CHAPTER. Linear Systems of Differential Equations. 6.1 Theory of Linear DE Systems. ! Nullcline Sketching. Equilibrium (unstable) at (0, 0)

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1 CHATER 6 inar Sysms of Diffrnial Equaions 6 Thory of inar DE Sysms! ullclin Skching = y = y y υ -nullclin Equilibrium (unsabl) a (, ) h nullclin y = υ nullclin = h-nullclin (S figur) = + y y = y Equilibrium (unsabl) a (, ) h nullclin = υ nullclin + y = (S figur) = y y =5y y Equilibrium (sabl) a (, ) h nullclin 5+ y = υ nullclin y = (S figur) 45

2 46 CHATER 6 inar Sysms of Diffrnial Equaions 4 = y y = y+ Equilibrium (sabl) a (, ) y h nullclin + y = υ nullclin y = (S figur) 5 = + y y = y Equilibrium (unsabl) a (, ) y h nullclin y = υ nullclin + y = (S figur) 6 = y y =5 y+ Equilibrium (sabl) a (, ) y h nullclin 5+ y = υ nullclin y = (S figur)! Skching Scond-rdr DEs = (a) ing y =, w wri h quaion as h firs-ordr sysm = y y =y (b) Th quilibrium poin is (, y)= (, )

3 SECTI 6 Thory of inar DE Sysms 47 (c) h nullclin + y = υ nullclin y = y (S figur) (d) From h dircion fild, h quilibrium poin (, y)= (, ) is sabl () A mass-spring sysm wih his quaion shows dampd oscillaory moion abou () is sabl 8 + = (a) ing y =, w wri h quaion as h firs-ordr sysm = y y = + y (b) Th quilibrium poin is (, y)= (, ) (c) hnullclin y = υ nullclin y = y (S figur) (d) From h dircion fild, h quilibrium poin (, y)= (, ) is unsabl () A mass-spring sysm wih his quaion nds o fly apar Hnc, () is unsabl 9 + = (a) ing y =, w wri h quaion as h firs-ordr sysm = y y = + (b) Th quilibrium poin is (, y)= (, )

4 48 CHATER 6 inar Sysms of Diffrnial Equaions (c) h nullclin = υ nullclin y = y (S figur) (d) From h dircion fild, h quilibrium poin (, y)= (, ) is sabl () A mass-spring sysm wih his quaion shows no damping and sady forcing; hnc, priodic moion abou an quilibrium is o h righ of h origin Hnc, () is sabl + + = (a) ing y =, w wri h quaion as h firs-ordr sysm = y y = y+ (b) Th quilibrium poin is (, y)= (, ) (c) h nullclin + y = υ nullclin y = y (S figur) (d) From h dircion fild, h quilibrium poin (, y)= (, ) is sabl () A mass-spring sysm wih his quaion shows havy damping Th forc movs h quilibrium wo unis o h righ of h origin Hnc, () is sabl! Braking u Sysms = + = 4 = = + = = = = = sin

5 SECTI 6 Thory of inar DE Sysms 49! Chcking I u 5 = lugging ()= 4 4 u ino h givn sysm asily vrifis: and 6 = Th fundamnal mari is and v()= = = 4 4 Th gnral soluion of his sysm is!= A is 4 By subsiuion, w vrify c ()= c + 4 ()= u saisfy h sysm Th fundamnal mari is Th gnral soluion is and v ()= c ()= c +

6 44 CHATER 6 inar Sysms of Diffrnial Equaions 7 = 4 By subsiuion, w vrify bg= u saisfy h sysm Th fundamnal mari is Th gnral soluion is 8 = bg= and v c bg= c + By subsiuion, w vrify u()= sin cos and v saisfy h sysm Th fundamnal mari is Th gnral soluion is! Uniqunss in h has lan sin cos bg= cos sin cos sin ()= c c sin + cos cos sin 9 Th dircion fild of = y, y = is shown W hav drawn hr disinc rajcoris for h si iniial condiions a(), y()= f a, f, a, f, a, f, a, f, a, f, a, f o ha alhough h rajcoris may (and do) coincid if on sars a a poin lying on anohr, hy nvr cross ach ohr 4 4 y 4 4

7 SECTI 6 Thory of inar DE Sysms 44 Howvr, if w plo coordina = () or y = y() for hs sam si iniial condiions w g h si inrscing curvs shown y = () y = y()! Vrificaion lugging ino yilds or panding his, w may wri v = = = = = which vrifis h soluion! Third-rdr Vrificaion To vrify u, v, w, you should follow h procdur carrid ou in roblm To show ha h vcor funcions u, v, w ar linar indpndn, s cu+ c v+ c w = c c + or if his wr o b wrin ou in scalar form c + =

8 44 CHATER 6 inar Sysms of Diffrnial Equaions c + c = c + c + c = c c + = Bcaus i was assumd ha hs quaions ar ru for all, hy mus hold for =, which yilds c = c + c = c + c = or c= c = c =! Eulr s hod umrics (a) Th IV + =, ()=, ( )= sudid in Eampl can b solvd numrically wih a spradsh using h following coding: A B C D E y d dy d d = C = * B = A + = B + * D = C + * E = C = * B Doing his rsuls in h following valus on d y d dy d

9 SECTI 6 Thory of inar DE Sysms 44 If h rang is coninud o = 4, hn h graphs ha corrspond o Figur 6 look lik h following y y y componn graph y componn graph y componn graph D y viw (b) Th IV =, ( 5)=, ( 5)= 5 sudid in Eampl 4 can b solvd numrically wih a spradsh using h following coding: A B C D E y d d = C = * B 5 * C = A + = B + * D = C + * E = C = * B 5 * C dy d

10 444 CHATER 6 inar Sysms of Diffrnial Equaions Doing his rsuls in h following valus on 5 4 y d d dy d If h rang is coninud o = 5, hn h graphs ha corrspond o Figur 64 look lik h following y y y phas porrai y componn graph

11 SECTI 6 Thory of inar DE Sysms 445 y componn graph D y viw (c) Th IV + = 5 cos, ()=, ( )= sudid in Eampl 5 can b solvd numrically wih a spradsh using h following coding: A B C D E y d d = C = * B + 5 * cos(a) = A + = B + * D = C + * E = C = * B + 5 * cos(a) Doing his rsuls in h following valus on dy d y d d dy d

12 446 CHATER 6 inar Sysms of Diffrnial Equaions If h rang is coninud o = 4, hn h graphs ha corrspond o Figur 65 look lik h following y y y phas porrai y componn graph y componn graph D y viw! aching Gams A 4 C 5 D 6 B! Finding Trajcoris 7 =, y = y Wri y dy d = y y = Sparaing variabls, yilds dy y = d

13 SECTI 6 Thory of inar DE Sysms 447 or ln y = ln + c ln c y = c y =± y = C whr C is an arbirary consan Hnc, h rajcoris consis of a family of smi-infini lins originaing a h origin Th quaions =, y = y show ha soluions mov along hs lins away from h origin as indicad in h figur (Compar wih h soluion o roblm ) All soluions ar furhr and furhr from h origin and go fasr and fasr h furhr away hy ar 8 = y, y = W wri hs quaions as dy d = y = y y Sparaing variabls, yilds h quaion in h diffrnial form ydy = d Ingraing, yilds y = + c, or + y = C whr C is an arbirary nonngaiv consan Hnc, h rajcoris consis of a family of circls cnrd a h origin Th quaions = y, y = show ha soluions mov along h rajcoris in h clockwis dircion as illusrad Kp in mind ha soluions do no all mov a h sam spd All circular pahs around h origin hav h sam priod, bu h pahs wih h largr radius mov a a fasr ra y

