Ch 2.1: Linear Equations; Method of Integrating Factors

Transcription

1 Ch.: Linar Equaions; Mhod of Ingraing Facors A linar firs ordr ODE has h gnral form d d f whr f is linar in. Eampls includ quaions wih consan cofficins such as hos in Chapr a b or quaions wih variabl cofficins: d d p g

2 Consan Cofficin Cas For a firs ordr linar quaion wih consan cofficins a b rcall ha w can us mhods of calculus o solv: d / d a b / a d b / a ln b / a a C b / a k a a d k ± C

3 Variabl Cofficin Cas: Mhod of Ingraing Facors W n considr linar firs ordr ODEs wih variabl cofficins: d d p g Th mhod of ingraing facors involvs mulipling his quaion b a funcion μ chosn so ha h rsuling quaion is asil ingrad.

4 Eampl : Ingraing Facor of Considr h following quaion: / Mulipling boh sids b μ w obain W will choos μ so ha lf sid is drivaiv of known quani. Considr h following and rcall produc rul: d d dμ d d d Choos μ so ha d μ μ d / [ μ ] μ μ μ μ μ

5 Eampl : Gnral Soluion of Wih μ w solv h original quaion as follows: [ ] C C d d d d d d / / 5 / 5 / 5 / / 5 5 μ μ μ

6 Mhod of Ingraing Facors: Variabl Righ Sid In gnral for variabl righ sid g h soluion can b found as follows: a g μ a d d a d d [ ] a d d a a aμ μ g a a a a g g d a g g d C a

7 Eampl : Gnral Soluion of W can solv h following quaion using h formula drivd on h prvious slid: Ingraing b pars Thus 5 5 a a a /5 g d C /5 5 d / /5 /5 /5 d /5 [ ] /5 /5 5 5 d 5 /5 /5 d 5 d C /5 /5 /5 / C 5 5 C /5

8 Eampl : Graphs of Soluions of Th graph on lf shows dircion fild along wih svral ingral curvs. Th graph on righ shows svral soluions and a paricular soluion in rd whos graph conains h poin 5. / C

9 Eampl : Gnral Soluion of W can solv h following quaion using h formula drivd on prvious slid: Ingraing b pars Thus 5 5 a a a /5 /5 g d C 5 d /5 5 d /5 /5 d /5 /5 d [ ] /5 /5 5 5 d [ ] /5 /5 / 5 5 C 5 C /5 C /5

10 Eampl : Graphs of Soluions of Th graph on lf shows dircion fild along wih svral ingral curvs. Th graph on righ shows svral ingral curvs and a paricular soluion in rd whos iniial poin on -ais sparas soluions ha grow larg posiivl from hos ha grow larg ngaivl as. / C /5

11 Mhod of Ingraing Facors for Gnral Firs Ordr Linar Equaion N w considr h gnral firs ordr linar quaion p g Mulipling boh sids b μ w obain d μ p μ g μ d N w wan μ such ha μ' pμ from which i will follow ha d d d d [ μ ] μ p μ

12 Ingraing Facor for Gnral Firs Ordr Linar Equaion Thus w wan o choos μ such ha μ' pμ. Assuming μ > i follows ha dμ p d p d ln μ μ k Choosing k w hn hav μ p d and no μ > as dsird.

13 Soluion for Gnral Firs Ordr Linar Equaion Thus w hav h following: Thn d p g p d d g p whr μ μ μ μ [ ] d p c d g c d g g d d whr μ μ μ μ μ μ μ

14 Eampl 4: Gnral Soluion of To solv h iniial valu problm firs pu ino sandard form: Thn and hnc 5 for 5 ln C C d C d C d g μ μ ln ln d d p μ

15 Eampl 4: Paricular Soluion of Using h iniial condiion and gnral soluion 5 ln C i follows ha C 5 ln or quivalnl ln / 5 5

16 Eampl 4: Graphs of Soluion of Th graphs blow show svral ingral curvs for h diffrnial quaion and a paricular soluion in rd whos graph conains h iniial poin. IVP : 5 Gnral Soluion : Paricular Soluion : 5 5 ln ln C

17 Ch.: Sparabl Equaions In his scion w amin a subclass of linar and nonlinar firs ordr quaions. Considr h firs ordr quaion d d f W can rwri his in h form d M N d For ampl l M - f and N. Thr ma b ohr was as wll. In diffrnial form M d N d If M is a funcion of onl and N is a funcion of onl hn M d N d In his cas h quaion is calld sparabl.

