Defiiio. A ordiary differeial equaio which is o liear is called a oliear ordiary differeial equaio. Example. The followig ordiary homogeeous differeial equaios are all oliear. (i) (ii) (iii) d y dy y =, d y dy y =, d y dy + 6y + 6y =. 3 Defiiio. To say ha y = g() is a soluio of differeial equaio dy d y F, y,,..., = o a ierval I meas ha F(, g(), g (),, g () ()) =, for every choice of i he ierval I. I oher words, a soluio, whe subsiued io he differeial equaio, makes he equaio ideically rue for i I. Iiial-value problem. A iiial-value problem associaed wih a firs order differeial equaio is of he form dy = f(,y), ε I y( ) = y, for some poi ε I. A iiial-value problem associaed wih a secod order differeial equaio has he form d y dy = f, y, wih iiial codiios y( ) = y, y ( ) = ξ, for some poi ε I. Iegral Equaios, ε I A iegral equaio is a equaio i which he ukow fucio, say u(), appears uder a iegral sig. A geeral example of a iegral equaio i u() is u() = f() + K(,s)u(s)ds where K(,s) is a fucio of wo variables called he kerel or ucleus of he iegral equaio. Accordig o Bocher [94], he ame iegral equaios was suggesed i 888 by du Bois- Reymod, alhough he firs appearace of iegral equaios is accredied o Abel for his hesis work o he Tauochroe, which was published i 83 ad 86. There is also he opiio ha such firs appearace was i Laplace s work i 78 as i shall make sese whe we speak of he iverse Laplace rasform. For example, he Laplace rasform of he give (kow) fucio f(), < <, is L{f()} = F(s) = e s f(), s > a

8 8 DIFFERENTIAL EQUATIONS provided ha he iegral coverges for s > a. So, if we are ow give F(s), say F(s) =, s >, s ad we are o fid he origial fucio (ow as ukow) f(), or he iverse Laplace rasform of F(s), i.e., f() = L {F(s)}, = s e s f(), he we are agais solvig above iegral equaio i (he ukow) f(). So i does make sese ha iegral equaios sared wih Laplace, sice he was, i he fial aalysis, afer recoverig he origial fucio f() from kowig F(s). I our above example, f() =. I he same vei, Fourier i 8 solved for he iverse f() of he followig Fourier rasform F(λ) of f(), < <, F{f} = F(λ) = e iλ f() as f() = F {F} = π e iλ F(λ)dλ. Hece i fidig he (ukow) f(), he solved a iegral equaio i f(). Wih such a explici soluio f(), i is o surprisig ha some hisorias cosider his Fourier (iverse rasform) resul as he firs very clear ad reachable soluio of a iegral equaio. Some problems have heir mahemaical represeaio appear direcly, ad i a very aural way, i erms of iegral equaios. Oher problems, whose direc represeaio is i erms of differeial equaios ad heir auxiliary codiios, may also be reduced o iegral equaios. Problems of a herediary aure fall uder he firs caegory, sice he sae of he sysem u() a ay ime depeds by defiiio o all he previous saes u( τ) a he previous imes τ, which meas ha we mus sum over hem, hece ivolve hem uder he iegral sig i a iegral equaio. We may he say ha such problems, amog ohers, have iegral equaios as heir aural mahemaical represeaio. The res of he examples are problems ha are formulaed i erms of ordiary or parial differeial equaios wih iiial ad/or boudary codiios ha are reduced o a iegral equaio or equaios. The advaage here is ha he auxiliary codiios are auomaically saisfied, sice hey are icorporaed i he process of formulaig he resulig iegral equaio. The oher advaage of he iegral equaio form is i he case whe boh differeial equaios as well as iegral equaios forms do o have exac, closed-form soluios i erms of elemeary kow fucios. CLASSIFICATION OF INTEGRAL EQUATIONS The mos of he iegral equaios fall uder wo mai caegories : Volerra ad Fredholm iegral equaios. A Volerra iegral equaio for he firs kid is of he form x f(x) = K (x, ξ)u( ξ)dξ, a ad a Volerra iegral equaio of he secod kid is of he ype x u(x) = f(x) = K (x, ξ)u( ξ)dξ. a A Fredholm iegral equaio of he firs ad secod kid are, respecively,

9 DIFFERENTIAL AND INTEGRAL EQUATIONS 9 b f(x) = K (x, ξ)u( ξ)dξ, a b u(x) = f(x) + K (x, ξ)u( ξ)dξ. a Iiial value problems reduced o Volerra Iegral Equaios Now, we shall illusrae i deail how a iiial value problem associaed wih a lieardiffereial equaio ad auxiliary codiios reduces o a Volerra iegral equaio. Example. Cosider he iiial-value problem dy = f(, y), ε I () y( ) = y () Iegraig () w.r.. from o, we wrie y() y( ) = f (s, y(s))ds y() = y + f (s, y(s)) ds (3) which is a Volerra iegral equaio of he secod kid. Coversely, he differeiaio of (3) gives dy = f(, y()) for all ε I (4) Furher, from (3), we wrie, o puig = boh sides, y( ) = y. (5) Tha is, y() give by (4) also saisfies he iiial codiio i (5). Example. Cosider he iiial value problem associaed wih he secod-order differeial equaio d y = λ y() + g(), () y() =, y () =. () We iegrae equaio () w.r.. over he ierval [, ]. We obai or dy y () = λ y ( ξ)dξ + g( ξ)dξ dy = λ u ( ξ)dξ + g( ξ)dξ, (3) usig oe iiial codiio give i (). Iegraig agai, we fid or ξ y() y() = λ y() = + ξ y(s)ds + dξ + ( s) g(s)ds + λ g(s)ds dξ ( s)y(s)ds

10 DIFFERENTIAL EQUATIONS y() = f() + λ K(,s) y(s)ds, (4) where f() = + ( s)g(s)ds is he o-homogeeous erm ad K(, s) = s is he kerel of Volerra iegral equaio (4). Iegral equaio (4) auomaically akes care of wo auxiliary codiios i (). Now, we shall cosider he iiial value problem associaed wih he geeral secod-order differeial equaio. Example 3. d y dy + A() + B()y() = g(), () y(a) = c, y (a) = c. () We wrie d y dy = A() B()y() + g(). We ow iegrae over he ierval (a, ) o obai dy A( ξ)y ( ξ)dξ B( ξ)y( ξ)dξ + g( ξ) ξ c = d Iegraig (3) agai, we obai This implies a a = [ A ( ξ)y( ξ) ] + A ( ξ)y( ξ)dξ B( ξ)y( ξ)dξ + g( ξ)dξ a a ) a a a = [ A ( ξ) B( ξ) ] y( ξ)dξ + g( ξ)dξ A()y() + ca(a) a. (3) y() c c ( a) = ( s)[a ( s) B(s)]y(s)ds a a + ( s)g(s)ds + c A(a)[ a. a a ] A(s)y(s)ds y() = [( s){a ( s) B(s)} A(s)]y(s)ds + f () (4) a where he o-homogeeous erm f() is f() = ( s) g(s)ds + ( a) [c (A(a)) + c ] + c. (5) a Equaio (4) is a Volerra iegral equaio of he secod kid of he ype y() = f() + K (,s)y(s)ds, (6) a i which he kerel K(, s) is give by K(,s) = ( s) [A (s) B(s)] A(s). (7)