14 448 CHATER 6 inar Sysms of Diffrnial Equaions! Compur Chck 9 Th compur phas porrai for roblm 7 and 8 ar shown y y Compur has orrai for roblm 7 Compur has orrai for roblm 8! Compur ab: Skw-Symmric arics (a) = y, y = y Trajcoris of his skw symmric sysm ar givn in h figur o ha rajcoris ar circls cnrd around h origin, and, hnc, h lngh of h vcor = (, y) is a consan (b) = ky, y =k Wri his sysm as h singl quaion + k = which has a gnral soluion of = Rcos( k δ ) Thn find y = k =R sin ( k δ ) Hnc, h lngh of any soluion vcor = (, y) is ()+ y ()= R cos( k δ)+ R sin( k δ )= R

15 SECTI 6 Thory of inar DE Sysms 449 Thrfor, h rajcoris of h sysm ar circls cnrd around h origin wih frquncy ω = k and priod π π k An opn-ndd graphic solvr can b usd o vrify hs facs! Th Wronskian Whn h Wronskian is no zro, h vcors ar linarly indpndn and form a fundamnal s (If h Wronskian of wo soluions is nonzro on I i will always b nonzro) W, = =, so h vcors form a fundamnal s W, = =5, so h vcors form a fundamnal s W, = =, so h vcors form a fundamnal s W, = = 8, so h vcors form a fundamnal s W, 6 W, cos sin = = cos + sin = sin cos b g, so h vcors form a fundamnal s cos sin = = cos + sin =, so h vcors form a fundamnal s sin cos! Suggsd Journal Enry 7 Sudn rojc

16 45 CHATER 6 inar Sysms of Diffrnial Equaions 6 inar Sysms wih Ral Eignvalus! Soluions in Gnral = 4 Th characrisic quaion of h sysm is p ()= 4 λ = λ + 5λ =, λ which has soluions λ =, λ = 5 Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v λ =5 = v Hnc, h gnral soluion is = 6 ()= + c c5 Th characrisic quaion of h sysm is p ()= λ = λ 8λ + 5=, 6λ which has soluions λ =, λ = 5 Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v λ = 5 = v Hnc, h gnral soluion is c c ()= + 5

17 = 4 = 5 = 4 Th characrisic quaion of h sysm is p ()= λ = λ 5λ + 6=, 4 λ SECTI 6 inar Sysms wih Ral Eignvalus 45 which has soluions λ =, λ = Finding h ignvcors corrsponding o ach ignvalu yilds λ= v= λ = v = Hnc, h gnral soluion is 5 8 Th characrisic quaion of h sysm is p ()= + ()= c c λ 5 = λ + λ 8=, 8 λ which has soluions λ =, λ = 8 Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v 4 5 λ = 8 = v Hnc, h gnral soluion is 5 c c 4 ()= Th characrisic quaion of h sysm is 5 p ()= λ = λ 6λ + 8=, λ

18 45 CHATER 6 inar Sysms of Diffrnial Equaions which has soluions λ =, λ = 4 Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v λ = 4 = v Hnc, h gnral soluion is 6 = 4 c c ()= + 4 Th characrisic quaion of h sysm is bg= 4 p λ = λ 4λ 5=, λ which has soluions λ =, λ = 5 Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v λ = 5 = v Hnc, h gnral soluion is 7 = c ()= c + 5 Th characrisic quaion of h sysm is p ()= λ = ( λ ) ( λ )=, λ which has soluions λ =, λ = Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v λ = = v

19 Hnc, h gnral soluion is 8 = 9 = SECTI 6 inar Sysms wih Ral Eignvalus 45 c c ()= + Th characrisic quaion of h sysm is p ()= λ = λ λ =, λ which has soluions λ =, λ = Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v λ = = v Hnc, h gnral soluion is + ()= c c Th characrisic quaion of h sysm is p ()= λ = λ λ =, λ which has soluions λ =, λ = Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v λ = = v Hnc, h gnral soluion is c c ()= +

20 454 CHATER 6 inar Sysms of Diffrnial Equaions 4 = 4 4 = = Th characrisic quaion of h sysm is 4 p ()= λ = λ 4=, 4 4λ which has soluions λ =, λ = Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v λ = v = Hnc, h gnral soluion is 4 c ()= c + Th characrisic quaion of h sysm is p ()= λ = λ + λ + =, 4λ which has soluions λ =, λ = Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v λ = = v Hnc, h gnral soluion is 5 8 c c ()= + Th characrisic quaion of h sysm is 5 p ()= λ = λ λ + 6 =, 8λ

21 = SECTI 6 inar Sysms wih Ral Eignvalus 455 which has soluions λ = 4, λ = 9 Finding h ignvcors corrsponding o ach ignvalu yilds λ= 4 = v λ = 9 v = Hnc, h gnral soluion is c c bg= + 9 Th characrisic quaion of h sysm is 4 p ()= λ = λ + λ =, 8 6λ which has soluions λ =, λ = Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v 4 λ = = v Hnc, h gnral soluion is 5 4 = c c 4 ()= + Th characrisic quaion of h sysm is 5 p ()= λ = λ 6λ + 8=, λ which has soluions λ =, λ = 4 Finding h ignvcors corrsponding o ach ignvalu yilds λ= = v λ = 4 = v

22 456 CHATER 6 inar Sysms of Diffrnial Equaions Hnc, h gnral soluion is! Rpad Eignvalus c ()= c Th characrisic quaion of h sysm is p ()= λ = λ λ + =, 4 λ which has soluions λ = and λ = wih on linarly indpndn ignvcor v = Th gnral soluion is, hrfor, c c whr u and u saisfy A I u ()= + + u u ( ) =, or 4 RST u = u which has on linarly indpndn quaion, u + u = Hnc, ()= RST UVW u u = u = + = k + k k c c k Bcaus h rm involving k is a mulipl of h firs rm, w hav c ()= c Th characrisic quaion of h sysm is p ()= λ = λ + λ + =, 8 5λ which has soluions λ = and λ = wih on linarly indpndn ignvcor v = RST, UVW UVW

23 Th gnral soluion is, hrfor, R bg= c cs whr u and u saisfy or + SECTI 6 inar Sysms wih Ral Eignvalus 457 T ( A+ I) u= u u = +, which has on linarly indpndn quaion, 4u + u = Hnc, u u = c c k U V W u u = k u = = + k k R S T U bg V W Bcaus h rm involving k is a mulipl of h firs rm, w hav! Soluions in aricular 7 Th characrisic quaion of h sysm is + R S T ()= c c + U V W p ()= λ = λ λ =, 5 4λ which has h soluions λ = and λ = Finding h ignvcors corrsponding o ach ignvalu, yilds λ= = v 5 λ = = v Th gnral soluion is c c 5 ()= +

24 458 CHATER 6 inar Sysms of Diffrnial Equaions lugging ino h iniial condiions ()= yilds c+ c = 5c + c = which givs c = and c = Th soluion of h IV is 8 Th characrisic quaion of h sysm is ()=F I H K F I + H K 5 p ()= λ = λ λ 4=, λ which has h soluions λ = and λ = 4 Finding h ignvcors corrsponding o ach ignvalu, yilds λ= = v λ = 4 = v Th gnral soluion is lugging ino h iniial condiions yilds c c ()= + 4 ()= c c = c + c = which givs c = and c = Th soluion of h IV is ()= 4