18 Eampl : Solving a Sparabl Equaion Solv h following firs ordr nonlinar quaion: d d Sparaing variabls and using calculus w obain d d d C C d Th quaion abov dfins h soluion implicil. A graph showing h dircion fild and implici plos of svral ingral curvs for h diffrnial quaion is givn abov.

19 Eampl : Implici and Eplici Soluions of 4 Solv h following firs ordr nonlinar quaion: Sparaing variabls and using calculus w obain Th quaion abov dfins h soluion implicil. An plici prssion for h soluion can b found in his cas: 4 d d C d d d d 4 4 C C C ± ± 4 4

20 Eampl : Iniial Valu Problm of 4 Suppos w sk a soluion saisfing -. Using h implici prssion of w obain Thus h implici quaion dfining is Using plici prssion of I follows ha 4 ± ± C C C C C C 4

21 Eampl : Iniial Condiion of 4 No ha if iniial condiion is hn w choos h posiiv sign insad of ngaiv sign on squar roo rm: 4

22 Eampl : Domain 4 of 4 Thus h soluions o h iniial valu problm d d 4 ar givn b 4 implici plici From plici rprsnaion of i follows ha and hnc domain of is -. No - ilds which maks dnominaor of d/d zro vrical angn. Convrsl domain of can b simad b locaing vrical angns on graph usful for implicil dfind soluions.

23 Eampl : Implici Soluion of Iniial Valu Problm of Considr h following iniial valu problm: cos Sparaing variabls and using calculus w obain ln d cos d d sin C cos d Using h iniial condiion i follows ha ln sin

24 Eampl : Graph of Soluions of Thus cos ln sin Th graph of his soluion black along wih h graphs of h dircion fild and svral ingral curvs blu for his diffrnial quaion is givn blow.

25 Ch.: Modling wih Firs Ordr Equaions Mahmaical modls characriz phsical ssms ofn using diffrnial quaions. Modl Consrucion: Translaing phsical siuaion ino mahmaical rms. Clarl sa phsical principls blivd o govrn procss. Diffrnial quaion is a mahmaical modl of procss picall an approimaion. Analsis of Modl: Solving quaions or obaining qualiaiv undrsanding of soluion. Ma simplif modl as long as phsical ssnials ar prsrvd. Comparison wih Eprimn or Obsrvaion: Vrifis soluion or suggss rfinmn of modl.

26 Eampl : Mic and Owls Suppos a mous populaion rproducs a a ra proporional o currn populaion wih a ra consan of.5 mic/monh assuming no owls prsn. Furhr assum ha whn an owl populaion is prsn h a 5 mic pr da on avrag. Th diffrnial quaion dscribing mous populaion in h prsnc of owls assuming das in a monh is p.5 p 45 Using mhods of calculus w solvd his quaion in Chapr. obaining p 9 k.5

27 Eampl : Sal Soluion of 7 A im a ank conains Q lb of sal dissolvd in gal of war. Assum ha war conaining ¼ lb of sal/gal is nring ank a ra of r gal/min and lavs a sam ra. a S up IVP ha dscribs his sal soluion flow procss. b Find amoun of sal Q in ank a an givn im. c Find limiing amoun Q L of sal Q in ank afr a vr long im. d If r & Q Q L find im T afr which sal is wihin % of Q L. Find flow ra r rquird if T is no o cd 45 min.

28 Eampl : a Iniial Valu Problm of 7 A im a ank conains Q lb of sal dissolvd in gal of war. Assum war conaining ¼ lb of sal/gal nrs ank a ra of r gal/min and lavs a sam ra. Assum sal is nihr crad or dsrod in ank and disribuion of sal in ank is uniform sirrd. Thn dq / d ra in - ra ou Ra in: /4 lb sal/galr gal/min r/4 lb/min Ra ou: If hr is Q lbs sal in ank a im hn concnraion of sal is Q lb/ gal and i flows ou a ra of [Qr/] lb/min. Thus our IVP is dq d r rq 4 Q Q

29 Eampl : b Find Soluion Q of 7 To find amoun of sal Q in ank a an givn im w nd o solv h iniial valu problm To solv w us h mhod of ingraing facors: or μ Q Q a dq d r / 5 Q 5 rq r / r 4 r / [ Q 5] r 4 r / d Q r / r / Q r / Q [ ] r / 5 C 5 C r /

30 Eampl : c Find Limiing Amoun Q L 4 of 7 N w find h limiing amoun Q L of sal Q in ank afr a vr long im: Q limq L [ Q 5] r / 5 lb lim 5 This rsul maks sns sinc ovr im h incoming sal soluion will rplac original sal soluion in ank. Sinc incoming soluion conains.5 lb sal / gal and ank is gal vnuall ank will conain 5 lb sal. Th graph shows ingral curvs for r and diffrn valus of Q. Q 5 r / r / Q

