IX. Orir Diffrtil Equtios A iffrtil qutio is qutio tht iclus t lst o rivtiv of uow fuctio. Ths qutios m iclu th uow fuctio s wll s ow fuctios of th sm vribl. Th rivtiv m b of orr thr m b svrl rivtivs prst. Grll spig iffrtil qutio is rprsttio of phsicl phomo whr th rivtivs corrspo to th "rts of chg" of th uow fuctio with rspct to th vribl. lssifictio of iffrtil qutios Orir iffrtil qutios ODE ivolv ol o ipt vribl or quivltl th uow fuctio is fuctio of sigl vribl. or mpl: t i A f t f t f E V m r orir iffrtil qutios. Prtil iffrtil qutios PDE coti fuctios of multipl vribls prtil rivtivs: t D t t z E z z V z m W will covr ol orir iffrtil qutios ODE. Orr of th iffrtil qutio is th highst rivtiv tht pprs i th qutio. Th bov mpls of th ODE r rspctivl sco first sco orr. Lir iffrtil qutio c b writt i th followig form: Tht ms ll th rivtivs icluig zroth o i.. r prst ol to th first powr thr o trms with mi orr.g. /. Not tht ll th trms cotiig th uow fuctio icluig its rivtivs r o th lft-h si. Th right-h si ol cotis th ipt vribl.
Homogous qutio is th lir iffrtil qutio whr ll th trms coti th uow fuctio its rivtivs. pig th bov ottio th qutio is homogous if I othr wors is homogous. O th othr h if th th qutio is ohomogous or complt. Solutio of iffrtil qutio is th fuctio.g. f t which stisfis th qutio iticll. or mpl: is th solutio of for costt bcus Grl solutio of th orr iffrtil qutio ivolvs rbitrr costts. To ssig vlus to ths costts itiol coitios iitil coitios bour coitios r. Prticulr solutio is solutio obti from th grl solutio b ssigig spcific vlus to th rbitrr costts. Spcil first orr qutios solutios A first orr ODE c b put ito th form: f or M N Grl solutio of first orr ODE cotis o rbitrr costt. Th bst w of solvig th ODE ps o its prticulr tp: thr r vrious spcil tps of ODE with spcil trics us to solv thm. W will cosir ol fw most prcticll importt spcil css.. Sprbl vribls If th ODE is of th followig form: f g or f g f g th it c b solv b sprtio of vribls itgrtio ovr ch vribl sprtl: f g g f or f g f g f f g g
Not tht spcil css of sprbl ODE: f or g c b itgrt irctl. Empl: i th grl solutio of th ODE: solutio for which =. This is st looig o-lir qutio but it is ctull sprbl:. Th fi th prticulr Itgrtig both sis: rrrgig l l l is th grl solutio. To fi th prticulr solutio plug i = = to fi th vlu of th costt : 6 substitut = bc to th grl solutio to gt:
Empl: Rioctiv c irrvrsibl first orr itics: N D Th rt of c is proportiol to th mout of rioctiv mtril t tim t: N t N t t N t t N t N t N l N t N t t is th grl solutio. With iitil coitio Nt= = N w gt th prticulr solutio: N t N t. Ect qutios Equtio whr M N M N I othr wors M N is ct iffrtil. Th qutio bcoms: whos solutio is or bcus this coms ow to: M N M N M N M whr ms itgrtig ovr whil pig costt vic vrs. This formul loos complict but th w it wors is rll ot:. irst ou solv
5 M b itgrtig M with rspct to. bcus th itgrtio is with rspct to ol th itgrl is trmi up to rbitrr fuctio of s. Thrfor: M. Now substitut this to th sco qutio: N solv for b itgrtig with rspct to which givs th formul bov. Empl: i grl solutio of si cos irst w will show tht it is ct qutio: si cos cos Th solv th first prt: si cos plug ito th sco prt: si ' si si si which givs th qutio for. Not tht ll th pt trms hv to ccl out bcus hs to p o ol! rom hr: ' Puttig it ll togthr: N M si which is th sir grl solutio.