11 DIFFERENTIAL AND INTEGRAL EQUATIONS Iegral equaio (5) is equivale o he give iiial value problem ad i akes care of auxiliary codiios i (). Exercise Obai he Volerra iegral equaio correspodig o each of he followig iiial value problems (a) y + λy = ; y() =, y () = (b) y + λy = ; y() =, y () = (c) y + y = si ; y() =, y () = (d) y y+ = ; y() =, y () = (e) y +λy = f(); y() =, y () = (f) y +y =, y() = y () = (g) y y 3y =, y() =, y () =.

12 DIFFERENTIAL EQUATIONS EXISTENCE THEOREM Le I deoe a ope ierval o he real lie < <, ha is, he se of all real saisfyig a < < b for some real cosas a ad b. The se of all complex-valued fucios havig k coiuous derivaives o I is deoed by C k (I). If f is a member of his se, oe wries f ε C k (I), or f ε C k o I. The symbol ε is o be read is a member of or belogs o. I is coveie o exed he defiiio of C k o iervals I which may o be ope. The real iervals a < < b, a b, a < b, ad a < b will be deoed by (a, b), [a,b], [a, b), ad (a, b], respecively. If f ε C k o (a,b), ad i addiio he righ-had kh derivaive of f exiss a a ad is coiuous from he righ a a, he f is said o be of class C k o [a, b). Similarly, if he kh derivaive is coiuous from he lef a b, he f ε C k o (a, b]. If boh hese codiios hold, oe says f ε C k o [a, b]. A oempy se S of pois of he real (, y) plae will be called coeced if ay wo pois of S ca be joied by a coiuous curve which lies eirely i S. A o-empy se S of pois of he y-plae is called ope if each poi of S is a ierior poi of S. A ope ad coeced se i he y-plae is called a domai. A poi P is called a boudary poi of a domai D if every circle aroud P coais boh pois i D ad pois o i D. A domai plus is boudary pois will be called a closed domai. If D is a domai i he real (, y) plae, he se of all complex-valued fucios f defied o D such ha all kh-order parial derivaives k f/ p y q (p+q = k) exis ad are coiuous o D is deoed by C k (D), ad oe wries f ε C k (D), or f ε C k o D. The ses C (I) ad C (D), he coiuous fucios o I ad D, will be deoed by C(I) ad C(D), respecively. Le D be a domai i he (, y) plae ad suppose f is a real-valued fucio such ha f ε C(D). The he ceral problem may be phrased as follows: Problem. To fid a differeiable fucio φ defied o a real ierval I such ha (i) (, ϕ() ε D ( ε I) (ii) d [ ()] ( ε I) This problem is called a ordiary differeial equaio of he firs order, ad is deoed by dy = f(,y). (E) If such a ierval I ad fucio ϕ exis, he ϕ is called a soluio of he differeial equaio (E) o I. Clearly if ϕ is a soluio of (E) o I, he ϕ ε C o I, o accou of (ii). I geomerical laguage, (E) prescribes a slope f(, y) a each poi of D. A soluio ϕ o I is a fucio whose graph [he se of all pois (, ϕ ()), ε I] has he slope f(, ϕ()) for each ε I.

13 EXISTENCE THEOREM 3 Suppose (, y ) is a give poi i D. The a iiial-value problem associaed wih differeial equaio (E) ad his poi (, y ) is formulaed i he followig way : Iiial-value Problem. To fid a ierval I coaiig ad a soluio ϕ = ϕ() of differeial equaio dy = f(, y) for (, y) ε D o I saisfyig he iiial codiio ϕ( ) = y. As see earlier, his iiial-value problem is compleely equivale o he fidig of a coiuous fucio ϕ = ϕ() o I saisfyig he iegral equaio ϕ() = y + f s, ϕ(s))ds, ε I. ( Remark. Give a coiuous fucio f(, y) o a domai D, he firs quesio o be aswered is Wheher here exiss a soluio of he differeial equaio dy = f(, y) for all ε I The aswer is YES, if ierval I is properly prescribed. Example. Cosider he differeial equaio dy = y, y() = wih = ε I. A soluio of his problem is ϕ() =. However, his soluio does o exis a =, alhough f(,y) = y is coiuous a =. This example shows ha ay geeral exisece heorem will ecessarily have o be of LOCAL aure aroud. Exisece i he large ca oly be assered uder addiioal codiios o f. Exisece i he large. O wha -rages does a soluio of iiial-value problem dy = f(,y), y( ) = y, ε I exis? Le E be a subse of (,y) space where E = {(,y), y }. Cosider I V P dy = f(, y), y() =. / A soluio of his IVP may exis for / ad icrease from o as goes from o /, he oe cao expec o have a exesio of soluios, y(), for > /. Uiqueess of soluios Le y be a scalar ad cosider he IVP dy = y /, y() =.

14 4 DIFFERENTIAL EQUATIONS This IVP has more ha oe soluio. Firs soluio : y() = for all. Secod soluio: I has oe parameer family of soluios defied by for c y() = c for c where c is a arbirary cosa wih c. Thus, soluio of his IVP is o uique. Defiiio ( -approximae soluio) Le f be a real-valued coiuous fucio o a domai D i he (,y) plae. A -approximae soluio of a ODE of he firs order dy = f(,y) o a -ierval I is a fucio ϕ ε C(I) such ha i) (, ϕ()) ε D for all ε I ii) ϕ ε C (I), excep possibly for a fiie se of pois S o I where ϕ () may have simple discoiuiies (i.e., a such pois of S, he righ ad lef limis of φ () exis bu are o equal), iii) ϕ () f(, ϕ()) < for ε I S. Remark () Whe =, he i will be udersood ha he se S is empy, i.e., S = φ. So (ii) holds for all ε I. () The saeme (ii) implies ha ϕ have a piecewise coiuous derivaive o I, ad we shall deoe i by ϕ ε C p (I). (3) Cosider he recagle y R = {(, y) : a, y y b, a >, b > } () abou he poi (, y ). y +b R Le f ε C o he recagle R. Sice he recagle R is a closed y se, so he coiuous fucio f o R is bouded. Le y b M = max f(,y) o R () Le α = mi (a, M b ) (3) a +a (Cauchy-Euler cosrucio of a approximae soluio). Theorem.. Le f ε C o he recagle R. Give >, here exiss a -approximae soluio ϕ of ODE of firs order dy = f(,y) () o he ierval I = { : α} such ha ϕ( ) = y, α beig some cosa. Proof. Le be give. We shall cosruc a approximae soluio for he ierval [, + α]. A similar cosrucio will defie i for [ α, ]. This approximae soluio will cosis of a polygoal pah sarig a (, y ), i.e., a fiie umber of sraigh-lie segmes joied ed o ed.