25 9 Th characrisic quaion of h sysm is SECTI 6 inar Sysms wih Ral Eignvalus 459 p ()= λ = ( λ ) ( λ )=, λ which has h soluions λ = and λ = Finding h ignvcors corrsponding o ach ignvalu, yilds λ= = v λ = = v Th gnral soluion is lugging ino h iniial condiions c c ()= + 5 ()= 4 yilds c = 5 and c = 4 Th soluion of h IV is 5 4 Th characrisic quaion of h sysm is ()= + = 5 4 p ()= λ 4 = λ + λ 6=, λ which has h soluions λ = and λ = Finding h ignvcors corrsponding o ach ignvalu, yilds 4 λ= = v λ = = v Th gnral soluion is lugging ino h iniial condiions c 4 ()= c + ()=

26 46 CHATER 6 inar Sysms of Diffrnial Equaions yilds 4c+ c = c + c = which givs c = and c = Th soluion of h IV is 5 5 ()=F H I K Th characrisic quaion of h sysm is F I + H K p ()= λ = λ λ =, λ which has h soluions λ = and λ = Finding h ignvcors corrsponding o ach ignvalu, yilds λ= = v λ = = v Th gnral soluion is lugging ino h iniial condiions + ()= c c ()= yilds c+ c = c + c = 5 which givs c = and c = Th soluion of h IV is ()= Th characrisic quaion of h sysm is F + 5I H K p ()= λ = λ + 5λ + 4=, λ

27 SECTI 6 inar Sysms wih Ral Eignvalus 46 which has h soluions λ = 4 and λ = Finding h ignvcors corrsponding o ach ignvalu, yilds λ=4 = v λ = = v Th gnral soluion is lugging ino h iniial condiions yilds c ()= 4 c + ()= 6 c+ c = c + c = 6 7 which givs c = and c = Th soluion of h IV is ()=F H I K Th characrisic quaion of h sysm is 7 F I + H K 4 p ()= λ = λ + 4λ =, 4 λ which has h soluions λ = and λ = 4 Finding h ignvcors corrsponding o ach ignvalu, yilds λ= = v λ =4 v = Th gnral soluion is ()= + c c 4

28 46 CHATER 6 inar Sysms of Diffrnial Equaions lugging ino h iniial condiions yilds ()= 4 c + c = c c = 4 which givs c = and c = Th soluion of h IV is 4 Th characrisic quaion of h sysm is ()= 4 p ()= λ = λ λ 5=, λ which has h soluions λ = 5 and λ = 7 Finding h ignvcors corrsponding o ach ignvalu, yilds λ=5 = v λ = 7 = v Th gnral soluion is lugging ino h iniial condiions yilds c ()= 5 c + ()= c+ c = c + c = 7 which givs c = and c = Th soluion of h IV is ()=F H I K F I + H K 5 7

29 5 Th characrisic quaion of h sysm is SECTI 6 inar Sysms wih Ral Eignvalus 46 p ()= λ = λ 5λ + 6=, 4 λ which has h soluions λ = and λ = Finding h ignvcors corrsponding o ach ignvalu, yilds λ= = v λ = v = Th gnral soluion is lugging ino h iniial condiions c ()= c + ()= yilds c + c = c c = which givs c = and c = Th soluion of h IV is ()= 6 Th characrisic quaion of h sysm is p ()= λ = λ λ =, λ which has h soluions λ = and λ = Finding h ignvcors corrsponding o ach ignvalu, yilds λ= = v λ = = v Th gnral soluion is lugging ino h iniial condiions c ()= c +

30 464 CHATER 6 inar Sysms of Diffrnial Equaions ()= yilds c + c = c + c = which givs c = and c = Th soluion of h IV is ()=! Craing w roblms 7 (a) An ampl is A = + a a b Th characrisic quaion of his mari is p( λ)= ( λ a) ( λ b), giving a doubl roo of a and a singl roo of b for h ignvalus Howvr, h ignvcor corrsponding o a is found by solving for, y, z in h quaion a a y a y b z = z or a + y = a ay = ay bz = az which implis z =, = α, y = In ohr words, i has only on (linarly indpndn) ignvcor,, Th ignvcor corrsponding o h singl ignvalu b is found by solving for, y, z in h quaion a a y b y b z z or = a + y = b ay = by bz = bz which implis =, y =, z = α In ohr words, h ign-vcor,,

31 SECTI 6 inar Sysms wih Ral Eignvalus 465 (b) An ampl is A = a a a Th characrisic quaion of his mari is p( λ)= ( λ a), giving a ripl roo of a for h ignvalus To find h ignvcor, solv for, y, z in h quaion a a a y a y z = z or a + y = a ay = ay az = az which implis y =, = α, z = β, α, β arbirary In ohr words, h wo (linarly indpndn) ignvcors ar,, and,,! n Indpndn Eignvcor 8 (a) Th ignvalu is λ =, wih an algbraic mulipliciy of W find h ignvcor(s) by plugging λ = ino h quaion Av = λ v and solving for h vcor v Doing his yilds h singl ignvcor c,, (b) From h ignvalu and ignvcor, on soluion has bn found ()= c (c) ow w solv for a scond soluion of h form ()= v+ u, whr v = (,, ) is h firs ignvcor, and u= au, u, u f is an unknown vcor lugging () ino h sysm = A and comparing cofficins of and yilds quaions for u, u, u, giving u =, u =, u = Hnc, w obain a scond soluion of ()= +

32 466 CHATER 6 inar Sysms of Diffrnial Equaions (d) To find a hird (linarly indpndn) soluion, w ry h spcific form bg= v+ u+ w! Soluions in Spac whr v and u ar vcors prviously found and w is h unknown vcor lugging his ino h sysm rsuls in h sysm of quaions ( A I) w = u W hn find w = aw, w, wf Solving his sysm yilds w =, w =, w = Hnc, w obain a hird soluion of + + bg= 9 Th characrisic quaion of h sysm is p ()= λ 4 λ = λ+ 6λ λ + 6=, 4 λ which has soluions λ =, λ =, and λ = Finding h ignvcors corrsponding o ach ignvalu, yilds Hnc, h gnral soluion is Th characrisic quaion of h sysm is p bg= λ = v = λ = v = λ = v = ()= c c c + + λ λ = λ + 7λ + 6=, λ

33 SECTI 6 inar Sysms wih Ral Eignvalus 467 which has soluions λ =, λ =, and λ = Finding h ignvcors corrsponding o ach ignvalu, yilds Hnc, h gnral soluion is! Spaial ariculars bg= λ = v = λ = v = 4 λ = v = c c c W find h ignvalus and ignvcors of h cofficin mari by h usual procdur, obaining Hnc, h gnral soluion is λ = v = λ = v = λ = v = ()= c c c + + Subsiuing his vcor ino h iniial condiion ()=,, yilds h hr quaions c c c = c c + + c c + c c = =

34 468 CHATER 6 inar Sysms of Diffrnial Equaions which has h soluion c =, c =, c = Hnc, h IV has h soluion ()= W find h ignvalus and ignvcors of h cofficin mari by h usual procdur, obaining λ= = v Hnc, h gnral soluion is λ = v = λ = v = ()= c c c + + Subsiuing his vcor ino h iniial condiion ()=, 4, yilds h hr quaions c + c = c + c = 4 c = which has h soluion c =, c =, c = Hnc, h IV has h soluion! Rpad Eignvalu Thory bg= + + (a) Th characrisic quaion of h mari is ( a λ) ( d λ) bc=, or λ ( a + d) λ + ( ad bc )= This has rpad roos if and only if h discrimina is zro (i, if and only if which simplifis o ( a d) + 4bc= ) ( a + d) 4( ad bc)=,

