September 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline

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1 Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Introucton to Ornar Dffrntal Equatons Larr artto Mchancal Engnrng AB Smnar n Engnrng Analss Sptmbr 7, 7 Outln Rvw numrcal solutons Bascs of ffrntal quatons Frst orr quatons Sparabl solutons Gnral soluton for lnar quaton Introucton to scon orr quatons Problms consr Bass of solutons onstant-coffcnt, homognous cas Rvw Numrcal Solutons Basc Dffrntal Equatons Gauss lmnaton s basc approach N pvotng stratgs to ruc roun-off rror n soluton Mofcatons of Gauss lmnaton Gauss-Joran somtms us for fnng nvrs of matr LU mtho gnrall prfrr Dos most of th lmnaton work wthout knowng th rght-han-s (b) vctor D ntgr vctor rqur for pvotng A ffrntal quaton s an quaton that contans rvatvs of a pnnt varabl,.g., () or u(, Dffrntal quaton soluton gvs () or u(, as a functon of npnnt varabl(s) Ornar ffrntal quatons (ODE) hav on npnnt varabl Partal ffrntal quatons (PDE) hav mor than on npnnt varabl Dfntons an Trms Dffrntal quatons hav bounar contons or ntal contons A gnral soluton to th ffrntal quaton s on whch can ft an bounar or ntal conton b ajustng constants n th soluton A soluton that satsfs th ffrntal quaton an th bounar or ntal contons s call a partcular soluton Mor Dfntons an Trms Th orr of a ffrntal quaton s th orr of th hghst rvatv n th quaton A lnar ffrntal quaton s on n whch th pnnt varabl an ts rvatvs all appar n lnar trms A homognous ffrntal quaton s on n whch all trms nvolv th pnnt varabl an ts rvatvs 6 ME A Smnar n Engnrng Analss Pag

2 Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Eampls of ODEs : Inpnnt : Dpnnt Applcatons Thr-orr, lnar, homognous Scon-orr, nonlnar, homognous Scon-orr, lnar, non-homognous Thr-orr, non-lnar, non-homognous sn( ) sn( cos( ) 7 Frst orr ffrntal quatons ar oftn us to mol rat procsss Nwton s coolng T/t = -k(t - T ) chmcal ractons, c /t = f(c,t) Nwton s scon law, F = ma las to scon orr quatons for mchancal sstms m t F Dflcton,, of rctangular bam ornt n rcton EI / = f() 8 Sparabl Forms Smpl ffrntal quatons can b wrttn as ntgrals Evn f numrcal quaratur s rqur ths s mor accurat than numrcal soluton of ODE f ( ) f ( ) f ( ) g( f g ( ) ( ) u h h( u) u 9 Lnar Frst-Orr Equaton Th soluton to th frst-orr quaton p Is gvn b th followng rsult h r h r whr h p Th constant,, rqurs th spcfcaton of th valu of at a partcular valu of ;.g., = at =, + Q(, = Is, + Q(, = an act form? From ffrntal of a functon of two varabls, f(,, s f P an Q satsf partal rvatv rlaton If f =, f = f f f f f, Q(, f P Q f f Q P f f Eact Form Q P If,, +Q(, = f W ma not know (or car) what f s, but w us f =, +Q(, to solv th ffrntal quaton W also know that, +Q(, = mans that f = or f = a constant W also know that P an Q ar rvatvs of ths mstrous f functon f f Q P, an Q(, onl f ME A Smnar n Engnrng Analss Pag

3 Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Eact Forms II Intgrat f =, + Q(, for constant f (f = ) f = constant,, bcaus f = f f, g( const f g( Q(,, const g( Q(,, h( const Eact Forms III Fnal quaton must b a functon of onl Intgrat ths quaton for g( g( Q(,, h( const g( g Q(, P (, ) const Substtut g( nto quaton for f f f, g( const const f f Eact Forms IV const, g( g( Q(,, const, Q(,, const ombn constants nto a sngl constant Obtan mplct rlatonshp btwn an Solvng Eact P + Q = const, Q(, const, Stp Intgrat, wth constant Stp Tak th rvatv of th stp rsult an subtract t from Q(, Stp Intgrat rsult of stp, that wll b a functon of onl, ovr Stp A rsults of stps an 6 Intgratng Factors Us to ntgrat, + Q(, = f P an Q ar not act Basc a s to fn a factor, F, that multpls th orgnal quaton: FP + FQ = Fn th F factor so that FP an FQ ar act Us tral an rror or FQ FP procss outln n Krszg to fn F 7 Frst-orr Equatons Frst orr rat quaton whr rat s proportonal to amount /t = -k = -k(t-t ) Gnral lnar frst orr quaton for (): / + f() = g() has clos form soluton shown blow s foun from ntal conton p f ( ) p p g( ) 8 ME A Smnar n Engnrng Analss Pag