15 EXISTENCE THEOREM 5 Sice f is coiuous o he closed recagle R = {(,y) : a, y y b, a >, b > } () So f is bouded ad uiform coiuous o R. Le M = max f(,y) (3) ad R α = mi(a, M b ). (4) The (i) α = a if M a b (fig.a) (ii) α = M b if M a b (fig..b) I he firs case, we ge a soluio valid i he whole ierval a, whereas i he secod of he ierval case, we ge a soluio valid oly o I, a sub-ierval f a. Slope(-M) Slope M y +b y y x a x x + a x a x α x x +α x +a (a) (b) Fig.. We cosider he secod case whe M a b. Sice f is uiformly coiuous o R, herefore, for give >, here exiss a real umber δ = δ = δ( ) > such ha y y + b Q M +α c P (,y ) T -M +a Fig.. Q f(,y) f(, y) (5) provided δ, y y δ (6) for (, y) ε R ad (, y) ε R.

16 6 DIFFERENTIAL EQUATIONS Now divide he ierval [, + α] io pars such ha < < < = + α ad max. k k mi (δ, δ ). (7) M Sarig from he poi C (, y ), we cosruc a sraigh-lie segme wih slope f(, y ) proceedig o he righ of uil i iersecs he lie = a some poi P (, y ). Here, slope of lie CP is f(, y ). This lie segme, CP, mus lie iside he riagular regio T bouded by he lies issuig from C wih slope M ad M, ad he lie = +α, as show i he figure (.) above because, we have, i his case, α = M b. (8) Now, a he poi P (, y ), we cosruc o he righ of a sraigh-lie segme wih slope f(, y ) upo he iersecio wih lie =, say a he poi P (, y ). Coiuig i his fashio, i a fiie umber of seps, he resula pah ϕ() will mee he lie = + α a he poi P (, y ). Furher, his polygo pah (fig..3) will lie eirely wihi he regio T. This ϕ is he required approximae soluio. y T (,y ) 3 +α +a o Fig..3 Aalyically, he soluio fucio ϕ() has he equaio ϕ() = ϕ( k ) + f( k+, ϕ( k )) ( k ) (9) for [ k, k ] ad k =,,,, ad φ( ) = y. From he cosrucio of he fucio φ, i is clear ha ϕ ε C p o [, +α], ad ha ϕ() ϕ( ) M () where, are i [, + α]. If is such ha k k, he equaios (7) ad () ogeher imply ha ϕ() ϕ( k ) M k M k k δ M. = δ. M From equaios (4), (5), (7) ad (9), we obai ϕ () f(, ϕ()) = f( k, ϕ( k )) f(, ϕ()), () where k k. This shows ha ϕ is a -approximae soluio, as desired. This complees he proof.

17 EXISTENCE THEOREM 7 Remark. Afer fidig a -approximae soluio of a IVP, oe may prove ha here exiss a sequece of hese approximae soluios which ed o a soluio. To achieve his aim, he oio of a equicoiuous se of fucios is required. Defiiio (Equicoiuous se family of fucios) Saeme. A se of fucios F = {f} defied o a real ierval I is said o be equicoiuous o I if, for give ay >, here exiss a real umber δ = δ = δ( ) >, idepede of f ε F, such ha f() f( ) < wheever < δ for, I. Noe. I his defiiio, he choice of δ does o deped o he member f of family F bu is admissible for all f i he family F. Theorem.. (Due o Ascoli). Saeme. O a bouded ierval I, le F = {f} be a ifiie, uiformly bouded, equicoiuous se of fucios. Prove ha F coais a sequece which is uiformly coverge o I. Proof. Le {r k }, k =,,, be all he raioal umbers prese i he bouded ierval I eumeraed/lised i some order. The se of umbers {f(r ) : f ε F} is bouded, hece here exiss a sequece of disic fucios { f }, f ε F, such ha he sequece f (r )} is coverge. { { f (r. Similarly, he se of umbers { f (r )} has a coverge subsequece )} Coiuiy i his way, a ifiie se of fucios ε F,, k =,,, is obaied which have he propery ha f } coverges a r, r,.., r k. { k Defie g = f. () Now, i will be show ha {g } is he required sequece which is uiformly coverge o I. Clearly, {g } coverges a each poi r k of he raioals o I. Thus, give ay >, ad each raioal umber r k ε I, here exiss a ieger N (r k ) such ha g (r k ) g m (r k ) < for a, m > N (r k ). () Sice he se F is equicoiuous, here exiss a real umber δ = δ = δ( ) >, which is idepede of f ε F, such ha f() f( ) <, (3) for < δ ad, ε I. We divide he ierval I io a fiie umber of subiervals I, I,, I p such ha he legh of he larges subierval is less ha δ, i.e., max.{l(i k ) : k =,,, p} < δ. (4) For each such subierval I k, choose a raioal umber r k ε I k. If ε I, he ε I k for some suiable k. Hece, by () ad (3), i follows ha g () g m () g () g ( r k ) + g ( r k ) g m ( r k ) + g m ( r k ) g m () < + + = 3, (5) provided ha m, > max {N ( r ), N ( r ),, N ( r p )}. This proves he uiform covergece of he sequece {g } o I, where g F for each N. This complees he proof. f k