31 Eampl : d Find Tim T 5 of 7 Suppos r and Q Q L. To find im T afr which Q is wihin % of Q L firs no Q Q L 5 lb hnc [ ] r /. Q 5 Q N % of 5 lb is.5 lb and hus w solv 5.5..T ln..t T 5 5 ln...t.4 min

32 Eampl : Find Flow Ra 6 of 7 To find flow ra r rquird if T is no o cd 45 minus rcall from par d ha Q Q L 5 lb wih r / Q 5 5 and soluion curvs dcras from 5 o 5.5. Thus w solv r ln..45r r 5 5 ln r 8.69 gal/min

33 Eampl : Discussion 7 of 7 Sinc siuaion is hpohical h modl is valid. As long as flow ras ar accura and concnraion of sal in ank is uniform hn diffrnial quaion is accura dscripion of flow procss. Modls of his kind ar ofn usd for polluion in lak drug concnraion in organ c. Flow ras ma b hardr o drmin or ma b variabl and concnraion ma no b uniform. Also ras of inflow and ouflow ma no b sam so variaion in amoun of liquid mus b akn ino accoun.

34 Eampl : Pond Polluion of 7 Considr a pond ha iniiall conains million gallons of frsh war. War conaining oic was flows ino h pond a h ra of 5 million gal/ar and is a sam ra. Th concnraion c of oic was in h incoming war varis priodicall wih im: c sin g/gal a Consruc a mahmaical modl of his flow procss and drmin amoun Q of oic was in pond a im. b Plo soluion and dscrib in words h ffc of h variaion in h incoming concnraion.

35 Eampl : a Iniial Valu Problm of 7 Pond iniiall conains million gallons of frsh war. War conaining oic was flows ino pond a ra of 5 million gal/ar and is pond a sam ra. Concnraion is c sin g/gal of oic was in incoming war. Assum oic was is nihr crad or dsrod in pond and disribuion of oic was in pond is uniform sirrd. Thn dq / d ra in - ra ou Ra in: sin g/gal5 6 gal/ar Ra ou: If hr is Q g of oic was in pond a im hn concnraion of sal is Q lb/ 7 gal and i flows ou a ra of [Q g/ 7 gal][5 6 gal/ar]

36 Eampl : a Iniial Valu Problm Scaling of 7 Rcall from prvious slid ha Ra in: sin g/gal5 6 gal/ar Ra ou: [Q g/ 7 gal][5 6 gal/ar] Q/ g/r. Thn iniial valu problm is dq d Q Q 6 sin 5 Chang of variabl scaling: L q Q/ 6. Thn dq d q 5 sin q

37 Eampl : a Solv Iniial Valu Problm 4 of 7 To solv h iniial valu problm q q / 5sin q w us h mhod of ingraing facors: μ q a / / / 5sin d Using ingraion b pars s n slid for dails and h iniial condiion w obain afr simplifing / / q 4 q cos / sin 7 cos 7 7 / / sin C

38 Eampl : a Ingraion b Pars 5 of 7 C d C d C d d d d d sin 7 cos 7 4 sin 5 sin 7 cos 7 8 sin sin 8 cos sin 6 7 sin 6 sin 8 cos sin 4 sin 4 cos cos 4 cos sin / / / / / / / / / / / / / / / / / /

39 Eampl : b Analsis of soluion 6 of 7 Thus our iniial valu problm and soluion is dq q d q 5sin 4 7 cos 7 q A graph of soluion along wih dircion fild for diffrnial quaion is givn blow. No ha ponnial rm is imporan for small bu dcas awa for larg. Also would b quilibrium soluion if no for sin rm. sin 7 /

40 Eampl : b Analsis of Assumpions 7 of 7 Amoun of war in pond conrolld nirl b ras of flow and non is los b vaporaion or spag ino ground or gaind b rainfall c. Amoun of polluion in pond conrolld nirl b ras of flow and non is los b vaporaion spag ino ground dilud b rainfall absorbd b fish plans or ohr organisms c. Disribuion of polluion hroughou pond is uniform.