35 SECTI 6 inar Sysms wih Ral Eignvalus 469 (b) Suppos a d and ( a d) + 4bc= Thn h singl ignvalu is a+ d To find h corrsponding ignvcor, w obain h quaions a d c v + b v = d a cv + b v = Solving hs quaions, yilds a f b d a v = v, v = (, ) (c) Wih on linarly indpndn ignvcor w obain on linarly indpndn soluion A scond soluion has h form of bg b g a+ d b = c d a + b bg = b g b g d a u a+ d a+ d W find u by solving ( A λ I) u= v, w obain u = and u = u! has orrais 4 = 6 Th ignvalus and vcors ar y v v λ =, v = (, ) λ = 5, v = (, ) 5 = 4 v v y Th ignvalus and vcors ar λ =, v = (, ) λ =, v = (, )

36 47 CHATER 6 inar Sysms of Diffrnial Equaions 6 = 5 y v v Th ignvalus and vcors ar λ =, v = (, ) λ = 4, v = (, ) 7 = 4 y v v Th ignvalus and vcors ar λ =, v = (, ) λ =, v = (, )! Vrificaion of Indpndnc 8 To show ()= ()= 4 4 ar linarly indpndn, w show ha h consans c and c for which c c = for all ar c= c = If his mus hold for all, i mus hold for =, which yilds h quaions whos soluion is c = c = c c c =! Adjoin Sysms 9 (a) Th ngaiv ranspos of h givn mari is simply h mari wih s in h plac of s, hnc h adjoin sysm is = w = ATw = w

37 SECTI 6 inar Sysms wih Ral Eignvalus 47 (b) Th firs qualiy is simply h produc rul for mari drivaivs Using h adjoin sysm, yilds b g, T T T T w = A w = w A and hnc, w T+ w T = w T + w T = (c) Th characrisic quaion of h mari is simply λ =, and hnc, h ignvalus ar +, Th ignvcor corrsponding o + can asily b found and is (, ) ikwis, h ignvcor for is (, ) Hnc, c ()= c + (d) () lugging in h iniial condiion ()= (, ), yilds c= c = So h soluion of h IV is ()=F I H K F I + H K F I = + H K If h iniial condiions ar w()= (, ), hn c =, c = o ha h iniial condiions ()= (, ) and w()= (, ) ar orhogonal vcors, and by h rsul in par (b) h wo rsuling soluions will always b orhogonal for all > So h soluion of h IV is Trajcoris ar orhogonal! Cauchy-Eulr Sysms w = + 4 (a) ()= λ v, whr λ is an ignvalu of A and v is a corrsponding ignvcor Thn or n h ohr hand, = λ λ v = λ λ v λ λ λ λ A = A v = Av = v bcaus v is an ignvcor of A Thrfor, = A

38 47 CHATER 6 inar Sysms of Diffrnial Equaions (b) W hav Th characrisic quaion is =, > p( λ)= ( λ) ( λ)+ 4 = whos ignvalus ar λ = and λ = and corrsponding ignvcors ar v = (, ) and v = (, ) From par (a), h gnral soluion is hn c c! Compur abs: rdicing has orrais ()= + 4 For ach of h linar sysms (a) (d) a fw rajcoris in h phas plan hav bn drawn Th analyic soluions ar hn compud (a) =, y =y y Solv ach of hs quaions individually, obaining = c and y = c Eliminaing yilds h rajcoris y c =, which is h family of hyprbolas shown (b) =, y =y y Solv ach of hs quaions individually, obaining = c and y c = Eliminaing yilds h rajcoris = c, which is h family of vrical lins For any saring poin a, yf h soluion movs asympoically owards a, f Th -ais is composd nirly of sabl quilibrium poins

39 SECTI 6 inar Sysms wih Ral Eignvalus 47 (c) = + y, y = + y = y+ c, bcaus y = y ; which is a family of sraigh lins in h phas plan wih slop and y-inrcp ( c, ) = y is a lin of unsabl qui- librium poins (d) = y, y = y W wri hs quaions as h singl quaion =, which has soluion = c + c Hnc, y = c c ow w add and subrac hs quaions, yilding or which is a family of hyprbolas wih as y! Radioaciv Dcay Chain + y = c y = c + y = k, y = and y = (S figur) 4 (a) Th amoun of iodin is simply dcrasing via radioaciv dcay; hnc, di =ki, d whr k is h dcay consan of iodin Work in Chapr showd ha h dcay consan is ln dividd by h half-lif of h marial; hnc, k = ln l Th amoun of non is incrasing wih h dcay of iodin, bu dcrasing wih is own radioaciv dcay, hnc, h quaion

40 474 CHATER 6 inar Sysms of Diffrnial Equaions d d = kik whr k = ln (b) In mari form, h quaions bcom I = k k k I Th ignvalus and vcors of his mari can asily b sn as k k λ=k v= λ = = k k v Hnc, h soluion! ulipl Comparmn iing ()= c k k k k + k c 4 (a) Calling () h numbr of pounds of sal in ank A and B, rspcivly, a im, h IV ha dscribs hs amouns is 6 5 =F I H K + F I H K, ()= 6, = F I H K F I H K ()= To solv his linar homognous sysm, w find h ignvalus and vcors of h cofficin mari which ar A = 6 6, λ=8 v= λ =4 = v

41 SECTI 6 inar Sysms wih Ral Eignvalus 475 (b) (c) (d) Hnc, h gnral soluion is = 8 c 4 c + lugging h iniial condiions ino his gnral soluion yilds 5 c c or c = 5 and c = 5 Hnc, = S figur for graph of h amoun of sal Th figur in par (b) indicas ha h amoun of sal in ank B is nvr as larg as h amoun of sal in ank A; i, B nvr cds A a any Th amoun of sal in ach ank gos o zro as pcd, i, B ; A + + ()= B A Tim, minus! Suggsd Journal Enry 44 Sudn rojc

42 476 CHATER 6 inar Sysms of Diffrnial Equaions 6 inar Sysms wih onral Eignvalus! Soluions in Gnral = Th characrisic quaion for h mari is λ + =, which has compl ignvalus ±i lugging i ino Av = λ v for λ, yilds h singl quaion v = i v Sing v = yilds v = i Thrfor, α =, β =, p =,, q =, Two linarly indpndn soluions ar hn obaind p ()= q= α cosβ α sin β cos sin ()= p+ q= + α sin β α cosβ sin cos Th gnral soluion is, hrfor, or wrin ou in componn form, = ()= c + c = c cos c + sin sin cos = ccos + csin y = c sin + c cos Th characrisic quaion for h mari is λ + 4λ + 5=, which has compl soluions λ, and λ = ±i lugging hs valus ino Av = λ v yilds h rspciv ignvcors ( i, ) Thrfor, α =, β =, p = (, ), q = b, g Two linarly indpndn soluions ar hn obaind bg = p q= α α cos β sin β cos sin α α bg = sin βp+ cos βq= sin cos +

43 Th gnral soluion is, hrfor, = 4 = bg= + = c c c SECTI 6 inar Sysms wih onral Eignvalus 477 cos + sin c + sin cos cos sin Th characrisic quaion for h mari is λ λ + 5=, which has compl soluions λ and λ = ± i lugging hs valus ino Av = λ v yilds h rspciv ignvcors (, ± i ) Thrfor, α =, β =, p = (, ), q = (, ) Two linarly indpndn soluions ar hn obaind bg = = α α cos βp sin βq cos sin bg Th gnral soluion is, hrfor, 6 5 = + = + α α sin βp cos βq sin cos ()= c + c = c cos sin + c sin cos Th characrisic quaion for h mari is λ 8λ + 7=, which has compl soluions λ and λ = 4 ±i lugging hs valus ino Av = λ v yilds h rspciv ignvcors ( ± i, 5 ) Thrfor, α = 4, β =, p = (, 5 ), q = (, ) Two linarly indpndn soluions ar hn obaind p q 4 ()= = 4 5 α cosβ α sin β cos sin ()= p+ q= α sin β α cosβ sin cos Hnc, h gnral soluion is ()= c + c = c cos sin c + sin + cos 5cos 5sin 4 4

