Summary: Solving a Homogeneous System of Two Linear First Order Equations in Two Unknowns
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1 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 ±, whr λ has as an gnvcor k a b : λ ( a bsn( ( asn( bcos( 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
2 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
3 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 6 And hy ach rspcvly has an gnvcor k, k Thrfor, a gnral soluon of hs sysm of dffrnal quaons s 6
4 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
5 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
6 as II ompl conjuga gnvalus If h coffcn mar A has wo dsnc compl conjuga gnvalus λ ± Also suppos k a b s an gnvcor (ncssarly has compl valud nrs of h gnvalu λ Thn h sysm A has a ralvalud gnral soluon λ ( a bsn( ( asn( bcos( λ cos( A ll dal: Smlar o wha w hav don bfor, frs hr was h complvalud gnral soluon n h form ( λ ( λ k k W flr ou h magnary pars by carfully choosng wo ss of coffcns o oban wo corrspondng ralvalud soluons ha ar also lnarly ndpndn: u v λ λ ( acos( bsn( ( asn( bcos( Th ralvalud 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 complvalud prsson (h fron half of h complvalud gnral soluon: k ( λ λ λ ( a b λ ( ( acos( asn( bcos( ( acos( bsn( λ ( a b(cos( sn( λ bsn( ( asn( bcos( Thn, u s jus h ral par of hs complvalud funcon, and v s s magnary par 8, Zachary S Tsng D
7 8, Zachary S Tsng D Eampl: Th characrsc quaon s r, gvng gnvalus r ± Tha s, λ and 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 8, Zachary S Tsng D 6 Eampl: 6, ( Th characrsc quaon s r r, gvng gnvalus r ± Thus, λ and Tak r and fnd on of s gnvcors: (A r I 6 ( 6 ( 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(
9 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(
10 as III Rpad ral gnvalu Suppos h coffcn mar A has a rpad ral gnvalus r, hr ar subcass ( 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
11 ( 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
12 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
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