41 Ch.4: Diffrncs Bwn Linar and Nonlinar Equaions Rcall ha a firs ordr ODE has h form ' f and is linar if f is linar in and nonlinar if f is nonlinar in. Eampls: ' - '. In his scion w will s ha firs ordr linar and nonlinar quaions diffr in a numbr of was including: Th hor dscribing isnc and uniqunss of soluions and corrsponding domains ar diffrn. Soluions o linar quaions can b prssd in rms of a gnral soluion which is no usuall h cas for nonlinar quaions. Linar quaions hav plicil dfind soluions whil nonlinar quaions picall do no and nonlinar quaions ma or ma no hav implicil dfind soluions. For boh ps of quaions numrical and graphical consrucion of soluions ar imporan.

42 Thorm.4. Considr h linar firs ordr iniial valu problm: d d p g If h funcions p and g ar coninuous on an opn inrval α β conaining h poin hn hr iss a uniqu soluion φ ha saisfis h IVP for ach in α β. Proof oulin: Us Ch. discussion and rsuls: μ g d whr μ μ p s ds

43 Thorm.4. Considr h nonlinar firs ordr iniial valu problm: d d f Suppos f and f/ ar coninuous on som opn rcangl α β γ δ conaining h poin. Thn in som inrval - h h α β hr iss a uniqu soluion φ ha saisfis h IVP. Proof discussion: Sinc hr is no gnral formula for h soluion of arbirar nonlinar firs ordr IVPs his proof is difficul and is bond h scop of his cours. I urns ou ha condiions sad in Thm.4. ar sufficin bu no ncssar o guaran isnc of a soluion and coninui of f nsurs isnc bu no uniqunss of φ.

44 Eampl : Linar IVP Rcall h iniial valu problm from Chapr. slids: 5 5 ln Th soluion o his iniial valu problm is dfind for > h inrval on which p -/ is coninuous. If h iniial condiion is - hn h soluion is givn b sam prssion as abov bu is dfind on <. In ihr cas Thorm.4. guarans ha soluion is uniqu on corrsponding inrval.

45 Eampl : Nonlinar IVP of Considr nonlinar iniial valu problm from Ch.: d d 4 Th funcions f and f/ ar givn b 4 f 4 f and ar coninuous cp on lin. Thus w can draw an opn rcangl abou - on which f and f/ ar coninuous as long as i dosn covr. How wid is rcangl? Rcall soluion dfind for > - wih 4

46 Eampl : Chang Iniial Condiion of Our nonlinar iniial valu problm is wih which ar coninuous cp on lin. If w chang iniial condiion o hn Thorm.4. is no saisfid. Solving his nw IVP w obain Thus a soluion iss bu is no uniqu. 4 4 f f 4 d d > ±

47 Eampl : Nonlinar IVP Considr nonlinar iniial valu problm / Th funcions f and f/ ar givn b / f / f Thus f coninuous vrwhr bu f/ dosn is a and hnc Thorm.4. is no saisfid. Soluions is bu ar no uniqu. Sparaing variabls and solving w obain / d d / c If iniial condiion is no on -ais hn Thorm.4. dos guaran isnc and uniqunss. ± /

48 Eampl 4: Nonlinar IVP Considr nonlinar iniial valu problm Th funcions f and f/ ar givn b Thus f and f/ ar coninuous a so Thm.4. guarans ha soluions is and ar uniqu. Sparaing variabls and solving w obain Th soluion is dfind on -. No ha h singulari a is no obvious from original IVP samn. f f c c d d

49 Inrval of Dfiniion: Linar Equaions B Thorm.4. h soluion of a linar iniial valu problm p g iss hroughou an inrval abou on which p and g ar coninuous. Vrical asmpos or ohr disconinuiis of soluion can onl occur a poins of disconinui of p or g. Howvr soluion ma b diffrniabl a poins of disconinui of p or g. S Chapr.: Eampl of. Compar hs commns wih Eampl and wih prvious linar quaions in Chapr and Chapr.

50 Inrval of Dfiniion: Nonlinar Equaions In h nonlinar cas h inrval on which a soluion iss ma b difficul o drmin. Th soluion φ iss as long as φ rmains wihin rcangular rgion indicad in Thorm.4.. This is wha drmins h valu of h in ha horm. Sinc φ is usuall no known i ma b impossibl o drmin his rgion. In an cas h inrval on which a soluion iss ma hav no simpl rlaionship o h funcion f in h diffrnial quaion ' f in conras wih linar quaions. Furhrmor an singulariis in h soluion ma dpnd on h iniial condiion as wll as h quaion. Compar hs commns o h prcding ampls.