44 478 CHATER 6 inar Sysms of Diffrnial Equaions 5 = 6 = Th ignvalus ar λ and λ =±i lugging hs valus ino Av = λ v yilds h rspciv ignvcors (, ± i ) Thrfor, α =, β =, p = (, ), q = (, ) Two linarly indpndn soluions ar hn obaind p ()= q= α cosβ α sinβ cos sin ()= p+ q= α sin β α cosβ sin cos Th gnral soluion is, hrfor, 4 bg= + = c c c + cos c + sin cos sin sin + cos Th ignvalus ar λ and λ =± i wih corrsponding ignvcors ( ± i, ) Thrfor, α =, β =, p = (, ), q = (, ) Two linarly indpndn soluions ar hn obaind ()= = αcosβp αsin βq cos sin ()= + = + αsin βp αcosβq sin cos 7 = Th gnral soluion is, hrfor, 4 ()= c + c = c cos sin c + cos + sin cos sin Th ignvalus ar λ and λ = ± i lugging hs valus ino Av = λ v yilds compl ignvcors (, i) Thrfor, α =, β =, p = (, ), q = (, )

45 8 = 9 = Two linarly indpndn soluions ar hn obaind Th gnral soluion is, hrfor, 5 SECTI 6 inar Sysms wih onral Eignvalus 479 bg = = α α bg = + = + cos βp sin βq cos sin p q α α sin β cos β sin cos bg= + = c c c cos c + + sin cos sin cos + sin Th ignvalus ar λ and λ =±i lugging hs valus ino Av = λ v yilds compl ignvcors ( ± i, ) Thrfor, α =, β =, p = (, ), q = (, ) Two linarly indpndn soluions ar h obaind p ()= q= α cosβ α sin β cos sin ()= p+ q= + α sin β α cosβ sin cos Th gnral soluion is, hrfor, 5 ()= c + c = c cos sin c + cos + sin cos sin Th ignvalus ar λ and λ = ±i lugging hs valus ino Av = λ v yilds compl ignvcors ( ± i, 5 ) Thrfor, α =, β =, p = (, 5 ), q = (, ) Two linarly indpndn soluions ar hn obaind p q ()= = 5 α cosβ α sin β cos sin ()= p+ q= 5 + α sin β α cosβ sin cos

46 48 CHATER 6 inar Sysms of Diffrnial Equaions Th gnral soluion is, hrfor, = ()= c + c = c cos sin c + cos + sin 5cos 5sin Th ignvalus ar λ and λ = ± i lugging hs valus ino Av = λ v yilds compl ignvcors ( ± i, ) Thrfor, α =, β =, p = (, ), q = (, ) Two linarly indpndn soluions ar hn obaind ()= = αcosβp αsin βq cos sin ()= + = + αsin βp αcosβq sin cos Th gnral soluion is, hrfor, = ()= c + c = c sin cos + c cos sin Th ignvalus ar λ and λ = ±i lugging hs valus ino Av = λ v yilds compl ignvcors (, ± i ) Thrfor, α =, β =, p = (, ), q = (, ) Two linarly indpndn soluions ar hn obaind p q ()= = α cosβ α sin β cos sin ()= p+ q= α sin β α cosβ sin cos Th gnral soluion is, hrfor, ()= c + c = c cos cos sin + c + sin sin + cos

47 4 = SECTI 6 inar Sysms wih onral Eignvalus 48 Th ignvalus ar λ and λ =± i lugging hs valus ino Av = λ v yilds compl ignvcors (, ± i ) Thrfor, α =, β =, p = (, ), q = (, ) Two linarly indpndn soluions ar hn obaind p ()= q= α cosβ α sin β cos sin p ()= + q= α sin β α cosβ sin cos Th gnral soluion is, hrfor,! Soluions in aricular =, ()= ()= c + c = c + cos c + sin cos sin sin + cos Th cofficin mari A has ignvalus λ= λ = ±i and corrsponding ignvcors ( ± i, ) Hnc, h wo linarly indpndn soluions obaind ar p q ()= = α cosβ α sin β cos sin ()= p+ q= + α sin β α cosβ sin cos lugging in ()= ()+ ()= + = c c c c yilds c = and c = Th soluion is, hrfor, ()= () ()= sin cos cos sin

48 48 CHATER 6 inar Sysms of Diffrnial Equaions 4 = 4, ()= Th cofficin mari A has ignvalus λ = λ =± i and corrsponding ignvcors (, i) Hnc, h wo linarly indpndn soluions obaind ar lugging in bg = = α α bg = + = + cos βp sin βq cos sin p q α α sin β cos β sin cos bg bg bg c c c c = + = + yilds c = and c = Th soluion is, hrfor, 5 =, ()= bg bg bg = = = cos sin sin + cos Th cofficin mari A has ignvalus λ = λ = ±i and corrsponding ignvcors ( i, ) Hnc, h wo linarly indpndn soluions obaind ar lugging in bg = = α α cos βp sin βq cos sin bg = p+ q= α α + sin β cos β sin cos bg bg bg = + = + c c c c yilds c = and c = Th soluion is, hrfor, bg bg = cos + sin = = cos

49 6 = 5, 5 4 ()= SECTI 6 inar Sysms wih onral Eignvalus 48 Th cofficin mari A has ignvalus λ = λ = ±i and corrsponding ignvcors ( 5, i) Th wo linarly indpndn soluions obaind ar lugging in bg = + = + 5 = p q= α α cos β sin β cos sin α α 5 bg sin βp cos βq sin cos bg bg bg 5 c c c c = + = + yilds c = and c = Th soluion is, hrfor,! Sysm 7 = (a) ()= () ()= 5cos cos sin 5 = 4 5sin = 5cos sin sin cos 5sin Solving h characrisic quaion λ λ = ( λ + ) λ + 4 = λ which has roos λ =, λ, λ =± i b g, (b) Solving for, y, z in h quaion y = y z z yilds = α, y =, z =, α arbirary, so h ignvcor corrsponding o λ = is =

50 484 CHATER 6 inar Sysms of Diffrnial Equaions (c) Solving for, y, z in h sysm y z = i yilds h ignvcor (,, i) Hnc, w idnify y z α =, β =, p = (,, ), q = (,, ) (d) Th wo linarly indpndn soluions obaind ar Wriing ou h gnral soluion ()= αcosβp αsin βq= cos sin ()= αsin βp+ αcosβq= sin + cos ()= c + c + c in scalar form, yilds ()= c y ()= c cos+ csin z ()= ccosc sin () Subsiuing h IC ()= c ()+ c ()+ c ()= c + c + c = yilds c =, c =, and c = Th soluion of h IV is, hrfor, or in coordina form + ()= ()+ ()= sin cos ()= y ()= sin z ()= cos

51 (f) Th rajcory of ( (), y (), z ()) SECTI 6 inar Sysms wih onral Eignvalus 485 in D spac is a hli (i, i roas around h in circls bu approachs h yz-plan) y z! Thrfold Soluions 8 = Th characrisic polynomial is givn by λ+ 4λ 6λ + 4 = ( λ ) λ b λ + g Hnc, h ignvalus ar λ =, λ and λ = ±i Subsiuing and + i ino h quaion Av = λ v yilds h ignvcors associad wih ach ignvalu Doing his, yilds λ= v=,, λ = + i v = i,, Thrfor, Th hr indpndn soluions ar ()= α =, β =, p = (,, ), q = (,, ) ()= αcosβp αsinβq= cos sin ()= p+ q= α sinβ α cosβ sin + cos