4 Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Estnc an Unqunss Important bcaus w can tr numrcal soluton of an ODE wth no soluton Eamn / = f(, wth ( ) = n a rgon < a an < b Drvat s boun: f(, K Equaton has a soluton n rgon < mn(a, b/k) Unqunss rqurs f/ M 9 Estnc an Unqunss Eampl: =, () = Hr w hav / = f(, = / wth ( = ) = = Rgon s < a an < b Drvatv s not boun at = = Thrfor hav no solutons: f(, K Attmpt soluton s = ln() +, but w cannot appl ths at = = Scon Orr Equatons Frst look at homognous lnar quatons Thn consr nonhomognous quatons Most nonlnar quatons rqur numrcal soluton p( ) q( ) t p( ) q( ) r( ) t,,, t Lnar Homognous n Orr p( ) q( ) t An lnar combnaton of two solutons, an, to ths quaton, = c + c, s also a soluton an prov ths b substtutng c + c combnaton nto orgnal quaton for whch an ar solutons Lnar Homognous n Orr II p( ) q( ) A bass of solutons for ths quaton s an two lnarl npnnt solutons an = c + c s a gnral soluton whr c an c can b us to ft ntal or bounar contons Intal contons spcf () an () Bounar contons spcf (a) an (b) Estnc an Unqunss p( ) q( ) wth K ' K Th ntal valu problm fn abov on th opn ntrval, I, fn as a < < b, has a unqu soluton f p() an q() ar contnuous on th ntrval, I, an s locat on th ntrval, I. Nt sl scusss lnar npnnc of solutons to ths ODE ME A Smnar n Engnrng Analss Pag

5 Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Lnar Inpnnc p( ) q( ) wth K ' K Th soluton to th ntal valu problm fn abov can b wrttn as () = k () + k () whr an ar lnarl npnnt For k () + k () =, w must hav k = an k = for an to b lnarl npnnt Wronsk Dtrmnant, W s call th Wronsk trmnant or Wronskan for th ODE unr scusson p( ) q( ) wth K ' K Th solutons to th ODE ar lnarl npnnt f thr s som pont,, n th soluton ntrval for whch W s not 6 onstant offcnts Smplst soluton s whn p() an q() ar constants Th ffrntal quaton for ths cas s shown blow Th soluton s shown on th nt sl c 7 onstant offcnt Soluton ODE solutons ar bas on soluton of charactrstc quaton + a + b = Solutons for ar roots of quaratc Two lnarl-npnnt ODE solutons ar = an = 8 onstant offcnt Soluton Two solutons, = an =, form bass for gnral soluton Gnral soluton s = + an show soluton s corrct b substtutng nto orgnal ODE Fn an from ntal or bounar contons 9 onstant offcnt Soluton Gnral soluton s lnar combnaton of lnarl-npnnt solutons = + = + Drvatvs of th nvual solutons ar = an = - ; = an = Us ths rvatvs (plus = an = ) to vrf nvual solutons Dos? t ME A Smnar n Engnrng Analss Pag

6 Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 ME A Smnar n Engnrng Analss Pag 6 onstant offcnt Soluton X Dfnton : haractrstc quaton X Ercs Fn th soluton to th followng ODE wth ntal contons that () = an () = Rpat th soluton to th ODE wth bounar contons that () = an () = Ercs Soluton (/) ODE has form scuss prvousl wth α = an β = / Soluton s = + whr,, Appl bounar contons to fn an Ercs Soluton (/) Frst cas () =, () = ) ( ' W can show that () = an () = Ercs Soluton (/) Scon cas () =, () = ) ( W can show that () = an () = hck Gnral Soluton (/) Plug soluton nto orgnal ODE 6 ' ' '

7 Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 ME A Smnar n Engnrng Analss Pag 7 hck Gnral Soluton (/) 7 6