Mh 225-4 Week 2 coninue.-.3; lo cover pr of.4-.5, EP 7.6 Mon Nov 4.-.3 Lplce rnform, nd pplicion o DE IVP, epecilly hoe in Chper 5. Tody we'll coninue (from l Wednedy) o fill in he Lplce rnform ble (on pge 2), nd o ue he ble enrie o olve liner differenil equion. Exercie ) (o review) Ue he ble o compue ) 4 5 co 3 2e 4 in 2 2 b) 2 2. 2 5 Exercie 2) (o review) Ue Lplce rnform o olve he IVP we didn' ge o lwednedy, for n underdmped, unforced ocillor DE. Compre o Chper 5 mehod. x 6 x 34 x = x = 3 x =
f, wih f Ce M F f e d for M verified c f c 2 f 2 c F c 2 F 2 2 n, n e ( 2 2 3 n! n ( co k in k coh k inh k e co k e in k f f f n, n f d f 2 f n f, n f co k 2 k in k 2 k 2 ( k 2 k 2 ( 2 k 2 ( k k 2 k 2 ( k 2 k 2 ( k 2 k 2 F f 2 F f f n F n f... f n F F F n F n F d 2 k 2 2 k 2 2 2 k 2 2
2 k 3 in k k co k 2 k 2 2 e f F e n e, n 2 n! n more fer exm! Lplce rnform ble work down he ble... 3) coh k = 3b) inh k = 2 k 2 k 2 k 2. Exercie 4) Recll we ued inegrion by pr on Wednedy o derive f = f f. Ue h ideniy o how ) f = 2 F f f, b) f = 3 F 2 f f f, c) f n = n F n f n 2 f... f n, n. d) f d = F. Thee re he ideniie h mke Lplce rnform work o well for iniil vlue problem uch we udied in Chper 5.
Exercie 5) Find 2 4 ) uing he reul of 4d. b) uing pril frcion. Exercie 6) Show n = n!, n, uing he reul of 4. n
Mh 225-4 Tue Nov 5.2-.3 Lplce rnform, nd pplicion o DE IVP, including Chper 5. Tody we'll coninue o fill in he Lplce rnform ble, nd o ue he ble enrie o olve liner differenil equion. One focu ody will be o review pril frcion, ince he ble enrie re e up preciely o how he invere Lplce rnform of he componen of pril frcion decompoiion. Exercie ) Check why hi ble enry i rue - noice h i generlize how he Lplce rnform of co k, in k re reled o hoe of e co k, e in k : e f F Exercie 2) Verify he ble enry n, n n! n by pplying one of he reul from yeerdy: f n = n f n f n 2 f... f n. Exercie 3) Combine,2, o ge n e n! n A hrder ble enry o undernd i he following one - go hrough hi compuion nd ee why i eem reonble, even hough here' one ep h we don' compleely juify. The ble enry i f F Here' how we ge i: F = f d d F = d d f e d = f e d d d f e d. I' hi l ep which i rue, bu need more juificion. We know h he derivive of um i he um of he derivive, nd he inegrl i limi of Riemnn um, o hi ep doe le eem reonble. The re i righforwrd: d d f e d = f e d = f.
For reonnce nd oher pplicion... Exercie 4) Ue f = F or Euler' formul nd e = 2 k 2 ) co k = 2 k 2 2 b) 2 k in k = 2 k 2 2 c) Then ue nd he ideniy 2 k 2 2 = 2 k 2 2 k 2 2 k 2 2 k 2 2 2 k 2 2 o verify he ble enry 2 k 2 2 = 2 k 2 in k co k. k 2 o how
Noice how he Lplce rnform ble i e up o ue pril frcion decompoiion. And be mzed how i le you quickly deduce he oluion o imporn DE IVP, like hi reonnce problem: Exercie 5) Ue Lplce rnform o wrie down he oluion o 2 x x = F in x = x x = v. wh phenomenon do he oluion o hi IVP exhibi? 5b) Ue Lplce rnform o olve he generl undmped forced ocillion problem, when x 2 x = F in x = x : x = v