52 486 CHATER 6 inar Sysms of Diffrnial Equaions 9 = Hnc, h gnral soluion is ()= c sin c c + cos + = cos sin Th characrisic polynomial of his sysm is λ λ = λ b + g = λ a a ccos csin c c cos + c sin wih ignvalus λ =, λ and λ = ±i Th ignvcors corrsponding o hs ignvalus ar λ = v =,, λ = + i v = i, i, Thrfor, α =, β =, p = b,, g, q =,, j f f Hnc, h gnral soluion = ()= c c + cos + cos + sin sin cos + c sin sin Th characrisic polynomial is givn by λ+ λ 7λ + 5= ( λ ) λ b λ + 5g cos cos sin

53 = SECTI 6 inar Sysms wih onral Eignvalus 487 Hnc, h ignvalus ar λ = and λ = λ = ± i Subsiuing and + i ino h quaion Av = λ v yilds h ignvcors associad wih ach ignvalu Doing his, yilds λ= v=,, λ = + i v =,, i Thrfor, Hnc h gnral soluion is Th ignvalus ar α =, β =, p = (,, ), q = (,, ) ()= c c c + cos + sin sin cos λ = and λ = λ = ±i Subsiuing and + i ino h quaion Av = λ v yilds h ignvcors associad wih ach ignvalu Doing his, yilds λ= v=,, λ = + i v = + i,, i Thrfor, α =, β =, p = (,, ), q =,, d h and so hr indpndn soluions ar ()= ()= αcosβp αsinβq= cos sin ()= αsinβp+ αcos βq= sin + cos

54 488 CHATER 6 inar Sysms of Diffrnial Equaions Hnc,! Tripl IVs = ()= c c +, ()= cos sin cos c + sin 5 6 Th ignvalus and vcors of h cofficin mari ar and wih α =, β =, # p = ()= = v =,, λ λ, λ = ± i; λ v = 5 + i, - + i, 6 sin + cos sin cos 5,, 6, q # =,, Th hr indpndn soluions ar lugging in h iniial condiions ()= αcosβp αsinβq= cos 6 sin 5 ()= p+ q= α sinβ α cosβ sin 6 + cos ()= c + c 6 + c 5 = yilds c =, c =, and c = Th soluion of h IV is = = cos sin cos sin 6cos

55 =, ()= Th ignvalus and vcors of h cofficin mari ar and λ SECTI 6 inar Sysms wih onral Eignvalus 489 = v =,, λ, λ =± i v =, ± i,,, hrfor, hr indpndn soluions ar ()= lugging in h iniial condiions yilds c ()= αcosβp αsinβq= cos sin ()= p+ q= α sinβ α cosβ sin + cos ()= c + c + c = = c = and c = Hnc, h soluion of h IV is = + + = =! ar of Indpndnc + cos + + cos sin sin + cos sin cos sin 4 Th Wronskian of wo vcor funcions is dfind as h drminan of h mari formd by placing h vcors as columns in h mari If h vcor funcions ar also soluions of a linar sysm of diffrnial quaions, hn h vcors ar linarly indpndn if and only if h

56 49 CHATER 6 inar Sysms of Diffrnial Equaions Wronskian is nonzro for any in h inrval of inrs In his problm, w obain h wo vcor soluions a ()= α b a cosβ α sin β b a ()= α b a + α sin β cosβ b formd from h ignvalus α ± i β and ignvcors p = aa, af, q = ab, bf of a mari W valua (), () whn =, yilding ()= aa, a f, ()= ab, b f Hnc, h Wronskian of () and () a = is W, ()= a b a b Bu h columns of his mari ar linarly indpndn and hus h Wronskian is nonzro Hnc, h vcors () and () ar linarly indpndn vcor funcions! ari Eponnial d 5 d d 4 A F = I+ A+ A + A + A4+ $ A A A A4 $ d!! 4!!! I HG K J = F I A HG K J = = A I+ A+ A+ $ A! 6 W diffrnia ()= A c, obaining ( )= A Ac lugging, ino = idniy A Ac= AAc A, yilds h 7 Givn A = and by compuing h powrs, yilds In gnral, w us h rul A =, A = A, A 4 = I, $ A n = I, A n+ = A

57 Hnc, h mari ponnial is A = I + + cosh sinh sinh cosh = Th gnral soluion o + = A can hn b wrin as SECTI 6 inar Sysms wih onral Eignvalus $ $ + $ = 4!! 5! $ $! 5! 4! c ()= c c cosh sinh c = cosh + sinh sinh cosh sinh cosh This could b wrin as a linar combinaion of vcors involving and bcaus cosh = +, sinh = 8 Givn and by compuing h powrs, yilds A A A 4 A = = I = =A = I and so on Hnc, h mari ponnial is A = I + cos = sin + sin cos +! Th gnral soluion can hn b wrin as bg= cos sin + $ = sin c c cos c = $ + $ 4!! 5! $ + $! 5! 4! cos c + sin sin cos

58 49 CHATER 6 inar Sysms of Diffrnial Equaions 5 9 A = 4 Th characrisic quaion is givn by λ + λ = Thus, by h Cayly-Hamilon horm, uliplying by A yilds h quaion Hnc, w solv for A and A o g Using hs quaions, w find A A A + A I= A+ A A = A = A+ I A = A + A Hnc, w approima h mari ponnial as! Skw-Symmric Sysms k = k A + = 5 5 = = = = I ! 6 4 W solv his using ignvalus and vcors Th characrisic quaion of h cofficin mari is λ k p( λ)= λ k k λ = + =, which has roos λ= λ =±ik lugging in iv ino h quaion k v ik k v v v givs kv = = ikv Sing v = yilds v = i Hnc, on of h conjuga ignvcors is v = = + i i +

59 SECTI 6 inar Sysms wih onral Eignvalus 49 W hn idnify α =, β = k, p = (, ), q = (, ) Th wo linarly indpndn vcor soluions ar hn αcosβp αsin βq cosk sin k αsin βp αcos βq sin k cosk ()= = ()= + = + Th gnral soluion is In componn form = c + c = c cosk sin k = c cosk + c sin k y = c sin k + c cos k + sin k c cosk To vrify ha h lngh of h soluion vcor is a consan for all, w wri h sysm as h singl quaion + k = whos gnral soluion is = Ccos( δ ) W hn find y = k =C sin ( k δ ) Hnc, h lngh of any soluion vcor = (, y) is ()+ y ()= C cos ( k δ)+ C sin ( k δ)= C cos ( k δ)+ sin ( k δ ) = C

60 494 CHATER 6 inar Sysms of Diffrnial Equaions! Compur ab: has orrai = 5, ()= y (, ) Th soluion of h IV is shown in h phas plan S figurs for plo of h -coordina and h y-coordina as a funcion of o ha hs graphs ar consisn wih h soluion in h phas plan has plan soluion y = 4 5, ()= y Th soluion of h IV in h phas plan is shown o ha h graphs of () and y () vr- 8 8 sus ar consisn wih h phas plan graph 8 () y ()

61 SECTI 6 inar Sysms wih onral Eignvalus 495! Coupld ass-spring Sysm W find h ignvalus and vcors of h mari k+ k k m m k k + k m m in rms of h k, k, k, m, m, bu his is rmly involvd so w l h paramr qual k= k = k = m= m = o g Finding h ignvalus using h compur algbra sysm apl, yilds h purly compl numbrs λ= λ = ±i and λ = λ4 = ±i wih corrsponding vcors λ= i v= i,, i, λ = i v =, i,, i Hnc, h ignvalus ar α ± i β, whr α =, β = and α =, β = Th corrsponding ignvcors ar α + iβ = i i = = + = + v p iq i i α + iβ = i = = + = i + v p iq i i

