Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm s quvaln o a scond ordr homognous lnar dffrnal quaon As a rsul, w know ha h gnral soluon conans wo lnarly ndpndn pars As wll, h soluon wll b conssd of som yp of ponnal funcons Thrfor, assum ha k r s a soluon of h sysm, whr k s a vcor of coffcns (of and Subsu and r k r no h quaon A, and w hav r k r A k r Snc r s nvr zro, w can always dvd boh sds by r and g r k A k W s ha hs nw quaon s acly h rlaon ha dfns gnvalus and gnvcors of h coffcn mar A In ohr words, n ordr for a funcon k r o sasfy our sysm of dffrnal quaons, h numbr r mus b an gnvalu of A, and h vcor k mus b an gnvcor of A corrspondng o r Jus lk h soluon of a scond ordr homognous lnar quaon, hr ar hr possbls, dpndng on h numbr of dsnc, and h yp of, gnvalus h coffcn mar A has 8, Zachary S Tsng D- - 9
Th possbls ar ha A has I Two dsnc ral gnvalus II ompl conjuga gnvalus III A rpad gnvalu A rlad no, (from lnar algbra, w know ha gnvcors ha ach corrsponds o a dffrn gnvalu ar always lnarly ndpndn from ach ohrs onsqunly, f r and r ar wo dffrn gnvalus, hn hr rspcv gnvcors k anf k, and hrfor h corrspondng soluons, ar always lnarly ndpndn 8, Zachary S Tsng D- -
as I Dsnc ral gnvalus If h coffcn mar A has wo dsnc ral gnvalus r and r, and hr rspcv gnvcors ar k and k Thn h sysm A has a gnral soluon r r k k Eampl: W hav alrady found ha h coffcn mar has gnvalus r and And hy ach rspcvly has an gnvcor k, k Thrfor, a gnral soluon of hs sysm of dffrnal quaons s 8, Zachary S Tsng D- -
8, Zachary S Tsng D- - Eampl:, ( Th characrsc quaon s r r (r (r Th gnvalus ar r and Thy hav, rspcvly, gnvcors For r, h sysm s (A r I (A I Solvng h boom quaon of h sysm:, w g h rlaon Hnc, k, For r, h sysm s (A r I (A I Solvng h frs quaon of h sysm:, w g h rlaon Hnc, k
8, Zachary S Tsng D- - Thrfor, a gnral soluon s Apply h nal valus, ( Tha s W fnd and, hnc w hav h parcular soluon
as II ompl conjuga gnvalus If h coffcn mar A has wo dsnc compl conjuga gnvalus λ ± µ Also suppos k a b s an gnvcor (ncssarly has complvalud nrs of h gnvalu λ µ Thn h sysm A has a ral-valud gnral soluon λ ( a µ bsn( µ ( a sn( µ b cos( µ λ cos( A ll dal: Smlar o wha w hav don bfor, frs hr was h compl-valud gnral soluon n h form ( λ µ ( λµ k k W flr ou h magnary pars by carfully choosng wo ss of coffcns o oban wo corrspondng ral-valud soluons ha ar also lnarly ndpndn: u v λ λ ( a cos( µ b sn( µ ( a sn( µ b cos( µ Th ral-valud gnral soluon abov s jus u v In parcular, mgh b usful o know how u and v could b drvd by pandng h followng compl-valud prsson (h fron half of h compl-valud gnral soluon: k ( λ µ λ λ ( a b λ ( µ ( acos( µ asn( µ bcos( µ ( acos( µ bsn( µ λ ( a b(cos( µ sn( µ λ bsn( µ ( asn( µ bcos( µ Thn, u s jus h ral par of hs compl-valud funcon, and v s s magnary par 8, Zachary S Tsng D- -
8, Zachary S Tsng D- - Eampl: Th characrsc quaon s r, gvng gnvalus r ± Tha s, λ and µ Tak h frs (h on wh posv magnary par gnvalu r, and fnd on of s gnvcors: (A r I Solvng h frs quaon of h sysm: (, w g h rlaon ( Hnc, b a k a b Thrfor, a gnral soluon s cos( sn( sn( sn( cos( cos( cos( sn( sn( cos(