51 Gnral Soluions For a firs ordr linar quaion i is possibl o obain a soluion conaining on arbirar consan from which all soluions follow b spcifing valus for his consan. For nonlinar quaions such gnral soluions ma no is. Tha is vn hough a soluion conaining an arbirar consan ma b found hr ma b ohr soluions ha canno b obaind b spcifing valus for his consan. Considr Eampl 4: Th funcion is a soluion of h diffrnial quaion bu i canno b obaind b spcifing a valu for c in soluion found using sparaion of variabls: d d c

52 Eplici Soluions: Linar Equaions B Thorm.4. a soluion of a linar iniial valu problm p g iss hroughou an inrval abou on which p and g ar coninuous and his soluion is uniqu. Th soluion has an plici rprsnaion μ g d whr μ μ p s ds and can b valuad a an appropria valu of as long as h ncssar ingrals can b compud.

53 Eplici Soluion Approimaion For linar firs ordr quaions an plici rprsnaion for h soluion can b found as long as ncssar ingrals can b solvd. If ingrals can b solvd hn numrical mhods ar ofn usd o approima h ingrals. Δ n k k k k ds s p g d g C d g whr μ μ μ μ μ

54 Implici Soluions: Nonlinar Equaions For nonlinar quaions plici rprsnaions of soluions ma no is. As w hav sn i ma b possibl o obain an quaion which implicil dfins h soluion. If quaion is simpl nough an plici rprsnaion can somims b found. Ohrwis numrical calculaions ar ncssar in ordr o drmin valus of for givn valus of. Ths valus can hn b plod in a skch of h ingral curv. Rcall h following ampl from Ch. slids: cos ln sin

55 Dircion Filds In addiion o using numrical mhods o skch h ingral curv h nonlinar quaion islf can provid nough informaion o skch a dircion fild. Th dircion fild can ofn show h qualiaiv form of soluions and can hlp idnif rgions in h -plan whr soluions hibi inrsing faurs ha mri mor daild analical or numrical invsigaions. Chapr.7 and Chapr 8 focus on numrical mhods.

56 Ch.5: Auonomous Equaions and Populaion Dnamics In his scion w amin quaions of h form ' f calld auonomous quaions whr h indpndn variabl dos no appar plicil. Th main purpos of his scion is o larn how gomric mhods can b usd o obain qualiaiv informaion dircl from diffrnial quaion wihou solving i. Eampl Eponnial Growh: r r > Soluion: r

57 Logisic Growh An ponnial modl ' r wih soluion r prdics unlimid growh wih ra r > indpndn of populaion. Assuming insad ha growh ra dpnds on populaion siz rplac r b a funcion h o obain ' h. W wan o choos growh ra h so ha h r whn is small h dcrass as grows largr and h < whn is sufficinl larg. Th simpls such funcion is h r a whr a >. Our diffrnial quaion hn bcoms r a r a > This quaion is known as h Vrhuls or logisic quaion.

58 Logisic Equaion Th logisic quaion from h prvious slid is r a r a > This quaion is ofn rwrin in h quivaln form d d r K whr K r/a. Th consan r is calld h inrinsic growh ra and as w will s K rprsns h carring capaci of h populaion. A dircion fild for h logisic quaion wih r and K is givn hr.

59 Logisic Equaion: Equilibrium Soluions Our logisic quaion is d d r r K > K Two quilibrium soluions ar clarl prsn: φ φ K In dircion fild blow wih r K no bhavior of soluions nar quilibrium soluions: is unsabl is asmpoicall sabl.

60 Auonomous Equaions: Equilibrium Solns Equilibrium soluions of a gnral firs ordr auonomous quaion ' f can b found b locaing roos of f. Ths roos of f ar calld criical poins. For ampl h criical poins of h logisic quaion d d r ar and K. Thus criical poins ar consan funcions quilibrium soluions in his sing. K

61 Logisic Equaion: Qualiaiv Analsis and Curv Skching of 7 To br undrsand h naur of soluions o auonomous quaions w sar b graphing f vs.. In h cas of logisic growh ha mans graphing h following funcion and analzing is graph using calculus. f r K

62 Logisic Equaion: Criical Poins of 7 Th inrcps of f occur a and K corrsponding o h criical poins of logisic quaion. Th vr of h parabola is K/ rk/4 as shown blow. [ ] 4 rk K K K r K f K K K r K K r f K r f s

63 Logisic Soluion: Incrasing Dcrasing of 7 No d/d > for < < K so is an incrasing funcion of hr indica wih righ arrows along -ais on < < K. Similarl is a dcrasing funcion of for > K indica wih lf arrows along -ais on > K. In his con h -ais is ofn calld h phas lin. d d r r > K

64 Logisic Soluion: Spnss Flanss 4 of 7 No d/d whn or K so is rlaivl fla hr and gs sp as movs awa from or K. d d r K

65 Logisic Soluion: Concavi 5 of 7 N o amin concavi of w find '': d d d f d f f f Thus h graph of is concav up whn f and f' hav sam sign which occurs whn < < K/ and > K. d d Th graph of is concav down whn f and f' hav opposi signs which occurs whn K/ < < K. Inflcion poin occurs a inrscion of and lin K/.