f, wih f Ce M F f e d for M verified c f c 2 f 2 c F c 2 F 2 2 n, n 2 2 3 n! n ( e ( co k in k coh k inh k e co k e in k 2 k 2 ( k 2 k 2 ( 2 k 2 ( k k 2 k 2 ( k 2 k 2 ( k 2 k 2 e f F f f f n, n f d f 2 f n f, n f co k F f 2 F f f n F n f... f n F F F n F n F d 2 k 2 2 k 2 2
2 k in k 2 k 3 in k k co k e n e, n 2 k 2 2 2 k 2 2 2 n! n more fer he miderm! Lplce rnform ble Exercie 6) Solve he following IVP. Ue hi exmple o recll he generl pril frcion lgorihm. x 4 x = 8 e 2 x = x =
Exercie 7) Wh i he form of he pril frcion decompoiion for 356 45 2 4 5 9 4 39 3 6 X = 3 3 2 4 2 4 7b) Hve Mple compue he precie pril frcion decompoiion. 7c) Wh i x = X? 7d) Hve Mple compue he invere Lplce rnform direcly.. 356 45 2 4 5 9 4 39 3 6 conver 3 3 2 4 2, prfrc ; 4 wih inrn ; ddble, fourier, fourierco, fourierin, hnkel, hilber, invfourier, invhilber, invlplce, invmellin, lplce, mellin, veble 356 45 2 4 5 9 4 39 3 6 invlplce 3 3 2 4 2,, ; 4 ()
Mh 225-4 Wed Nov 6.4-.5 The following Lplce rnform meril i ueful in yem where we urn forcing funcion on nd off, nd when we hve righ hnd ide "forcing funcion" h re more compliced hn wh undeermined coefficien cn hndle. We will coninue hi dicuion on Fridy, wih few more ble enrie including "he del (impule) funcion". f wih f Ce M F f e d for M commen u uni ep funcion e for urning componen on nd off =. f u e F more compliced on/off f f d F G "convoluion" for invering produc of Lplce rnform The uni ep funcion wih jump = i defined o be, u =,. I grph i hown below. Noice h hi funcion i clled he "Heviide" funcion in Mple, fer he peron who populrized i (mong lo of oher ccomplihmen) nd no becue i' hevy on one ide. hp://en.wikipedi.org/wiki/oliver_heviide wih plo : plo Heviide, = 3..3, color = green, ile = `grph of uni ep funcion` ; grph of uni ep funcion.6.2 3 2 2 3 Noice h echniclly he vericl line hould no be here - more precie picure would hve olid poin, nd hollow circle,, for he grph of u. In erm of Lplce rnform inegrl definiion i doen' cully mer wh we define u o be.
Then, ; i.e. u =, ; i.e. nd h grph h i horizonl rnlion by o he righ, of he originl grph, e.g. for = 2: plo Heviide 2, =..5, color = green, ile = `grph of u(-2)` ;.6.2 grph of u(-2) 2 3 4 5 Exercie ) Verify he ble enrie u uni ep funcion e for urning componen on nd off =. f u e F more compliced on/off
Exercie 2) Conider he funcion f which i zero for 4 nd wih he following grph. Ue lineriy nd he uni ep funcion enry o compue he Lplce rnform F. Thi hould remind you of homework problem from he ignmen due omorrow - lhough you're ked o find he Lplce rnform of h ep funcion direcly from he definiion. In your nex week' homework ignmen you will re-do h problem uing uni ep funcion. (Of coure, you could lo check your nwer in hi week' homework wih hi mehod.) 2 2 3 4 5 6 7 8 plo Heviide Heviide 2 2 Heviide 4, =..8, color = green ; wih inrn : lplce Heviide Heviide 2 2 Heviide 4,, ;
Exercie 3) Explin why he decripion bove led o he differenil equion iniil vlue problem for x x x =.2 co u x = x = 3b) Find x. Show h fer he pren op puhing, he child i ocilling wih n mpliude of excly meer (in our linerized model).
