LINEAR 2 nd ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

Transcription

1 Diol Bgyoko (0) I.INTRODUCTION LINEAR d ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS I. Dfiiio All suh diffril quios s i h sdrd or oil form: y + y + y Q( x) dy d y wih y d y d dx dx whr,, d r oss. If h idpd vril is im, h h followig oio is of usd:... y + y + y Q( ), wih dy d. d y.. y d y. d Ths quios ppr o dily sis : Nwo s d lw of lds o quios of his yp. So dos h sudy of h moio of sprig (wih or wihou dmpig). Th rsisor (R), iduor (L), d pior (C) sysm lso lds o his quio. So do my mhil virios, iludig h os for h suspsio sysms of rs, ruks, vs,. I h s of lroi or mhil virios, h vril is of h im x&& + x& + x Q() dx x & d x && d x d Drivig dmpig or Trm rsis (pplid rm(propor. xrl o vloiy or for) spd) For h RLC irui, h lri hrg q ( ) sisfis ( ) dq( ) d q L + R + q d d () F()

2 Diol Bgyoko (0) II. Cop of Lir Idpd for Fuios I grl, wo fuios () ( ) f f (i.., umr s opposd o ig fuio h vris) d r lirly dpd of hir rio is os os, f d f! r lirly dpd.!! Exmpls: () () Ay wo os fuios r lirly dpd. Two fuios r lirly idpd of hir rio is o os : si os os, os ( ) os if os. Ky Poi: Th sm wy i r d r j r dd o rprs rirry vor i h x y r r r pl, V V i + j, h sm wy lirly idpd soluios r dd o x V y rprs h grl soluio of d, rd ordr D.E. II. SOLUTION OF A d ORDER DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS II. Soluio of h Homogous Equio: susiuio Th grl soluio o x& + x& + x 0 is oid s follows. dx S x, x&, d & x. Susiu h i h D.E. d (W divid y us i is diffr from zro!) Solv h rsulig qudri quio for o g: + whr 4 disrimi

3 Diol Bgyoko (0) By viru of h fudml horm of lgr, d lwys xis (o o sid hy hd o rl or h hy hv o lwys diffr). Cs if Cs if + A B A + B No h for A d, Β r o lirly idpd. W oi sod fuio y mullyig y. A lo of physis is hidd i h ov li. For d rl --- h oudry odiios of ld o h disrdig of Β s i h rprss xpoil growh if > 0. For 0 (o dmpig) 4 is of giv d d r oh osillory fuios. [ os( ) + i ( ) ] i A A A si. O g similr rsuls for! O gs soluio h is rl fuio y kig ( lir omiio): Aos ( ) + B ( ) si For 0 A lwys rl fuio d of is omplx Th im dpd rl pr lds o rgy loss.

4 Diol Bgyoko (0) MORE PHYSICS: x&& + x& + x 0 This quio dsris mhil, lril, d ohr virios or osillios for whih hr is o drivig for. If 0, hr is o dmpig, i.., o rsis o h virios. I his s (i.., 0), h osillios r hos of simpl hrmoi moio for 0. (To hv osillios, h disrimi hs o giv if 0.) & x + x & x 0 x of h form x& ω x whih dfis simpl hrmoi moio, for 0. Th rsulig sysm is osrviv d is ol rgy (Ek+Ep) is os. For 0,, + Α No wll h i his s, d for < 0, o hs. L ω. A olly quivl soluio is oid y kig lir omiio of osi d si fuios s follows: i + Β i ( Α osω + Β siω ). Ky Physis Pois I h s whr hr is dmpig, h ol rgy is o logr os. (Th xrl for ig of h sysm is o osrviv.) Furhr, h frquy of h osillios is diffr from h o w hv if is zro. 4

5 Diol Bgyoko (0) Th iroduio of dmpig o oly uss h ol rgy o o osrvd, u lso modifis h frquy of h osillio! Idd, 4 ω 4 ω0 4 I grl, o hs ω ω 0, giv h is posiiv for 0. If h dmpig rm is vry lrg, h ( 4) posiiv! I h s, hr r o osillios! ( Α ) + Β wih & rl. II. Soluios of Ihomogous Lir d O.D.E. wih Cos Coffiis Th grl soluio of x& + x& + x Q( ), whr,, d r oss d whr () 0, Q is ig + is h grl soluio of homogous quio. is (i.., y) priulr soluio of h ihomogous quio. W disussd ov how o g. W disuss low mhod of gig. Our mhod for oiig is guidd y h form of Q ( ). Cosul h xook for furhr illusrios wih xmpl prolms. Cs : Q() Cos 5

6 Diol Bgyoko (0) S X k. Th g & & d. Susiu hm i h D.E. o s h & k is idd soluio of x& + x& + x k. Rmmr h for ihomogous diffril quios, w g priulr soluios y wy w! Cs : Q() P () k, whr P () is polyomil of dgr S S S S Th lul () () () k if k k & & d if k u k k if k. Susiu hm i h D.E. Mov ll o-zro rms o h lf of h quio. Divid y h o-zro xpoil rm (i.., k ). Wh rmis is polyomil i (or x, if x is h vril) h is idilly zro. Group oghr ll h os rms (hir sum is h offii of 0 ). Group oghr ll h rms ivolvig,,,., rspivly. Th oly wy polyomil idilly zro is for h sum of h offiis of vry powr of h vril (0,,,.) o zro. So, qu h offiis of h h ordr moomils qul o zro o g h sysm of lgri quios o solvd o g α 0, α, α,. d hrfor o g S () α + α + α... α 0 + Clrly, o hs o firs solv h homogous quio, x& + x& + x 0, o g.d for mpig o g priulr soluio. Cuio If Q() ois os(k) or si(k), h rll h hy r h rl d imgiry prs of ik. H, s Q() P() ik d prods s idid i h 6

7 Diol Bgyoko (0) ov. Afr oiig S() ik, h k is rl pr (for os(k) or imgiry pr (for si(k) o oi h orr priulr soluio, X, of h diffril quio. As prviously sd rpdly, you r o rd h xook for xmpls d illusrios h r workd ou d h o ovrd i lss if w r o kp p wih our prs roud h pl. 7