66 Logisic Soluion: Curv Skching 6 of 7 Combining h informaion on h prvious slids w hav: Graph of incrasing whn < < K. Graph of dcrasing whn > K. Slop of approimal zro whn or K. Graph of concav up whn < < K/ and > K. Graph of concav down whn K/ < < K. Inflcion poin whn K/. Using his informaion w can skch soluion curvs for diffrn iniial condiions.

67 Logisic Soluion: Discussion 7 of 7 Using onl h informaion prsn in h diffrnial quaion and wihou solving i w obaind qualiaiv informaion abou h soluion. For ampl w know whr h graph of is h sps and hnc whr changs mos rapidl. Also nds asmpoicall o h lin K for larg. Th valu of K is known as h carring capaci or sauraion lvl for h spcis. No how soluion bhavior diffrs from ha of ponnial quaion and hus h dcisiv ffc of nonlinar rm in logisic quaion.

68 Solving h Logisic Equaion of Providd and K w can rwri h logisic ODE: d K rd Epanding h lf sid using parial fracions K Thus h logisic quaion can b rwrin as A K / K d rd / K Ingraing h abov rsul w obain B ln ln r C K A B K B A K

69 Solving h Logisic Equaion of W hav: ln ln r C K If < < K hn < < K and hnc ln ln r C K Rwriing using propris of logs: ln or K r C K whr r K K r C K c r

70 Soluion of h Logisic Equaion of W hav: K r K for < < K. I can b shown ha soluion is also valid for > K. Also his soluion conains quilibrium soluions and K. Hnc soluion o logisic quaion is K r K

71 Logisic Soluion: Asmpoic Bhavior Th soluion o logisic ODE is K r K W us limis o confirm asmpoic bhavior of soluion: lim lim K lim r K K Thus w can conclud ha h quilibrium soluion K is asmpoicall sabl whil quilibrium soluion is unsabl. Th onl wa o guaran soluion rmains nar zro is o mak. K

72 K r K Eampl: Pacific Halibu of L b biomass in kg of halibu populaion a im wih r.7/ar and K kg. If.5K find a biomass ars lar b h im τ such ha τ.75k. a For convninc scal quaion: K Thn K and hnc K K r K K kg

73 Eampl: Pacific Halibu Par b of b Find im τ for which τ.75k..95 ars.75.5 ln τ τ τ τ τ K K K K K K K K K K K K K r r r r r K K K K

74 Criical Thrshold Equaion of Considr h following modificaion of h logisic ODE: d d r r > T Th graph of h righ hand sid f is givn blow.

75 Criical Thrshold Equaion: Qualiaiv Analsis and Soluion of Prforming an analsis similar o ha of h logisic cas w obain a graph of soluion curvs shown blow. T is a hrshold valu for in ha populaion dis off or grows unboundd dpnding on which sid of T h iniial valu is. S also laminar flow discussion in. I can b shown ha h soluion o h hrshold quaion is d d r r > T T T r

76 Logisic Growh wih a Thrshold of In ordr o avoid unboundd growh for > T as in prvious sing considr h following modificaion of h logisic quaion: d d r r > and T K < T Th graph of h righ hand sid f is givn blow. < K

77 Logisic Growh wih a Thrshold of Prforming an analsis similar o ha of h logisic cas w obain a graph of soluion curvs shown blow righ. T is hrshold valu for in ha populaion dis off or grows owards K dpnding on which sid of T is. K is h carring capaci lvl. No: and K ar sabl quilibrium soluions and T is an unsabl quilibrium soluion.

78 Ch.6: Eac Equaions & Ingraing Facors Considr a firs ordr ODE of h form M N Suppos hr is a funcion ψ such ha ψ M ψ N and such ha ψ c dfins φ implicil. Thn ψ ψ d d M N ψ d d and hnc h original ODE bcoms d ψ d [ φ ] Thus ψ c dfins a soluion implicil. In his cas h ODE is said o b ac. [ φ ]

79 Thorm.6. Suppos an ODE can b wrin in h form M N whr h funcions M N M and N ar all coninuous in h rcangular rgion R: α β γ δ. Thn Eq. is an ac diffrnial quaion iff M N R Tha is hr iss a funcion ψ saisfing h condiions iff M and N saisf Equaion. ψ M ψ N