18 8 DIFFERENTIAL EQUATIONS Remark. The exisece of a soluio o he iiial-value problem, wihou ay furher resricio o he fucio f(, y) is guaraeed by he followig Cauchy-Peao heorem. Theorem.3. (kow as Cauchy-Peao Exisece heorem). Saeme. If f ε C o he recagle R, he here exiss a soluio ϕ ε C of he differeial equaio dy = f(,y) o he ierval α for which ϕ( ) = y, where R = {(,y) : a, y y b, a >, b > }, α = mi (a, M b ), M = max f(,y) o R. Proof. Le { }, =,,, be a moooically decreasig sequece of posiive real umbers which eds o as. By heorem., for each such, here exiss a approximae soluio, say ϕ, of ODE dy = f(, y) () o he ierval α wih ϕ ( ) = y. () I is beig give ha α = mi (a, M b ) (3) M = max f(,y) for (,y) ε R (4) R = {(,y): a, y y b, a >, b> }. (5) Furher, from heorem., i follows ha ϕ () ϕ ( ) M (6) for, i [, +α]. Applyig (6) o = ad sice, we kow ha α M b, (7) i follows ha ϕ () y < b for all i < α. (8) This implies ha he sequece {ϕ } is uiformly bouded by y + b. Furher, (6) implies ha he sequece {ϕ } is a equicoiuous se. Hece, by he heorem., here exiss a subsequece { φ }, k =,,.., of {ϕ }, covergig uiformly o he ierval [ α, k + α] o a limi fucio φ, which mus be coiuous sice each φ is coiuous. Now, we shall show ha his limi fucio φ is a soluio of () which mees he required specificaios. For his, we wrie he relaio defiig φ as a approximae soluio i a iegral form, as follows : φ () = y + [f(s, φ (s)) + (s)] ds (9) where (s) = φ (s) f(s, φ (s)) () a hose pois where φ exiss, ad (s) =, oherwise.

19 EXISTENCE THEOREM 9 Because φ is a -approximae soluio, so (s). () Sice f is uiformly coiuous o R, ad φ φ uiformly o [ α, + α] as k, i follows ha f(, φ k ()) f(, φ()) uiformly o [ α, + α], as k. Replacig by k i (9), oe obais, i leig k, φ() = y + k f(s, φ(s))ds. () From (), we ge φ( ) = y (3) ad upo differeiaio, as f is coiuous, dφ = f(, φ()). (4) I is clear from (3) ad (4) ha φ is a soluio of ODE () hrough he poi (, y ) o he ierval α of class C. This complees he proof of he heorem. Remarks. () If uiqueess of soluio is assured, he choice of a subsequece i heorem. is uecessary. () I ca happe ha he choice of a subsequece is uecessary eve hough uiqueess is o saisfied. Cosider he example dy = y /3. () There are a ifiie umber of soluios sarig/issuig a he poi C(,) which exis o I = [,]. For ay c, c, he fucio φ c defied by c 3 / φ c () = ( c) () c < 3 is a soluio of () o I. If he cosrucio of heorem. is applied o equaio (), oe fids ha he oly polygoal pah sarig a he pah C(,) is φ. This shows ha his mehod cao, i geeral, give all soluios of (). Theorem.4. Le f ε C o a domai D i he (, y) plae, ad suppose (, y ) is ay poi i D. The here exiss a soluio φ of dy = f(,y) for (, y) ε D, y( ) = y () o some -ierval coaiig i is ierior. Proof. Sice domai D is ope, here exiss a r > such ha all pois whose disace from C(, y ) is less ha r, are coaied i D.

20 DIFFERENTIAL EQUATIONS D R Le R be ay closed recagle coaiig C(, y ), ad coaied i his ope circle of radius r. The heorem. applied o () o R gives he required resul.

21 UNIQUENESS OF SOLUTIONS 3 We cosider he followig examples. UNIQUENESS OF SOLUTIONS Example : Cosider he iiial value problem dy = y /3 i [, ], y() =. Hece f(, y) = y /3 is a coiuous fucio. There are wo soluios of i, amely, y (), 3 y () = i [, ]. 7 Example : Cosider he iiial value problem dy = y / i [, ] y() =. I his problem f(,y) = y / is coiuous. This problem also have wo soluios, amely, y (), y () = i [, ] 4 Example 3: Cosider he iiial value problem dy = y /3 i [, ], y() =. Here f(, y) = y /3 is a coiuous fucio. Furher y (), 3 / y () =, 3 are wo soluios of he above iiial value problem. The above examples show ha somehig more ha he coiuiy of f (,y) i he differeial equaio dy = f(, y) i D is required i order o guaraee ha a soluio passig hrough a give poi (, y ) D be uique. A simple codiio which permis oe o imply uiqueess of soluio is he Lipschiz codiio, defied below.

22 DIFFERENTIAL EQUATIONS Defiiio: Suppose f(, y) is defied i a domai D i he (, y) plae. If here exiss a cosa K > such ha f(, y ) f(, y ) K y y for every pair of pois (, y ) ad (, y ) i D, he f (,y) is said o saisfy a Lipschiz codiio w.r.. y i D. The cosa K is called he Lipschiz cosa. Noaios: () The fac ha f(,y) saisfy Lipschiz codiio is expressed as f Lip i D. () If, i addiio f C i D, we wrie as f (C, Lip) i D. Remark. If f Lip i D, he f is uiformly coiuous i y for each fixed, alhough ohig is implied cocerig he coiuiy of f w.r... Defiiio(Covex se): A se D R is said o be covex se if D coais he lie segme joiig ay wo pois i D. f Theorem (3.): Le f(, y) be such ha exiss ad is bouded for all (, y) D, where D is a y domai or closed domai such ha he lie segme joiig ay wo pois of D lies eirely wihi D, The f saisfies a Lipschiz codiio, (wih respec o y) i D, where he Lipschiz cosa is give by K = lub (,y) D f (, y) y. f (, y) Proof: Sice exiss ad is bouded for all (, y) D, here exiss a cosa K (K > ) y such ha f (, y) lub = K. () (,y) D y Moreover, by he mea value heorem of differeial calculus, for ay pair of pois (, y ), (, y ) i D here exiss ξ, y < ξ < y, such ha f (, ξ) f(, y ) - f(, y ) = (y y ), () y for (, ξ) D. Thus f(, y ) - f(, y ) = y y y y lub (,y) = K y y, f (, ξ ) y D f (, y) y This implies f(, y ) - f(, y ) K y y, (3) for all (, y ), (, y ) i D. This shows ha f(, y) saisfies a Lipschiz codiio i D ad K is he Lipschiz cosa. Remark : The sufficie codiio of he above heorem (3.) is o ecessary for f (, y) o saisfy a Lipschiz codiio i D. Tha is, here exiss fucio f(, y) such ha f saisfy a Lipschiz codiio i D bu such ha he hypohesis of heorem (3.) is o saisfied.