62 496 CHATER 6 inar Sysms of Diffrnial Equaions Th four linarly indpndn soluions ar hn bg α α = cos βp sin βq= cos sin bg α α = sin βp+ cos βq= sin + cos bg Th gnral soluion is = p q= α α cos β sin β cos sin = + 4 bg sin cos ()= c + c + c + c 4 4 lugging his ino h iniial condiions ()=, ()=, ()= d, and 4 ()= w g c =, c =, c =, and c 4 = Finally, bcaus =, and y =, w hav h dsird rsul! Suggsd Journal Enry 4 Sudn rojc ()= cos cos y ()= cos + cos

63 SECTI 64 Uncoupling a inar DE Sysm Uncoupling a inar DE Sysm! Uncoupling Homognous inar Sysms = Th cofficin mari has ignvalu and ignvcors λ= v=, λ = v =, arics of ignvcors ar Thrfor, A= 5 = and = 5 Hnc, ransforming from o h nw variabls w = = yilds h uncoupld sysm w = w w = w = Solving his uncoupld sysm yilds w ()= c and w ()= c Th soluion of h original sysm is c ()= w()= c c = Th cofficin mari has ignvalu and ignvcors λ = v =, λ = v =, + c arics of ignvcors ar = and = 4

64 498 CHATER 6 inar Sysms of Diffrnial Equaions = Thrfor, A= 4 = Hnc, ransforming from o h nw variabls w = yilds h uncoupld sysm w = w and w =w Solving his uncoupld sysm yilds w ()= c and w ()= c Th soluion of h original sysm is c ()= w = c c = Th cofficin mari has ignvalu and ignvcors arics of ignvcors ar Thrfor, λ = v =, λ = v =, + c = and = A= = Hnc, ransforming from o h nw variabls w = yilds h dcoupld sysm w = w and w =w Solving his dcoupld sysm yilds w ()= c and w ()= c Th soluion of h original sysm is 4 = 4 c ()= w()= c c = Th cofficin mari has ignvalu and ignvcors λ = v =, λ = 5 v =, + c arics of ignvcors ar = and = 4

65 5 = Thrfor, A= 4 4 SECTI 64 Uncoupling a inar DE Sysm 499 = 5 Hnc, ransforming from o h nw variabls w = yilds h uncoupld sysm w = w and w = 5 w Solving his uncoupld sysm yilds w c 5 ()= and w ()= c Hnc, h soluion of h original sysm is 5 c ()= w()= c c = Th cofficin mari has ignvalu and ignvcors arics of ignvcors ar Thrfor, = λ = v =, + c 5 5 λ =4 v =, A= 5 and = 5 5 = 4 Hnc, ransforming from o h nw variabls w = yilds h dcoupld sysm w = w and w =4 w Solving his dcoupld sysm yilds w c 4 ()= and w ()= c Hnc, h soluion of h original sysm is 6 = c ()= w()= c c c = + 4 Th cofficin mari has ignvalu and ignvcors λ = v =, λ = v =, 4 arics of ignvcors ar = and =

66 5 CHATER 6 inar Sysms of Diffrnial Equaions 7 = Thrfor, A= = Hnc, ransforming from o h nw variabls w = yilds h uncoupld sysm w =w and w = w Solving his uncoupld sysm yilds w ()= c and w ()= c Hnc, h soluion of h original sysm is c ()= w()= c c c = + Th cofficin mari has ignvalu and ignvcors λ= v=,, λ = v =,, λ = v =,, arics of ignvcors ar Thrfor, = A= = and = = = Hnc, ransforming from o h nw variabls w = yilds h dcoupld sysm w = w, w =, and w = Solving his uncoupld sysm yilds w ()= c, w ()= c, and w ()= c Th soluion of h original sysm is In scalar form ()= w()= c c c c c c = + + = c c c y = c + c z = c + c

67 8 = Th cofficin mari has ignvalu and ignvcors λ= v=,, λ = v =,, λ = v =,, arics of ignvcors ar Thrfor, = A = = SECTI 64 Uncoupling a inar DE Sysm 5 and = = Hnc, ransforming from o h nw variabls w = yilds h dcoupld sysm w =, w = w, and w = w Solving his dcoupld sysm yilds w ()= c, w ()= c, and ()= c Hnc, h soluion of h original sysm is w ()= w()= =c, = c, and = c + c! Uncoupling onhomognous inar Sysms 9 = + c c c c c = + + Th ignvalus ar and, and hir wo indpndn ignvcors ar, and, W form h marics W chang h variabl w = and = =, o yild h dcoupld sysm w = + w

68 5 CHATER 6 inar Sysms of Diffrnial Equaions or w = w+ and w =w Solving hs, yilds w ()= c and w ()= c Thus + = sin ()= w()= c c c c = + + Th ignvalus ar and 4, and hir wo indpndn ignvcors ar, and, W form h marics W chang h variabl w or = and = =, o yild h dcoupld sysm + = w w 4 w = w+ sin w =4w sin Solving hs, yilds Thus = + w ()= c cos + sin 5 5 w ()= c 4 c ()= w()= cos + sin 5 5 c4 = c c cos sin 4 5 cos + sin Th ignvalus ar and, and hir wo indpndn ignvcors ar, and, W form h marics W chang h variabl w = and = =, o yild h dcoupld sysm w w = +

69 SECTI 64 Uncoupling a inar DE Sysm 5 or Solving hs, yilds w = ( ) w = w + ( + ) w ()= + c 4 w ()= c 4 8 Thus = ()= w()= + c 4 c c = 4 8 c Th ignvalus ar 6 and, and hir wo indpndn ignvcors ar 4, and, W form h marics W chang h variabl w or = 4 and = 5 4 =, o yild h dcoupld sysm w 6 w = w = 6w+ w = w 5 Solving hs, yilds w 6 ()= c 6 6 w ()= c + +

70 54 CHATER 6 inar Sysms of Diffrnial Equaions Thus = ()= w()= c c c c + + = W us apl o firs find h ignvalus and ignvcors, yilding λ= 5 v=,, λ = v =,, λ = v =,, 6 Hnc, W hn find so and Th dcoupld sysm is = = A = = w = Dw+ f

71 or 4 = w w w = w 5 SECTI 64 Uncoupling a inar DE Sysm 55 w w + w c Solving hs hr quaions individually yilds w 5 4 c F I ()= + HG 5 5 K J w 7 c F ()= I HG K J F I HG K J 4 ()= Transforming back yilds h soluion ()= w(), which urns ou o b + = w w w w w w + w = + w+ w + w, F 878 I c HG K J F 77 9 I c HG K J F 7 a f I HG K J 5 c = c = = c + c + c + + W us apl o firs find h ignvalus and ignvcors, yilding λ= v=,,, λ = v =,,, λ = v =,,, λ = v =,,, 4 4

72 56 CHATER 6 inar Sysms of Diffrnial Equaions Hnc, Thn find and so and Hnc, h dcoupld sysm is = = T = = A = or w w w = w 4 w = Dw+ f Solving hs hr quaions individually yilds w w w w 4 w()= c w()= c w()= c + w ()= c