80 Eampl : Eac Equaion of 4 Considr h following diffrnial quaion. Thn and hnc From Thorm.6. Thus d d N M 4 4 ODE is ac 4 N M 4 4 ψ ψ 4 4 C d d ψ ψ

81 Eampl : Soluion of 4 W hav and I follows ha Thus B Thorm.6. h soluion is givn implicil b 4 4 ψ ψ 4 4 C d d ψ ψ k C C C 4 4 ψ k 4 ψ c 8

82 Eampl : Dircion Fild and Soluion Curvs of 4 Our diffrnial quaion and soluions ar givn b d d c A graph of h dircion fild for his diffrnial quaion along wih svral soluion curvs is givn blow.

83 Eampl : Eplici Soluion and Graphs 4 of 4 Our soluion is dfind implicil b h quaion blow. 8 c In his cas w can solv h quaion plicil for : 8 c 4 ± 7 c Soluion curvs for svral valus of c ar givn blow.

84 Eampl : Eac Equaion of Considr h following diffrnial quaion. cos sin Thn M cos N sin and hnc M cos N ODE is ac From Thorm.6. ψ M cos Thus ψ N sin cos d sin C ψ ψ d

85 Eampl : Soluion of W hav and I follows ha Thus B Thorm.6. h soluion is givn implicil b sin cos N M ψ ψ sin cos C d d ψ ψ k C C C sin sin ψ k sin ψ c sin

86 Eampl : Dircion Fild and Soluion Curvs of Our diffrnial quaion and soluions ar givn b cos sin sin c A graph of h dircion fild for his diffrnial quaion along wih svral soluion curvs is givn blow.

87 Eampl : Non-Eac Equaion of Considr h following diffrnial quaion. Thn and hnc M To show ha our diffrnial quaion canno b solvd b his mhod l us sk a funcion ψ such ha ψ M ψ N Thus M N N ODE is no ac d / C ψ ψ d

88 Eampl : Non-Eac Equaion of W sk ψ such ha and Thn Thus hr is no such funcion ψ. Howvr if w incorrcl procd as bfor w obain as our implicil dfind which is no a soluion of ODE. N M ψ ψ / C d d ψ ψ k C C C / / /??? ψ c

89 Eampl : Graphs of Our diffrnial quaion and implicil dfind ar c A plo of h dircion fild for his diffrnial quaion along wih svral graphs of ar givn blow. From hs graphs w s furhr vidnc ha dos no saisf h diffrnial quaion.

90 I is somims possibl o convr a diffrnial quaion ha is no ac ino an ac quaion b mulipling h quaion b a suiabl ingraing facor μ: For his quaion o b ac w nd This parial diffrnial quaion ma b difficul o solv. If μ is a funcion of alon hn μ and hnc w solv providd righ sid is a funcion of onl. Similarl if μ is a funcion of alon. S for mor dails. Ingraing Facors N M N M μ μ μ μ μ μ μ N M N M N M μ μ N N M d d

91 Eampl 4: Non-Eac Equaion Considr h following non-ac diffrnial quaion. Sking an ingraing facor w solv h linar quaion dμ M N dμ μ μ μ d N d Mulipling our diffrnial quaion b μ w obain h ac quaion which has is soluions givn implicil b c

92 Ch.7: Numrical Approimaions: Eulr s Mhod Rcall ha a firs ordr iniial valu problm has h form d f d If f and f/ ar coninuous hn his IVP has a uniqu soluion φ in som inrval abou. Whn h diffrnial quaion is linar sparabl or ac w can find h soluion b smbolic manipulaions. Howvr h soluions for mos diffrnial quaions of his form canno b found b analical mans. Thrfor i is imporan o b abl o approach h problm in ohr was.

93 Dircion Filds For h firs ordr iniial valu problm f w can skch a dircion fild and visualiz h bhavior of soluions. This has h advanag of bing a rlaivl simpl procss vn for complicad quaions. Howvr dircion filds do no lnd hmslvs o quaniaiv compuaions or comparisons.

94 Numrical Mhods For our firs ordr iniial valu problm f an alrnaiv is o compu approima valus of h soluion φ a a slcd s of -valus. Idall h approima soluion valus will b accompanid b rror bounds ha nsur h lvl of accurac. Thr ar man numrical mhods ha produc numrical approimaions o soluions of diffrnial quaions som of which ar discussd in Chapr 8. In his scion w amin h angn lin mhod which is also calld Eulr s Mhod.

