HIGHER ORDER DIFFERENTIAL EQUATIONS

Transcription

1 Prof Enriqu Mtus Nivs PhD in Mthmtis Edution IGER ORDER DIFFERENTIAL EQUATIONS omognous linr qutions with onstnt offiints of ordr two highr Appl rdution mthod to dtrmin solution of th nonhomognous qution givn in th following riss Th inditd funtion (), is solution of th ssoitd homognous qution Dtrmin sond solution of th homognous qution prtiulr solution of th inhomognous ED ; ; ; ; Problms for group disussion: Mk onvining dmonstrtion tht th sond ordr qution u b 0; m, b,, onstnt lws hs t lst on solution of th form, whr m is onstnt Two Eplin wh E D st point must hv, onsquntl, sond solution of th m m form or form, whr m mr onstnts OMOGENEOUS LINEAR EQUATIONS WIT CONSTANT COEFFICIENTS d W hv sn tht th first ordr linr qution, u 0, whr is onstnt, hs th rnging ; ponntil solution - thrfor s nturl to tr to dtrmin if thr r ponntil solutions - homognous linr qutions of highr ordr tp: n n n n 0 0 () Whr th offiints, i 0,,, n r rl onstnts 0 To our surpris, ll i solutions of th qution () r ponntil funtions or r formd from ponntil funtions n

2 Prof Enriqu Mtus Nivs PhD in Mthmtis Edution Mthod of solution: strt with th spil s of th sond ordr qution + b + = m m m m 0 () If w tr solution of th form, thns, so tht m m m th qution () boms:, m bm 0, or m m m bm 0 As nvr zro whn hs rl vlu, th onl w tht th ponntil funtion stisfis th diffrntil qution is hoosing m suh tht it is root of th qudrti qution m bm 0 () This qution is lld uilir qution or hrtristi qution of th diffrntil qution () Emin thr ss: th solutions of th uilir qution orrsponding to distint rl roots, rl qul roots ompl onjugt roots m CASE I: distint rl roots: If qution () hs two distint rl roots, m m, rrivd t two solutions, m Ths funtions r linrl indpndnt on m -, thrfor, form fundmntl st Thn, th gnrl solution of qution () in this intrvl is m m () CASE II: Rl Estt qul Whn m m w nssril ponntil onl solution, qudrti formul, b m bus th onl w solution of th qution is: m m m m m m m is b 0 m () Aording to th Thus, sond In this qution w tk tht b m Th gnrl solution is thrfor m m (6)

3 Prof Enriqu Mtus Nivs PhD in Mthmtis Edution CASE III: ompl onjugt roots m r ompl, w n writ i i,, m If 0 p > 0 th r rl, i m m, whr Thr is no forml diffrn ( i ) ( i ) btwn this s s I, hn, owvr, in prti it is prfrrd to work with rl funtions not ompl ponntil With this objt using Eulr's formul: i os i sn, tht is rl numbr Th onsqun of this formul is tht: i os i sn, -i os i sn, (7) whr w hv usd os (- ) os ( ) sn (- ) sn ( ) Not tht if th first dd thn subtrt th two qutions (7), w obtin rsptivl: i -i i -i os, i sin ( ( i ) i ) As is solution to qution () for n hoi of th onstnts, if, obtin th solutions: ( i ) ( i ) ( i ) ( i ) i i But os i i i sin Aordingl, th rsults dmonstrt tht th lst two rl funtions sn os r solutions of th qution () Morovr, ths solutions form fundmntl, thrfor -, th gnrl solution is: os os sin sin Sond ordr diffrntil qutions Solv th following diffrntil qutions: 0

4 Prof Enriqu Mtus Nivs PhD in Mthmtis Edution SOLUTION: I prsnt th uilir qutions, roots orrsponding gnrl solutions m m m ; m m m 0 n: 0 0 SOLUTION: m m 0 m 0 m m 0 n: 0 SOLUTION: m 0 m - i, m - i, m os sin n: Initil vlu problm Solv th initil vlu problm 0; (0) -, 0 SOLUTION: Th roots of th uilir qution m m 0 m i, m i so tht os B ppling th ondition ( 0 ) sin 0, w s tht os 0 sn 0 W diffrntit th bov qution thn ppling (O) = w gt or ; thrfor, th solution is: os sn

5 Th two diffrntil qutions, p 0 k 0 Prof Enriqu Mtus Nivs PhD in Mthmtis Edution, k rl, r importnt in pplid mthmtis For th first, th uilir qution m k 0 hs imginr roots m k i m k i Aording to qution (8), with 0 k, th gnrl solution is os k sn k (9) Auilir qution th sond qution, m k 0, hs distint rl roots m k m k ; thrfor its gnrl solution is k -k (0) Not tht if w hoos thn in 0, prtiulr solutions w k snh k -k osh k k -k snh k insmuh s osh k r linrl indpndnt in n rng of th is, n ltrntiv form of th gnrl solution of p 0 is osh k snh k ighr-ordr Equtions In gnrl, to solv diffrntil qution of ordr n s n n n n 0 0 () Whr n: i, i 0,,, n r rl onstnts, w must solv polnomil qution of dgr n n nm n m m m 0 0 () If ll th roots of qution () r rl distint, th gnrl solution of qution () is m m mn n It's diffiult to summriz th nlogous ss II III bus th roots of n uilir qution of dgr grtr thn two n our in mn ombintions For mpl, quinti ould hv fiv distint rl roots, or thr distint rl roots two ompl, or four rl ompl, fiv rls but qul, but two quls fiv rls, so on Whn m is root of n qution k multipliit uilir dgr n (i roots quls k), on n show tht th solutions r linrl indpndnt

6 Prof Enriqu Mtus Nivs PhD in Mthmtis Edution,, m m m k m Finll, rmmbr tht whn th offiints r rl, ompl roots of uilir qution lws ppr in onjugt pirs Thus, for mpl, ubi polnomil qution m hv two ompl zros t most Third-ordr diffrntil qution Rsolv 0 + ~ - = 0 SOLUTION: In rviwing m 0 m w should not tht on of its roots is If w divid m m ight m, m m m m m, m m m m Thus, th gnrl solution is m w s tht thn th othr roots r - - Fourth-ordr diffrntil qution d d Rsolv 0 SOLUTION: Th uilir qution is m m 0 m 0 roots m m i m i m Thus, ording to th s II, th solution is: hs th i -i i -i Aording to Eulr's formul, w n writ th grouping os sn i -i in th form With hng in th dfinition of th onstnts qull, i -i n b prssd in th form os sn gnrl solution is Aordingl, th os sin os sin 6

7 Prof Enriqu Mtus Nivs PhD in Mthmtis Edution Gnrl Eriss Nonhomognous linr qutions with onstnt offiints of ordr two highr For h of th following E D finds th gnrl solution: d d d d d d d d d d d d