Picure for he wing: plo plo. in, =.. Pi, color = blck : plo2 plo Pi in, = Pi..2 Pi, color = blck : plo3 plo Pi, = Pi..2 Pi, color = blck, lineyle = 2 : plo4 plo Pi, = Pi..2 Pi, color = blck, lineyle = 2 : plo5 plo., =.. Pi, color = blck, lineyle = 2 : plo6 plo., =.. Pi, color = blck, lineyle = 2 : diply plo, plo2, plo3, plo4, plo5, plo6, ile = `dvenure he winge` ; 3 dvenure he winge 2 2 2 4 6 8 2 4 6 8 2 3 Alerne pproch vi Chper 5: ep ) olve for. ep 2) Then olve nd e x = y for. x x =.2 co x = x = y y = y = x y = x
f, wih f Ce M F f e d for M verified c f c 2 f 2 c F c 2 F 2 2 n, n 2 2 3 n! n ( e ( co k in k coh k inh k e co k e in k e f u f u f f f n, n 2 k 2 ( k 2 k 2 ( 2 k 2 ( k k 2 k 2 ( k 2 k 2 ( k 2 k 2 F e e F e F f 2 F f f n F n f... f n
f d F f 2 f n f, n f co k 2 k in k 2 k 3 in k k co k e n e, n F F n F n F d 2 k 2 2 k 2 2 2 k 2 2 2 k 2 2 2 n! n f g d F G f wih period p e p p f e d Lplce rnform ble
Mh 225-4 Fri Nov 8.5, EP7.6 Tody we finih dicuing Lplce rnform echnique: Impule forcing ("del funcion")...ody' noe. Convoluion formul o olve ny inhomogeneou conn coefficien liner DE, wih pplicion o inereing forced ocillion problem...ody' noe. Lplce ble enrie for ody: f wih f Ce M F f e d for M commen u uni ep funcion e for urning componen on nd off =. f u e F more compliced on/off e uni impule/del "funcion" f g d F G convoluion inegrl o inver Lplce rnform produc EP 7.6 impule funcion nd he operor. Conider force f cing on n objec for only on very hor ime inervl, for exmple when b hi bll. Thi impule p of he force i defined o be he inegrl p f d nd i meure he ne chnge in momenum of he objec ince by Newon' econd lw m v = f m v d = m v = f d = p Since he impule p only depend on he inegrl of f, nd ince he exc form of f i unlikely o be known in ny ce, he eie model i o replce f wih conn force hving he me ol impule, i.e. o e f = p d, where d, i he uni impule funcion given by = p.
d, =,,,. Noice h d, d = d =. Here' grph of d 2,., for exmple: 6 2 3 4 Since he uni impule funcion i liner combinion of uni ep funcion, we could olve differenil equion wih impule funcion o-conruced. A fr Lplce rnform goe, i' even eier o ke he limi for he Lplce rnform d,, nd hi effecively model impule on very hor ime cle. d, = u u d, = e e = e e. In Lplce lnd we cn ue L'Hopil' rule (in he vrible ) o ke he limi : lim e e = e e lim = e. The reul in ime pce i no relly funcion bu we cll i he "del funcion" nywy, nd viulize i funcion h i zero everywhere excep =, nd h i i infinie = in uch wy h i inegrl over ny open inervl conining equl one. A explined in EP7.6, he del "funcion" cn be hough of in rigorou wy liner rnformion, no funcion. I cn lo be hough of he derivive of he uni ep funcion u, nd hi i conien wih he Lplce ble enrie for derivive of funcion. In ny ce, hi led o he very ueful Lplce rnform ble enry uni impule funcion e for impule forcing
Exercie ) Revii he wing from Wednedy' noe nd olve he IVP below for x. In hi ce he pren i providing n impule ech ime he child pe hrough equilibrium poiion fer compleing cycle. x x =.2 2 4 6 8 x = x =. wih plo : plo plo. in, =.. Pi, color = blck : plo2 plo Pi in, = Pi..2 Pi, color = blck : plo3 plo Pi, = Pi..2 Pi, color = blck, lineyle = 2 : plo4 plo Pi, = Pi..2 Pi, color = blck, lineyle = 2 : plo5 plo., =.. Pi, color = blck, lineyle = 2 : plo6 plo., =.. Pi, color = blck, lineyle = 2 : diply plo, plo2, plo3, plo4, plo5, plo6, ile = `Wednedy dvenure he winge` ; 3 3 Wednedy dvenure he winge 2 4 6 8 2 4 6 8 2 impule oluion: five equl impule o ge me finl mpliude of meer - Exercie : f.2 Pi um Heviide k 2 Pi in k 2 Pi, k =..4 : plo f, =..2 Pi, color = blck, ile = `lzy pren on Fridy` ; 3 3 lzy pren on Fridy 2 4 6 8 2 4 6 8 2 Or, n impule = nd noher one =. g.2 Pi 2 in 3 Heviide Pi in Pi : plo g, =..2 Pi, color = blck, ile = `very lzy pren` ;
3 3 very lzy pren 2 4 6 8 2 4 6 8 2 Convoluion nd oluion o non-homogeneou phyicl ocillion problem (EP7.6 p. 499-5) Conider mechnicl or elecricl forced ocillion problem for x, nd he priculr oluion h begin re: x b x c x = f x = x =. Then in Lplce lnd, hi equion i equivlen o 2 X b X c X = F X 2 b c = F X = F 2 F W. b c Becue of he convoluion ble enry f g d F G convoluion inegrl o inver Lplce rnform produc he oluion i given by x = f w = w f = w f d. where w = W. The funcion w i clled he "weigh funcion" of he differenil equion, becue he oluion x i ome or of weighed verge of he he force f beween ime nd, where he weighing fcor re given by w in ome or of convolued wy. Thi ide generlize o much more compliced mechnicl nd circui yem, nd i how engineer experimen mhemiclly wih how propoed configurion will repond o vriou inpu forcing funcion, once hey figure ou he weigh funcion for heir yem. The mhemicl juificion for he generl convoluion ble enry i he end of ody' noe, for hoe who hve udied iered double inegrl nd who wih o undernd i.