. Ict qutios: Itgrtig fctors whr M N M N but wh multipli b som fuctio th qutio bcoms ct. i..: M N is th ct qutio. Not tht w c multipl th qutio b whtvr w wt without chgig it. Th fuctio is cll th itgrtig fctor. Sic with th itgrtig fctor th qutio is ct: M N th sm mtho s for ct qutios pplis. Th tric howvr is to fi th pproprit itgrtig fctor. I grl w will gt prtil iffrtil qutio for th itgrtig fctor i.. much mor complict qutio th th ODE w origill tri to solv. Th ol hop is tht th itgrtig fctor ps ol o o of th vribls: ithr or or tht w somhow ow or c guss th itgrtig ftor. Empl: solv This qutio is ict bcus W will tr to loo for itgrtig fctor tht ps ol o i.. so tht: th: If th itgrtio fctor is i ipt of ll s hv to ccl out: thr must b ol lft i th qutio. W s tht this is i th cs os ot p o w hv iffrtil qutio for it which is sprbl: 6
7 Itgrtig both sis: l l l l tig potil: Now bc to th qutio: is ct s w c chc: To solv follow th procur for ct qutios: th Th grl solutio is:
. Lir qutios first orr Lir qutio of th first orr: P Q whr P Q r fuctios of c lws b solv. Thr r svrl ws to o it:. This qutio hs itgrtig fctor tht ps ol o : th th qutio c b writt s ct qutio: P with solutio or: P Q Q P Q P P Q P This is grl formul tht will lws wor s log s ou c o th itgrls.. With th bov itgrtig fctor w c solv this s ct qutio problm.. Aothr w to solv it is to us th grl tctics for lir qutios w will iscuss i mor til shortl whr th grl solutio of th homogous qutio is fou prticulr solutio of th complt qutio. Th homogous problm: is sprbl: P P which ms it rltivl s th prticulr solutio of th complt qutio P Q c b fou for mpl b th mtho of utrmi cofficits ctull ol o cofficit hr pli i th t sctio. Empl: solv cos 8
ollowig th first mtho: th itgrtig fctor: cos l l th: cos cos si th solutio is: si You c tr to vrif b pluggig it bc ito th qutio Lir iffrtil qutios W hv lr stt tht lir iffrtil qutio i grl of orr is writt s: which w cll complt qutio. Th corrspoig homogous qutio is: W lr ow how to hl th first orr qutio. Thrfor w will cosir ol highr orr qutios i prticulr th sco orr which r prcticll th most importt. Th fumtl thorm for solvig lir iffrtil qutios is s follows: To fi th grl solutio of lir iffrtil qutio of th orr > with :. irst fi th grl solutio of th corrspoig homogous qutio. This is cll complmtr or homogous solutio. Rmmbr this solutio will hv rbitrr costts for orr qutio.. i o prticulr solutio of th complt qutio i.. fuctio P tht stisfis th complt qutio. 9
. Th grl solutio of is th th suprpositio of th grl solutio of th homogous qutio prticulr solutio of th complt qutio P : Agi this is grl solutio which ms it will hv rbitrr costts. Ths c b trmi for prticulr problm from th iitil or bour coitios. P Lir iffrtil qutios with costt cofficits Wht w hv stt so fr pplis grll to lir ODE. Th most importt tp of ths qutios r lir ODE with costt cofficits. Tht ms tht ll th fuctios r just costts. I othr wors w hv qutio of th tp: omplmtr homogous solutio. or lir ODE with costt cofficits th solutio to th homogous problm is lws fou followig simpl grl rul: s tril solutio of th form whr is som costt which w o ot ow but will fi b substitutig this tril solutio to th bov homogous qutio. This ils polomil qutio for : which i grl will hv roots.. If th roots r ll istict multiplicit th th grl solutio of th homogous problm is whr r rbitrr costts which must b thr s w ow.