23 UNIQUENESS OF SOLUTIONS 3 (a) Cosider he fucio f defied by f(, y) = y, () where D is he recagle defied by D = {(,y) a, y b} () we oe ha f(, y ) f(, y ) = y y y y a y - y (3) for all (, y ) ad (, y ) i D. (Θ y - y y y ) f Thus f(,y) saisfies a Lipschiz codiio i D. However, he parial derivaive does o exiss y a ay poi (, ) D for which. (b) Cosider he fucio f(, y) give by f(, y) = y () i he recagle R = { (, y), y }. () We fid f(, y ) f(, y ) y - y y y y y (3) i R. So f(, y) saisfies a Lipschiz codiio, wih Lipschiz cosa. However, he parial f derivaive does o exis a ay poi (, ) i R for, as y lim f (, + h) f (,) h h lim h. = h h lim h = (4) h h does o exis. Remark : The wo iiial value problems, discussed i he begiig of his lesso, did o have a uique soluio, ad his ca ow be aribued o he failure of he Lipschiz codiio a =. Defiiio: The series u () is said o coverge uiformly o a fucio u() o he ierval = a b if is sequece of parial sums, {S ()}, coverges uiformly o u() o he ierval a b. Weiersrass M-es : Suppose {u ()} is a sequece of real valued fucios defied o he ierval a b, ad suppose u () M for all =,, 3,.. ad for all [a, b]. The he series = u () coverges uiformly o he ierval [a, b] if he series = M of posiive real umbers coverges.

24 4 DIFFERENTIAL EQUATIONS Noe : The exisece proof give i Cauchy-Peao exisece heorem (.) is usaisfacory i he respec ha here is o cosrucive mehod give for obaiig a soluio o he iiial value problem. I paricular, if f(, y) saisfies a Lipschiz codiio i addiio o is coiuiy, a relaively simple ye a very useful mehod exiss, kow as he mehod of successive approximaios, which deduces he exisece ad uiqueess of a soluio of he give iiial value problem. This mehod is also called Picard ieraio mehod ad which is give i he followig heorem. Theorem 3. : (Picard Lidelof Theorem) Saeme: Le D be a domai of he y-plae. Le (, y ) be a ierior poi of D. Le recagle R = {(, y) : a, y y b, a >, b > } lie wihi D. Le f(, y) be a real valued fucio which is coiuous i D, saisfies a Lipschiz codiio (w.r.. y) i D ad M = max f(, y) i R. The, here exiss a uique soluio φ = φ() of he iiial value problem dy = f(, y) i D, y( ) = y o he closed ierval α, where α = mi{a, M b }. Proof : The give iiial value problem is equivale o he followig Volerra iegral equaio y() = y + f(s, y(s)) ds, () for a. Thus, a soluio of he give I V P o α mus saisfy () ad coversely. Now, we defie a sequece {φ k } of successive approximaios (Picard ieras) of he problem by he recurrece formulas φ ( ) =, y φ k+ () = y + f (s, φ (s)) ds, () k o he ierval α. Here, k =,,,.. We shall be cosiderig he ierval [, + α] oly. Similar argumes hold for he ierval [ α, ]. Firsly, i will be show ha (i) every φ k () exiss o [, + α], (ii) φ k C, ad (iii) φ k () y M( ), (3) for [, + α] ad for all k. We shall prove i by he mahemaical iducio. Obviously φ, beig he cosa fucio, saisfies hese codiios. Now, we assume ha ϕ k does he same ad we shall prove he requiremes for φ k+. By assumpio, f(, φ k ()) is defied ad coiuous o he ierval [, + α]. Hece, by formula (), φ k+ () exiss o [, + α], φ k+ () C s here, ad φ k+ () y = f(s, φ k (s)) ds

25 UNIQUENESS OF SOLUTIONS 5 f(s, φ k (s)) ds = M( ). (4) Therefore hese properies are shared by he fucio φ k () by iducio, for all k. Secodly i shall be proved ha he sequece {φ k }of fucios coverges uiformly o a coiuous fucio φ = φ() o I. For his, we defie k () = φ k+ () - φ k (), (5) for [, + α]. From equaios () ad (5), we wrie k () = {f(s, φ k (s)) f(s, φ k- (s)}ds K f(s, φ k (s)) f(s, φ k- (s) ds φ k (s) - φ k- (s) ds = K k- (s) ds, (6) where K is a Lipschiz cosa ad we have used he fac ha f Lip o R. Equaio (3) gives for k =, () = φ () - φ () = φ () y M( ), ad a easy iducio (lef as a exercise o a reader) o (6) implies ha k+ M ( αk) k ()., (Θ α) (7) K (k + )! for all k ad [, + α]. This shows ha he erms of he series = k k() are majorized by hose of he power series for M [e αk ], ad herefore, by he Weiersrass M es for uiform covergece, he series K k= k() is uiformly coverge o [, + α]. Thus, he series φ () + [φ k+ () - φ k ()], (8) k= is absoluely ad uiform coverge o he ierval [, + α]. Cosequely, he sequece of is parial sums S + () = φ () + [φ k+ () - φ k ()] k= = φ (), (9)

26 6 DIFFERENTIAL EQUATIONS is absoluely ad uiformly coverge o he ierval [, + α], o a limi fucio, say φ(),which is coiuous o [, + α]. Thirdly, i will be show ha his limi fucio φ() is a soluio of desired problem i equaio (). As φ is coiuous, so f(s, φ(s)) exiss for s [, + α] ad [f(s, φ(s)) f(s, φ k (s))] ds f(s, φ(s)) f(s, φ k (s)) ds K φ(s) - φ k (s) ds () as f saisfies he Lipschiz codiio o R. Sice φ k φ uiformly o [, + α], so φ(s) - φ k (s) as k () uiformly o he ierval [, + α]. Combiig () ad (), i follows ha f(s, φ k (s)) f(s, φ(s)) o [, + α] () uiformly as k. Cosequely equaios (), (9) ad () imply ha, o akig k, we ge a oce φ() = y + f(s, φ(s)) ds. (3) This proves ha φ() is a soluio of iegral equaios (), ad, herefore, a soluio of he give iiial value problem o he ierval [, + α]. Fially, we shall prove ha soluios of () is uique. If possible, suppose ha Ψ = Ψ() is aoher soluio of iegral equaios (). The Ψ() = y + f(s, Ψ(s)) ds, (4) o I. Le N = Max φ() - Ψ() o I. (5) From equaios (3) o (4), we obai φ() - Ψ() = {f(s, φ()) - f(s, Ψ(s))}ds K f(s, φ()) - f(s, Ψ(s) ds φ(s) - Ψ(s) ds. (6) Usig (5) i (6), we ge φ() - Ψ() KN( ), for [, + α]. (7) By he repeaed use of (6) ad (7), we obai K N( ) φ() - Ψ(), [, + α] (8)!