95 Eulr s Mhod: Tangn Lin Approimaion For h iniial valu problm f w bgin b approimaing soluion φ a iniial poin. Th soluion passs hrough iniial poin wih slop f. Th lin angn o soluion a iniial poin is hus f Th angn lin is a good approimaion o soluion curv on an inrval shor nough. Thus if is clos nough o w can approima φ b f

96 Eulr s Formula For a poin clos o w approima φ using h lin passing hrough wih slop f : Thus w cra a squnc n of approimaions o φ n : whr f n f n n. For a uniform sp siz h n n- Eulr s formula bcoms n n n n n f f f M f K n h f n n n

97 Eulr Approimaion To graph an Eulr approimaion w plo h poins n n and hn connc hs poins wih lin sgmns. whr f f n n fn n n n n n

98 Eampl : Eulr s Mhod of For h iniial valu problm 9.8. w can us Eulr s mhod wih h. o approima h soluion a....4 as shown blow. 4 f f f f h h h h

99 Eampl : Eac Soluion of W can find h ac soluion o our IVP as in Chapr.: k 49 d.d 49 ln 49. C 49. k k 49. ± C

100 Eampl : Error Analsis of From abl blow w s ha h rrors ar small. This is mos likl du o round-off rror and h fac ha h ac soluion is approimal linar on [.4]. No: ac appro Prcn Rlaiv Error ac Eac Appro Error % Rl Error

101 Eampl : Eulr s Mhod of For h iniial valu problm 4 w can us Eulr s mhod wih h. o approima h soluion a and 4 as shown blow. 4 M f f f f h h h h Eac soluion s Chapr.:

102 Eampl : Error Analsis of Th firs n Eulr appros ar givn in abl blow on lf. A abl of approimaions for is givn on righ. S for numrical rsuls wih h Th rrors ar small iniiall bu quickl rach an unaccpabl lvl. This suggss a nonlinar soluion. Eac Appro Error % Rl Error Eac Appro Error % Rl Error Eac Soluion : 7 4 4

103 Eampl : Error Analsis & Graphs of Givn blow ar graphs showing h ac soluion rd plod oghr wih h Eulr approimaion blu. Eac Appro Error % Rl Error Eac Soluion : 7 4 4

104 Gnral Error Analsis Discussion of 4 Rcall ha if f and f/ ar coninuous hn our firs ordr iniial valu problm f has a soluion φ in som inrval abou. In fac h quaion has infinil man soluions ach on indd b a consan c drmind b h iniial condiion. Thus φ is h mmbr of an infini famil of soluions ha saisfis φ.

105 Gnral Error Analsis Discussion of 4 Th firs sp of Eulr s mhod uss h angn lin o φ a h poin in ordr o sima φ wih. Th poin is picall no on h graph of φ bcaus is an approimaion of φ. Thus h n iraion of Eulr s mhod dos no us a angn lin approimaion o φ bu rahr o a narb soluion φ ha passs hrough h poin. Thus Eulr s mhod uss a succssion of angn lins o a squnc of diffrn soluions φ φ φ of h diffrnial quaion.

106 Error Analsis Eampl: Convrging Famil of Soluions of 4 Sinc Eulr s mhod uss angn lins o a squnc of diffrn soluions h accurac afr man sps dpnds on bhavior of soluions passing hrough n n n For ampl considr h following iniial valu problm: / φ 6 Th dircion fild and graphs of a fw soluion curvs ar givn blow. No ha i dosn mar which soluions w ar approimaing wih angn lins as all soluions g closr o ach ohr as incrass. Rsuls of using Eulr s mhod for his quaion ar givn in.

107 Error Analsis Eampl: Divrgn Famil of Soluions 4 of 4 Now considr h iniial valu problm for Eampl : Th dircion fild and graphs of soluion curvs ar givn blow. Sinc h famil of soluions is divrgn a ach sp of Eulr s mhod w ar following a diffrn soluion han h prvious sp wih ach soluion sparaing from h dsird on mor and mor as incrass. 4

108 Error Bounds and Numrical Mhods In using a numrical procdur kp in mind h qusion of whhr h rsuls ar accura nough o b usful. In our ampls w compard approimaions wih ac soluions. Howvr numrical procdurs ar usuall usd whn an ac soluion is no availabl. Wha is ndd ar bounds for or simas of rrors which do no rquir knowldg of ac soluion. Mor discussion on hs issus and ohr numrical mhods is givn in Chapr 8. Sinc numrical approimaions idall rflc bhavior of soluion a mmbr of a divrging famil of soluions is hardr o approima han a mmbr of a convrging famil. Also dircion filds ar ofn a rlaivl as firs sp in undrsanding bhavior of soluions.