Exercie 2. Le' ply he reonnce gme nd prcice convoluion inegrl, fir wih n old friend, bu hen wih non-inuoidl forcing funcion. We'll ick wih our erlier wing, bu conider vriou forcing periodic funcion f. x x = f x = x = ) Find he weigh funcion w. b) Wrie down he oluion formul for x convoluion inegrl. c) Work ou he pecil ce of X when f = co, nd verify h he convoluion formul reproduce he nwer we would've goen from he ble enry 2 k in k 2 k 2 2 in co d ; co in d ; #convoluion i commuive d) Then ply he reonnce gme on he following pge wih new periodic forcing funcion...
We worked ou h he oluion o our DE IVP will be Exmple ) A qure wve forcing funcion wih mpliude nd period 2. Le' lk bou how we cme up wih he formul (which work unil = ). wih plo : x = in f d Since he unforced yem h nurl ngulr frequency =, we expec reonnce when he forcing funcion h he correponding period of T = 2 = 2. We will dicover h here i he poibiliy w for reonnce if he period of f i muliple of T. (Alo, forcing he nurl period doen' gurnee reonnce...i depend wh funcion you force wih.) f 2 n = n Heviide n Pi : plo plo f, =..3, color = green : diply plo, ile = `qure wve forcing nurl period` ; qure wve forcing nurl period 2 3 ) Wh' your voe? I hi qure wve going o induce reonnce, i.e. repone wih linerly growing mpliude? x in f d : plob plo x, =..3, color = blck : diply plo, plob, ile = `reonnce repone?` ;
Exmple 2) A ringle wve forcing funcion, me period f2 f d.5 : # hi niderivive of qure wve hould be ringle wve plo2 plo f2, =..3, color = green : diply plo2, ile = `ringle wve forcing nurl period` ; 2) Reonnce? ringle wve forcing nurl period 2 3 x2 in f2 d : plo2b plo x2, =..3, color = blck : diply plo2, plo2b, ile = `reonnce repone?` ; Exmple 3) Forcing no he nurl period, e.g. wih qure wve hving period T = 2. f3 2 2 n = n Heviide n : plo3 plo f3, =..2, color = green : diply plo3, ile = `ou of phe qure wve forcing` ; 3) Reonnce? ou of phe qure wve forcing 5 5 2 x3 in f3 d : plo3b plo x3, =..2, color = blck : diply plo3, plo3b, ile = `reonnce repone?` ;
Exmple 4) Forcing no he nurl period, e.g. wih priculr wve hving period T = 6. f4 n = Heviide 6 n Heviide 6 n : plo4 plo f4, =..5, color = green : diply plo4, ile = pordic qure wve wih period 6 ; pordic qure wve wih period 6 5 5 4) Reonnce? x4 in f4 d : plo4b plo x4, =..5, color = blck : diply plo4, plo4b, ile = `reonnce repone?` ;
Hey, wh hppened???? How do we need o modify our hinking if we force yem wih omehing which i no inuoidl, in erm of worrying bou reonnce? In he ce h hi w modeling wing (pendulum), how i i geing puhed? Precie Anwer: I urn ou h ny periodic funcion wih period P i (poibly infinie) uperpoiion P of conn funcion wih coine nd ine funcion of period P, 2, P 3, P,.... Equivlenly, hee 4 funcion in he uperpoiion re, co, in, co 2, in 2, co 3, in 3,... wih ω = 2. Thi i he P heory of Fourier erie, which you will udy in oher coure, e.g. Mh 35, Pril Differenil Equion. If he given periodic forcing funcion f h non-zero erm in hi uperpoiion for which n = (he nurl ngulr frequency) (equivlenly P n = 2 ), here will be reonnce; oherwie, no reonnce. We could lredy hve underood ome of hi in Chper 5, for exmple Exercie 3) The nurl period of he following DE i (ill) T = 2 forcing funcion below i T = 6. Noice h he period of he fir nd h he period of he econd one i T = T = 2. Ye, i i he fir DE whoe oluion will exhibi reonnce, no he econd one. Explin, uing Chper 5 uperpoiion ide. ) x x = co in. 3 b) x x = co 2 3 in 3.