27 UNIQUENESS OF SOLUTIONS 7 K ad for all =,, 3,. Sice, he series [, + α], herefore, (! ) of posiive erms coverges for K ( ) for all [ o, o + α] (9)! as. From equaios (8) ad (9), we fid φ() - Ψ() for all + α This implies φ() = Ψ() for all + α () hus, soluio φ = φ() of he give iiial value problem is uique he ierval [ + α]. We ca carry hrough similar agreemes o he ierval [ α, ]. This complees he proof of he heorem. Remark: A sigifica ad useful feaure of his proof is ha he sequece {φ ()} coverges uiformly o φ(), which is he uique soluio of he give iiial value problem. The proof is also cosrucive i ha i provides a mehod of obaiig approximae soluios o ay required degree of accuracy. However, from his poi of view, i is rarely of pracical value as he ieraes usually coverge oo slowly o be useful. Furher, he above heorem is oly a local exisece heorem i ha, whaever he origial ierval of defiiio, a, i geeral he soluio is oly guaraeed o exis i a smaller ierval, α, where α a. dy Example. Cosider he iiial value problem = y, y() =. Soluio. Iegraig over he ierval [, ], we obai y() = + y(s)ds, () which is a volerra iegral of he secod kid. Le φ () =. () The, by Picard s mehod φ () = + φ (s) ds = + ds = +, (3) φ () = + = + = + + φ 3 () = + φ (s)ds ( + s) ds, (4)! φ (s)ds

28 8 DIFFERENTIAL EQUATIONS = + s ( + s +! 3 )ds = (5)! 3! Coiuig like his, we shall obai 3 φ () = (6)! 3!! Takig he limi as, we ge lim φ () = e. (7) This implies ha φ() = e (8) is he uique soluio of he give iiial value problem, by he Picard s mehod of successive approximaios. Example : Solve he iiial value problem dy = y, y() = By Picard mehod. Soluio. The correspodig iegral equaio is y() = + s y(s)ds. () Picard s ieraes are φ () =, () φ () = + s. ds 3 = + 3 φ () = + 3 = + 3 φ 3 () = +, (3) + s 3 s + ds 3! 3 3 s 3 s = + + 3! 3 + Μ φ () = + s φ - (s) ds +, (4)! 3 s 3 3! ds 3 3 3, (5)

29 UNIQUENESS OF SOLUTIONS = + + 3! The exac soluio ca easily be see o be! 3 3. (6) / 3) φ() = e, (7) o which he above approximae soluios coverge. Example 3: Solve he iiial-value problem dy = (y + ), y() =, by Picard s mehod. ( 3 Soluio. The iegral equaio, equivale o he above iiial value problem is y() = + s(y(s) s + ) ds. () The approximae soluios are φ () =, φ () = + s(3 s ) ds 3 = s s φ () = + s 3 + ds, = s s s φ 3 () = + s ds 8 4 = ad so o. The exac soluio ca be easily foud o be / φ() = + e = () (3) 8 -, (4) Example 4: Solve he differeial equaio dy = y, y() =, by he mehod of successive approximaios o. (5) 384 Soluio : We wrie f(, y) = y ad he iegral equaio correspodig o he iiial value problem is y() = + y(). () The successive approximaios are, give by,

30 3 DIFFERENTIAL EQUATIONS Thus φ () =, φ () = + φ - (), () for =,, 3,.. φ () =, (3) φ () = + s ds = +, (4) φ () = + = + s s + ds s s + 4 ds = + +. (5).4 We shall esablish by iducio ha φ () = + +! +..+!, (6) for all. for =,, ; we have already checked he relaio (6). Suppose ha. The This implies φ - () = + φ () = + + s! s + 3 = + s s + = !!! s! s !!! s ds. () φ () = + +! + 3! +..+.! (8) Therefore, by he priciple of mahemaical iducio, he equaliy (6) is rue for all =,, 3, Moreover, we observe ha φ () is he parial sum of he firs ( + ) erms of he ifiie series expasio of he fucio φ() = e. (9) ds (7)

31 UNIQUENESS OF SOLUTIONS 3 Furher his series coverges for all real. This meas ha φ () φ(), for all real. Hece, he fucio φ is he required soluio. Example 5: Solve he iiial value problem dy = -y, y() = by he mehod of successive approximaios. Soluio : The iegral equaio equivale o he give iiial value problem is y() = + -y(s) ds = - y(s) ds. () The successive approximaios give by Picard s mehod are φ () =, φ + () = - φ (). for =,, () We fid φ () = - ds =, (3) φ () = - ( s)ds = + φ 3 () = -, (4)! s s + ds! 3 = + -. (5)! 3! By he iducio, i may be verified ha (lef as a exercise o he readers) 3 φ () = ( ). (6)!! 3!! We observe ha φ () is he parial sum of he firs ( + ) erms of he ifiie series expasio of he fucio φ() = e -. (7) Furher his series coverges for all real. This meas ha φ () φ() = e -, for all. Hece he fucio φ, give i (7), is he soluio of he give problem. Example 6: Solve he iiial value problem dy = y, y() = by he mehod of successive approximaios. Soluios : The give iiial value problem is equivale o he iegral equaio

32 3 DIFFERENTIAL EQUATIONS y() = + s. y(s) ds. () The successive approximaios are φ () =, ϕ + () = + s φ (s) ds. () we fid φ () = + s. ds (3) = +, φ () = + s( + s ) ds = + + φ 3 () = + 4, (4)! s s 4 + s + ds! 4 6 = (5)! 3! From he iducio, we shall fid (lef as a exercise) φ () = (6)! 3!! we visualize ha φ () is he parial sum of he firs ( + ) erms of he ifiie series expasio of he fucio φ () = e. (7) Furher his series coverges uiformly for all real. This meas ha φ () φ() = e for all. Hece he fucio φ() is he required soluio of he give problem. Example 7: Solve he iiial value problem dy = y, y() = by he mehod of successive approximaios. Soluios : The give problem is equivale o he iegral equaio y() = + y(s) ds. () The successive approximaios are give by φ () =, φ + () = + φ (s) ds. () we fid φ () =, (3)