. If som roots r rpt.g. hs multiplicit hs multiplicit tc. ou must multipl th corrspoig potil with polomil of th orr tc with rbitrr cofficits. Th solutio th loos li this: D D D Not tht i ithr cs w will hv th right umbr i.. of th rbitrr costts. Empl: i th grl solutio of m. t m >. Tril solutio t givs: m t t i m m i Th grl solutio is th: which c lso b rwritt s it it D cost D si t This is th hrmoic oscilltor problm which is immsl importt throughout phsics/phsicl chmistr. Empl: i grl solutio of 8 6 Substitutig tril solutio : 8 6 Hr w hv th root with multiplicit of. Th grl solutio will b:
omplt o-homogous solutio. W will iscuss two mthos of fiig prticulr solutio of th complt qutio.. Mtho of utrmi cofficits spcil right h si. This mtho is pplicbl if th fuctio o th right h si of th complt ODE hs prticulr form. Bs o this form w c costruct tril solutio with som uow costts much li th tril solutio for th homogous problm. Th costts r trmi b substitutig to th qutio solvig th rsultig lgbric qutios. Th form of th right h si ssum tril solutios r giv i th followig tbl. or smbols D p q r costts is positiv itgr ll giv. or th tril solutio b c b tc. r rbitrr costts to b trmi p cos p si p cos p D si p Assum tril solutio p cos p b si p p p cos q p si q cos q D si q q cos q bsi q p p cos p si p p p p cos p D D D D D si p cos q si q p cos q D D D D D si q Sums of of th bov p cos p si p p cos q si q Sums of corrspoig tril solutios
Ths tril solutios r pplicbl ol if o trm i thm pprs lso i th solutio to th homogous qutio. If trm of th ssum tril solutio is prt of th homogous solutio th th tril solutio must b multiplis b th smllst positiv itgrl powr of so tht o of th trms i th tril solutio logr ppr i th homogous o. Empl: forc mp lir hrmoic oscilltor: m. A cos t o t t Rwrit th qutio first to simplif:. B t t o cos t whr w hv fi: m m B o A o m Grl solutio of th homogous qutio:. t t b substitutig tril solutio t givs: with solutios: i i ~ whr ~ Th grl solutio of th homogous qutio is: i ~ t i ~ t t th solutio cs potill to zro mpig. or cos ~ t D This loos li th simpl hrmoic oscilltor cpt: t D sit th frquc is shift from to ~ ~
This is i fct th mp oscilltor prt: if thr ws o forc right h si th oscilltios woul go to zro with tim. Now to fi th prticulr solutio of th complt problm w c us th fct tht th right h si hs cosi i it w c loo for solutio i th tril form of: cos t bsi t Sic th homogous solutios lso hv th mpig potil thr is o gr tht this solutio is coti i th homogous o v if ~. Th: Substitutig: t si t b cos t si t b cos t t cos t bsi t cos t bsi t t bsi t si t b cos t cos t bsi t B cos t cos collctig trms with si cos: I othr wors: b cos t b bsi t b B B cos t I th spcil cs th solutio is obviousl vr simpl b B /. If w gt ftr som oig but strightforwr lgbr: b Th solutio of th complt qutio thrfor is: B B ~ ~ t B B D t D sit cos t cos si t It woul hv b lot sir to wor with th compl potil rthr th cos. Lt s t th right h si i th form: p i t i.. ssum tril solutio i th form p it th t th rl prt.
Doig th rivtivs substitutig s bfor: it it it i B it i B B i Th prticulr solutio is: Tig rl prt: B R i it B p it p it i R B i cos t i si t B B cos t si which givs us ctl th sm rsult s bfor but th clcultio ws lot simplr. Empl: Solv Th homogous solutios is: 8si cos si Now w hv th sm fuctio si i this prssio s o th right h si. Th tril solutio thrfor hs to b multipli b th smllst powr of to giv us somthig iffrt. I this cs th first powr of os th tric th tril solutio is: Substitutig ito th iffrtil qutio: cos bsi cos b si 8si Solvig for b: b Ls to th grl solutio: cos si si 5
6. Mtho of vritio of costts I this mtho w us th homogous complmtr solutio to fi th prticulr complt solutio. Not tht this wors for grl lir iffrtil qutio ot ol th os with costt cofficits s log s w c com up with th solutio to th homogous problm. Lt th solutio to th homogous -th orr lir ODE: b: To fi th prticulr solutio of th complt qutio: rplc th costts b fuctios. Now w to fi ths fuctios so tht stisfis th complt qutio. To trmi fuctios w to impos coitios. O coitio is th complt qutio itslf but th rmiig os c b t t will. Th coitios tht l to th simplst clcultios r: Whr th lst coitio is th coitio tht th complt iffrtil qutio is stisfi. This l to th sstm of qutios for th rivtivs of from which th c b obti b itgrtio. I prticulr for th sco orr qutio w hv:
Empl: Homogous solutio is cos cos si Th w ssum th prticulr solutio to th complt qutio i th form: cos si whr r ow fuctios to b trmi. Diffrtitig: si cos cos si th itiol coitio w us is: Diffrtitig gi: cos si cos si si cos Substitutig ito th qutio: si cos cos rom th two qutios: cos si si cos cos w fi Itgrtig: t th solutio: l cos cos si si cos l / cos 7