8, Zachary S Tsng D- - Eampl:, ( Th characrsc quaon s r r, gvng gnvalus r ± Thus, λ and µ Tak r and fnd on of s gnvcors: (A r I ( ( Solvng h scond quaon of h sysm: (, w g h rlaon ( Hnc, b a k Th gnral soluon s sn( sn( cos( cos( sn( cos( cos( sn( sn( cos(
8, Zachary S Tsng D- - 7 Apply h nal valus o fnd and : cos( sn( sn( cos( ( Thrfor, and onsqunly, h parcular soluon s sn( cos( sn( sn( sn( cos( cos( sn( cos(
as III Rpad ral gnvalu Suppos h coffcn mar A has a rpad ral gnvalus r, hr ar sub-cass ( If r has wo lnarly ndpndn gnvcors k and k Thn h sysm A has a gnral soluon k r k r No: For marcs, hs possbly only occurs whn h coffcn mar A s a scalar mulpl of h dny mar Tha s, A has h form α α α, for any consan α Eampl: Th gnvalu s r (rpad Thr ar ss of lnarly ndpndn gnvcors, whch could b rprsnd by any nonzro vcors ha ar no consan mulpls of ach ohr For ampl k, Thrfor, a gnral soluon s k 8, Zachary S Tsng D- - 8
( If r, as usually dos, only has on lnarly ndpndn gnvcor k Thn h sysm A has a gnral soluon k r (k r η r Whr h scond vcor η s any soluon of h nonhomognous lnar sysm of algbrac quaons (A r I η k Eampl: 7, ( Th gnvalu s r (rpad Th corrspondng sysm s (A r I 7 Boh quaons of h sysm ar, w g h sam rlaon Hnc, hr s only on lnarly ndpndn gnvcor: k 8, Zachary S Tsng D- - 9
8, Zachary S Tsng D- - N, solv for η: η I has soluon n h form η η η hoos η, w g η / A gnral soluon s, hrfor, Apply h nal valus o fnd and Th parcular soluon s
Summary: Solvng a Homognous Sysm of Two Lnar Frs Ordr Equaons n Two Unknowns Gvn: A Frs fnd h wo gnvalus, r, and hr rspcv corrspondng gnvcors, k, of h coffcn mar A Dpndng on h gnvalus and gnvcors, h gnral soluon s: I Two dsnc ral gnvalus r and r : r r k k II Two compl conjuga gnvalus λ ± µ, whr λ µ has as an gnvcor k a b : λ ( a µ bsn( µ ( a sn( µ b cos( µ λ cos( III A rpad ral gnvalu r: ( Whn wo lnarly ndpndn gnvcors s k r k r ( Whn only on lnarly ndpndn gnvcor s k r (k r η r No: Solv h sysm (A r I η k o fnd h vcor η 8, Zachary S Tsng D- -
Ercss D-: Rwr h followng scond ordr lnar quaon no a sysm of wo quaons y y y Thn: (a show ha boh h gvn quaon and h nw sysm hav h sam characrsc quaon (b Fnd h sysm s gnral soluon 7 Fnd h gnral soluon of ach sysm blow 7 8 7 8 Solv h followng nal valu problms 8, ( 9, (, ( 8 8, (, ( 8, Zachary S Tsng D- -
9, (, (, ( For ach of h nal valu problms #8 hrough #, how dos h soluon bhav as? 7 Fnd h gnral soluon of h sysm blow, and drmn h possbl valus of α and β such ha h nal valu problm has a soluon ha nds o h zro vcor as α, ( 7 β Answrs D-: (a r r, (b 7 cos( sn( cos( sn( cos( sn( 8, Zachary S Tsng D- -
8, Zachary S Tsng D- - sn( sn( cos( cos( sn( cos( 7 8 sn cos sn cos 9 9 8 8 79 8sn( 79 cos( 79 sn( 79 cos( 8 For #8 and 9, ( lm For #,,,, and, h lms do no s, as ( movs nfnly far away from h orgn For #, 9 ( lm 7 7 ; h parcular soluon wll nd o zro as provdd ha, whch can b achvd whnvr h nal condon s such ha α β (, α β, ncludng h cas α β