33 UNIQUENESS OF SOLUTIONS 33 φ () = + ds =, (4) φ () = + s ds = + [(s ) + ] ds. (5) Here, i is coveie o have iegrad occurrig i he successive approximaios i powers of (s ) raher ha i powers of s (Θ = ad o zero). Therefore, (5) gives (s ) ( ) φ () = + s + = + ( ) +. (6)! φ 3 () = + (s ) + (s ) + 3 ( ) ( ) = + ( ) +! 3!. (7) By iducio, we shall obai (exercise) 3 ( ) ( ) ( ) φ () = + ( ) ! 3!! (8) We oe ha φ () is he parial sum of he firs ( + ) erms of he ifiie series expasio of he fucio φ() = e -. (9) Moreover his series coverges for all real. Therefore, φ () φ() = e - for all. Hece he fucio ϕ(), give i (8), is he required soluio of he give problem. ds

34 34 DIFFERENTIAL EQUATIONS 4 THE -h ORDER DIFFERENTIAL EQUATION Le be a posiive ieger. Le f, f,...,f be real valued coiuous fucios defied o some domai D of he real (, y, y,,y ) space, which is ( + ) dimesioal. As before, I ad y i = y i () for I. A sysem of ordiary differeial equaios of he firs order is of he ype (i ormal form) or wrie as dy = f (, y, y, y 3,,y ) dy = f (, y, y,,y ) dy = f (, y, y,,y ) i dy = f i (, y, y,,y ), i. Iiial value problem: Le (, y, y,, y ) D. The iiial value problem cosiss of fidig differeiable fucios φ (), φ (), φ () defied o a real ied I such ha (i) (, φ (), φ (),.., φ ()) D for all I dφ i ( ) (ii) = f i (, φ (), φ (),.., φ ()) for all I (iii) φ i i ( ) = y for i. Remark: The resuls so for obaied (for he case = ) ca be carried over successfully he sysem of differeial equaios (5). Le R be he dimesioal real Euclidea space wih is elemes y = ( y, y,,y ), y i R for i. I developig he heory of sysem of differeial equaios (5), we shall eed o use a coveie measure of he magiude (or orm) of y = ( y, y,,y ). I is deoed by y ad defied as y = y + y + + y = i= y i. We prefer o use his orm. The disace bewee wo pois y ad y of R is defied o be y y. Here, y is a o egaive real umber. Noe : Oher defiiios for he magiude for a vecor y R are y = i= y i,

35 THE -h ORDER DIFFERENTIAL EQUATION 35 y = max. ( y i ) All are, of course, equivale. Resul (Normed liear Space) : The liear space R is a ormal liear space i which he orm fucio : R R, saisfies he followig properies (i) y = if ad oly if y = ; (ii) y ; (iii) y + y y + y (iv) αy = α y for all y, y, y i R ad α R is a scalar. Defiiio (Meric Space): Le M be a o empy se. A fucios ρ: M x M R is said o be a meric o M if i saisfies he followig properies (i) ρ (x, y) = ρ (y, x) (ii) ρ (x, y) = x = y (iii) ρ (x, y) ρ (x, z) + ρ (z, y) The, he pair (M, ρ) is called a meric space or we simply say ha M is a meric space wih meric ρ. Noe : ρ is also ermed as disace fucio. Resul : The ormed liear space R is a meric space wih meric ρ iduced by is orm ad defied as ρ (y, y ) = y y. Defiiio : A sequece of vecors {y k } i R is said o be coverge if i is coverge w. r.. his disace fucio. Remark : Sequece {y k } is coverge iff each of he compoe sequece { y i k }, i, is coverge. Defiiio : A Baach space is a complee ormed liear space, complee as a derived / iduced meric space. Defiiio : Lipschiz codiios i R Suppose a vecor fucio f(, y) is defied o a domai D of he (, y) space wih y R ad I. If here exiss a cosa K > such ha f(, y ) f(, y ) K y y holds for every pair (, y ) ad (, y ) i D, he he vecor fucios f (,y) is said o saisfy a Lipschiz codiio w. r.. he variable y i D, ad oe wrie, as before, f Lip i D. Cosa K is called Lipschiz cosa. Noe :- The admissible values of K deped o he orm i he f space as well as he orm i he y space. Defiiios : - approximae soluio i R Suppose f C o a domai D i he (, y) space. A -approximae soluio of he vecor differeial equaio dy = f(, y); y R, (a, b) = I o a ierval I is a vecor fucios φ() C o I such ha

36 36 DIFFERENTIAL EQUATIONS (i) (, φ()) D for I (ii) φ C o I excep possibly for a fiie se of poi S o I. (iii) φ () f(, φ() for I S. Defiiio: A soluio of vecor differeial equaio o a ierval I is a vecor fucio φ = (φ, φ,..,φ ) defied o I saisfyig (i) (, φ()) D for I dφ (ii) = f(, φ ()) for I. Defiiio: (Equi-coiuous family) Le E R be a subse. A family F = {f} of fucios f(y) defied o E is said o be equicoiuous if, for each >, here exiss δ = δ > such ha f(y ) f(y ) wheever y, y E, f F ad y - y δ. Observaios : () The choice of δ does o deped o f (y) bu is valid for all admissible fucios f (y) i he family F. () The mos frequely ecouered equicoiuous families F will occur whe all f F saisfy he Lipschiz codiio o he (, y) space w.r.. y. Here, here exiss a Lipschiz cosa K for all f F ad we may choose δ = K, for give >. Remarks: () I erms of he defiiios iroduced above, all he previous heorems discussed i chapers -3, are valid for he vecor differeial equaio dy = f(, y), y R if, i heir saemes ad proofs, y is replaced by he word vecor y, f is replaced by he word vecor fucio f ad he magiude is udersood i he sese of orm, defied above for vecors. () The Ascoli heorem (.) is valid for vecors also. Therefore, i will be assumed from ow owards ha hese heorems sad proved for he more geeral vecor differeial equaio dy = f(, y), I ad y R. Now, we shall wrie he sysem of differeial equaios iroduced earlier i vecor oaio. To achieve his, we iroduce he vecor y wih compoes y i (i =,,,), so ha y () y () y() = y (). dy The derivaive of a vecor valued fucio y() is, ad is defied o be he vecor(or colum- dy i ( ) marix/vecor) whose compoes are, (i =,,.,), so ha

