important aoilaesxuinerfuuitiemubeuenm .nu?ahire.ouanwiod u.sn Why : differential Note: within only possibility Noted Essential ingredient equation

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1 I) Motivtion ( Ch 1 textbook ) 2 Wh re we here? Stud objects tht evolve within described b reiltions continuous quntities ( ultimtel dismte toms! ) Mmticl description model [ Process Wht u most modeling series steps importnt to processes Wht importnt mible should 1 How deeuibe sstem 9 oilesxuinerfuuitiemubeuenm nu?hireounwiod use 1 Note models tpicll be usn improved b ohinvin nlzing correcting even Noted time not but ver onl possibilit ex intuitive mn spce exmples long d! Essentil ingredient eqution differentil 2

2 Exmple Ide rte ten known reltion between chnge one vrible nd vribles mselves " LH d x2+ E d poihon s}me s unknown function reltion Time between xlt ) duldt t 3 nd Definition A Is n eqution differentil eqution ( b E which contins derivtives ) one ( or more ) dependent vribles with respect to one l or more ) independent vribles nb If onl one independent vrible onl ordinr derivtives ORDINARY diff eq UNRNOWNISI Is A FUNCTION OF 1 VARIABLE mb Ff more thn one independent prtil derivtives D PARTIAL diff eg vrible UNKNOWN ( s ) Is FUNCTION SEVERAL VARIABLE NATH 647 3

3 Is Definition The (e) ORDER high # den ppering in it diff eg derivtive ( s Notes ltl dt doth IR Exmple 1 CREDIT del evolution credit crd " Step 1 discover D E t hnd modeling problem Alt ) mount crd mone fter t on The credit ers Evolution Alt ) * compound interest 25% APY * Minimum pment ech month Toger n give d if 025 A or $420 I er nuthouse 420

4 A Steps solve model weknow solution Here eqution Alt ) C eds t 5 Check dte oes Ceo t 025 Alt ) t 0 ( e > 0 025L e NIH we did lern soon how to compute such siduhens! C is nbiwotto determine solution need dditionl informtion blnce out time 0 A LO ) $ C e C Then Alt ) e i 25 t Question Answer nen After When Is lon pid f ( Alt )oi 10 ers Ltl is heting ou owe $5578 5

5 1 Wednesd August 24 Chpter 1 DI Direction field grphicl ( Recll ) from previous nlsis lecture cc blnce mel p 025 A 420 +Ceo Alt ) 1680 Solution curves ^ Alt ) *e #* $1680 Slope given t n point b D E Red problem form { Diff eq fit ) > t Initil condition lt g kngn quntities Is clled n initil vlue problem ( Iup ) 5t

6 2 Q When is blnce incresing 1 stble / decresing DA g 025A 420 I Depends onl $1680 on got d sign dte! daq An 7 > s > s > df > }A4I6&o s > L o d±o It If A < 1680 it will decrese ( until it reches zero nd I stop ping ) If If A S 1680 it will increse towrds infinit A 1680 it will re

7 Direction field 3 Generl concept ds flti We b t ) cn behvior vis short drwing segments with slope grid points in ( t solutions lt ) ) plne flti ) E mple_ 2 Freefll Legend Newton observes n pple flling from tree B Step interesting quntities I hlb ) height pple I u elb ) velocit pple hv Newton s in " lw rte chnge velocit ( ccelertion ) prepred ll forces pplied to to net effect object M lt ) m d I F The constnt proportionlit is mss ( m ) 3

8 Wht forces re pplied to * grvit pushes down * ir resistnce pushes up object? 4 D Fl m o Toger m * m mss object * g ccelertion due to g k 98 Mls * grvit ner se level dr coefficient legends on du mg v shpe ) Step Cn we s smth before solving? Trmple n me d 10kg hg Is I When Is velocit stble linuesingleesi? A Sign dldt! In prticulr dug 0 if 98 0 he O or wlt ) 49 m

9 5 Not this is constnt solution ODE! clled equilibrium solution or Direction here field terminl velocit hgmls NMI # or * x*x * # e * If v< hsmls this vlue it will increse towrds If s h9m1s it will decrese towrd this vlue In both cses h9 pproches Mls Steptf Solution write t velocit eqution in form du It 02 ( 49 v ) 5

10 hg dvldt n re 49 Itlnlvltl 02 ± enll 491 ni Recll chin It ) n g [ lnlvltl B integrting both sides we hstf find 02 ln new C 2 t lt ) vlt ) h9 exp ( C 02 ± ec e ) vlt ) 49 + c e Ott where c is n tec rbitrr constnt Note method vlid [ for ll 1st order E sme e?+cet jie 6

11 N Frid August 24 1 Recp modeling Exmples l ingredients for ) simple IN order models Ke proportionlit between rte chnge nd some function quntit D d proportionl to dt K 2 otff " " Y d kg 3 Logistic growth dttr K N ( Nur ) useful for dinr modeling Rte increse # infected people proportionl to product # infected b # Newton s lw cooling or treting Rte in n vhne object temperture proportionl to between its difference temperture nd surrounding temperture K ( Tf F) d 1

12 Grphicl nlsis direction ^ t to\ t xxtg IF tod * > solutions ll tngent to direction field? Higher order models 9 1 ZI Free m fll with vrible hlt ) DI dhdt dr m height Hooke s lw nd hrmonic oscilltor M tmm R Displcement spring ; s ttched to mss * Restoring force Fspr ;ng ( x go ) stiffness constnt length t red * Newton s 2 " " lw lnegled friction moth kln x )

13 Note 3 This is Morles ODE equivlent to 1st order sstem or u d n { tht it is mdt K ( x xo ) E) Direction fields; grphicl Tke n t order D E d flt solutions ) A solution Is function tt lt tht ) such 4 lh flt ltl ) for ll t The slope or direction field cousins line segments on smll grid eith slope mf( t ) n Cse fltil 2 / x t x* x i tlt +* x* > c A solution Is lws tngentil to slope field Grphicl method to find or pproximte n INTEGRAL CURVE representing Solution 3

14 Nt it f does not depend on t fl ) 4 Exmple es ( ) \ Notion st *** ** * $#/ x * t * INTEGRAL t equilibrium ( t ) Cste I solutions curve fl )0 equilibrtion ( Note solutions rech mxlmin t equilibrium points) Notion stbilit It is unstble In exmple An equilibrium " if solution ner if tend w fwom it bove { it " solution is stble + I Is stble is unstble tend towrds it Anor exmple Use computer " lb