37 THE -h ORDER DIFFERENTIAL EQUATION 37 dy dy dy =. dy Similarly, we defie he vecor valued fucio f(, y), a specified fucio of he vecor y ad real variable, o be he vecor whose compoes are f i (, y), i, so ha f (, y) f (, y) f(, y) = f (, y) I follows a oce ha firs order sysem meioed earlier ca be wrie i he compac vecor form dy = f(, y). Special case (Liear Sysem) : Whe he firs order sysem is liear, he fucios f i (, y, y,,y ) are of he paricular form f i (, y, y,,y ) = k= a ik () y k, i i which a ik () are coiuous fucios o I. I his case, his sysem is, i fac, dy = a () y + a () y + a 3 () y 3 + a () y. dy = a () y + a () y + a 3 () y 3 + a () y. Μ dy = a () y + a () y + a 3 () y 3 + a () y. This liear sysem ca be pu i he compac vecor form dy = A() y where he marix A() is of ype ad a() a ()...a () A() = a () a ()...a () a () a ()...a () ad

38 38 DIFFERENTIAL EQUATIONS y = y y y dy wih = dy dy dy. I his case, f (,y) saisfies a Lipschiz codiio o he ( + ) dimesioal regio D, where D = {(, y) a b, y R, y < }. (Here, D is o a domai, sice i is o ope). I fac, for (, y ) ad (, y ) i D, f(, y ) f(, y ) K y y [a, b] wih K = max. aik() : i= k =,,..., The above meioed liear sysem is, herefore, expressible as dy i = a ik () y k, i. k= Resul I: For he liear sysem of differeial equaios dy α = a αk () y k, α. k= where he fucios a αk () C o [a, b], here exiss oe ad oly oe soluio φ() of his sysem o [a, b] passig hrough ay poi (, y ) D. Resul II: Le he fucios a ij (), (i, j =,,.), be coiuous o a ope ierval I, which may be ubouded. The here exiss o I oe ad oly oe soluio ϕ of he vecor differeial equaio saisfyig he iiial codiio φ( ) = y, I, y <. Noe : For he proofs of hese resuls, he reader is advised o cosul he book by Coddigo ad Leviso. Reducio of h order ODE o a firs order vecor differeial equaio This ca be achieved by iroducig variables o represe he derivaives appearig i he give h order ODE. Now, we cosider a differeial equaio of h order of he ype z () = f(, z, z (), z (),.., z () ) Where z (j) j d z = (j =,,..,), is a real variable o ierval I, z ad f are scalars (ad o j vecor), ad he fucio f is defied o a domai D of he real ( + )-dimesioal space. Problem : To fid a fucio φ = φ() defied o a real -ierval I possessig derivaives here such ha (i) (, φ(), φ () (), φ () (),.φ ( -) ()) D for all I (ii) φ () () = f(, φ(), φ () (), φ () (),.φ (-) ()) for all I.

39 THE -h ORDER DIFFERENTIAL EQUATION 39 If such a ierval I ad a fucio φ() exis, he φ() is said o be a soluio of he give h order differeial equaio o he ierval I. If φ is a soluio, clearly φ C o I. Noe ha φ is o a vecor here. Iiial value problem : Le (, z, z,,z ) D. The he problem of fidig a soluio φ of he give h order ODE o a ierval I coaiig such ha φ( ) = z, φ () ( ) = z,..,φ (-) ( ) = z is called a iiial value problem. Remark : The heory of he soluio of h order ODE ca be reduced o he heory of a sysem of firs order differeial equaios. For his, we make he subsiuios z = y z () = y z () = y 3 Μ z (-) = y The, we ge he followig sysem of firs order differeial equaios i ukows y, y,., y. dy = y dy = y 3 dy = y dy = f(, y, y,., y ). This sysem, i ur, is equivale o he followig firs-order vecor differeial equaio dy = f(, y), where y y 3 y y y =, f =. y y f (, y, y,..., y ) Theorem (4.) : Cosider he h order ordiary differeial equaio d z dz d z = f, z,,..., where he fucio f is coiuous ad saisfies Lipschiz codiio i a domai D of real ( + )- dimesioal space. Le (, z, z,,z ) be a poi of D. Prove ha here exiss a uique soluio φ() of he give h order differeial equaio such ha

40 4 DIFFERENTIAL EQUATIONS φ( ) = z, φ () ( ) = z,..,φ (-) ( ) = z, defied o some ierval aroud. Proof : Cosider he subsiuios y = z, y dz =, y 3 dz =,.,y = We shall ow prove ha, he give h order differeial equaio is equivale o he followig sysem of firs order ordiary differeial equaios. dy = y dy = y 3 Μ d z () dy = y dy = f(, y,y,..,y ). () Le φ = φ() be a soluio of he give h order iiial value problem. We defie φ = φ, φ dφ =,,φ d φ =. (3) Le φ y = φ φ, (4) be a vecor of fucios. The y is a soluio of he firs order sysem () which saisfies he iiial codiios φ ( ) = z, φ ( ) = z,, φ m ( ) = z m. (5) Coversely, ow i is assumed ha a vecor fucio y = () which saisfies iiial codiios i (5). The dφ dφ dφ 3 = φ = φ 3 d φ = = φ 4 3 d φ = 3 φ φ Μ φ is a soluio of firs order sysem

41 THE -h ORDER DIFFERENTIAL EQUATION 4 ad or Μ dφ dφ = φ d φ = = f(, φ, φ,φ 3,.,φ ), (6) d φ = f(, φ dφ d φ d φ,,,, ). (7) This shows ha y = φ is a soluio of he give h order ordiary equaio ad his soluio, usig (5), saisfies he iiial codiios φ ( ) = z dφ, ( ) = z d φ,.., ( ) = z. (8) This esablishes he desired equivalece. This complees he proof. Remark : I also shows ha he firs compoe φ of he vecor φ φ y = φ is a soluio of he give h order ordiary differeial equaio. Now (, z, z,,z ) is a poi of D, here exiss a ( + )-dimesioal recagle, say R, abou his poi such ha he fucio f saisfies he saed hypohesis i he recagle R. Thus, he sysem () of he firs order equaios saisfies all he hypohesis(ecessary) i he recagle R. So, here exiss a uique soluio (φ, φ,,φ ) of sysem () which saisfies he codiios i (8) ad his soluio is defied o some sufficiely small ierval aroud. Thus, if we se φ = φ, he above show equivalece gives he desired coclusio. Remark : I is hus clear ha all saemes proved abou he sysem () of firs-order equaios carry over direcly o saemes abou he h order ordiary differeial equaio. I paricular, we sae he followig heorem abou liear equaio. Theorem (4.) : Cosider he liear ordiary differeial equaio d y d y dy a () + a () +..+ a - () + a () y = F() where a,a,.,a ad F are coiuous fucios o he ierval a b ad a () o a b. Le be a poi of he ierval a b ad le z, z,,z be real cosas. Prove ha here exiss a uique soluio φ = φ() of he above ODE such ha φ( ) = z, φ () ( ) = z,..,φ (-) ( ) = z.