73 4 SECTI 64 Uncoupling a inar DE Sysm 57 Transforming back, yilds h soluion ()= w(), which urns ou o b w w w4 w w w w w w4 = = +, w w + w! Working Backwards 4 4 = c c 4 4 = c + c = c c + = c + c + 5 Givn ignvalus ar and and rspciv ignvcors ar (, ) and (, ), w form h marics and hn form h diagonal mari = and = D =, whos diagonal lmns ar h ignvalus W hn us h rlaion D= A, prmuliply by, and posmuliply by, yilding A = D = 4! Jordan Form 6 (a) Th sysm = 4 = has a solubl ignvalu of and only on indpndn ignvcor v =, W find h gnralizd ignvcor w ha saisfis h quaions ( A I) w = v, or w = w or w+ w =, which yilds w = and w = Hnc, w =, W now form h mari

74 58 CHATER 6 inar Sysms of Diffrnial Equaions = = vw and compu = A= 4 = (b) Transforming from o h nw variabls u = yilds h nw sysm u = u + u and u = u Solving his sysm yilds ()= c, u c c u Th soluion of h original sysm is! Suggsd Journal Enry 7 Sudn rojc u bg= + c c c c ()= ()= c a + f = + + c + c + c

75 SECTI 65 Sabiliy and inar Classificaion Sabiliy and inar Classificaion! Classificaion Vrificaion = (saddl poin) 4 Th mari has ignvalus and Bcaus i has a las on posiiv ignvalu, i is unsabl As h ignvalus ar ral and hav opposi signs, h origin is a saddl poin = (cnr) Th mari has ignvalus ±i Bcaus h ral par is zro, h origin is sabl, bu no asympoically sabl a quilibrium poin Th origin is a cnr = (sar nod) Th mari has ignvalus and Bcaus boh ignvalus ar ngaiv, h origin is an asympoically sabl quilibrium poin Also h mari has wo linarly indpndn ignvcors (in fac vry vcor in h plan is an ignvcor), and hnc, h origin is a sar nod 4 = (dgnra nod) Th mari has ignvalus and Bcaus boh ignvalus ar ngaiv, h origin is an asympoically sabl quilibrium poin Also hr iss only on linarly indpndn ignvcor corrsponding o h ignvalu; hnc, h origin is a dgnra nod 5 = (nod) 4 Th mari has ignvalus and 5, which mans h origin is an unsabl quilibrium poin Th fac ha h roos ar ral and unqual mans h origin is a nondgnra nod 6 = (spiral sink) Th mari has ignvalus ± i Bcaus h ral par of h ignvalus is ngaiv, h origin is an asympoically sabl quilibrium poin Th fac ha h ignvalus ar compl wih ngaiv ral pars also mans h origin is a spiral sink

76 5 CHATER 6 inar Sysms of Diffrnial Equaions! Undampd Spring 7 + ω = Dno = and =! ; h quaion bcoms!! = ω Th cofficin mari has ignvalus ±iω, so h origin (, ) is a cnr poin and hus classifid as nurally sabl! Dampd Spring 8 m!! + b! + k=! = y Th scond-ordr quaion can b wrin as h linar sysm! y! = k b y m m Th drminan of h cofficin mari is k, which is assumd posiiv Hnc, h mari is m nonsingular and = y = is an isolad quilibrium poin Th ignvalus of his sysm ar h roos of λ k b λ = b λ + λ + = m m m m b kg, which ar 4 λ = b+ b mk m 4 and λ = b b mk m From hs roos, w s ha whn b >, rgardlss of h valus of m >, and k >, h roos will ihr b ral and ngaiv or compl wih ngaiv ral pars In ihr cas, h origin is asympoically sabl Bcaus only whn h hr paramrs m, k, and b ar posiiv is considrd, h origin will always b asympoically sabl, which is h naur of ral sysms wih fricion! n Zro Eignvalu 9 (a) If λ = and λ, hn A is a singular mari bcaus A = A λ I = Hnc, h rank of A is lss han Bu h rank of A is no bcaus if i wr i would b h mari of all zros, which would hav boh ignvalus Th rank of A is, which mans h krnl of A consiss of a on-dimnsional subspac of R, a lin hrough h

77 SECTI 65 Sabiliy and inar Classificaion 5 origin Bu h krnl of A is simply h ac of soluions of A =, which ar h quilibrium poins of!= A W us h soluion of h form c a c ()= c y = b + λ d o find h quilibrium poins W compu h drivaivs and s hm o zro Sing! = y! =, yilds h quaion c! c λ λ, d ()= = which implis c = Th poins ha saisfy! = y! = ar h poins y ()= = c a b, which consiss of all mulipls of a givn vcor (i, a lin hrough h origin) (b) If a soluion sars off h lin of quilibrium poins, hn c If λ >, h scond rm λ c c d bcoms largr and largr Hnc, h soluion movs farhr and farhr away from h lin of quilibrium poins n h ohr hand, if λ <, h scond rm bcoms smallr and smallr, h soluion movs owards h lin! Zro Eignvalu Eampl = y y (a) Th characrisic quaion of his sysm is λ λ = Th corrsponding ignvcors ar λ = and λ = υ =, and υ =, b g ; i yilds ignvalus (b) Sing = y =, w s ha all poins on h lin = y ar quilibrium poins, and hus (, ) is no an isolad quilibrium poin (c) W s = and!y = + y, yilding ()= c and y = c + y Hnc, y ()= c + c

78 5 CHATER 6 inar Sysms of Diffrnial Equaions whr c and c ar arbirary consans In vcor form, his is ()= c c y = + (d) Bcaus ()= c, h soluions mov along vrical lins (or don mov a all) To amin his furhr, assum w sar a an iniial poin ( (), y() )= a, yf Finding consans, c and c, yilds h soluion ()= y ()= + y a a which says ha saring a any poin, yf, h soluion movs vrically approaching h 45-dgr lin and h poin a, f! Boh Eignvalus Zro Whn boh ignvalus of A ar zro, such as in h marics or, h soluion is! Srang Sysm I a c c c c a c ()= c b + d = b + d f (a) Th characrisic quaion of his sysm is λ + λ =, yilding λ = and λ = Th corrsponding ignvcors can b sn o b υ =,, υ =, (b) Sing = y =, y w s ha all poins on h lin y = ar quilibrium poins, and hus (, ) is no an isolad quilibrium poin Also from h diffrnial quaions, = y, Sampl rajcoris of a singular sysm

79 SECTI 65 Sabiliy and inar Classificaion 5 w s ha soluions mov along rajcoris on 45-dgr lins Abov h lin y =, = y = y < and h movmn is downward and o h lf Blow h lin y =, movmn is upward and o h righ This oucom is shown in h phas plan (S h figur) o ha h soluions blow h quilibrium lin approach h lin bcaus h rajcoris mov along h 45-dgr lins, bu h quilibrium lin gos up by lss han 45 dgrs, and h soluions abov h quilibrium lin mov down owards h lin (c) I is normally assumd ha h mari A is nonsingular so normally (, ) is an isolad quilibrium poin! Srang Sysm II = ohing movs; all rajcoris ar poins! Srang Sysm III 4 = k Th characrisic quaion of his sysm is k λ = ( λ k) ( λ + )= λ and, hnc, h roos ar λ = k, λ = (a) k (, ) implis ha h origin (, ) is an asympoically sabl nondgnra nod (b) k = implis ha h origin (, ) is an asympoically sabl sar nod (c) (d) k (, ) implis ha h origin (, ) is an asympoically sabl nondgnra nod k = implis ha h mari is singular; hnc, h origin is no an isolad quilibrium poin (all rajcoris of his sysm mov vrically owards h ais) () k (, ) implis h origin is an unsabl saddl poin

80 54 CHATER 6 inar Sysms of Diffrnial Equaions! Bifurcaion oin 5 Th characrisic quaion of is λ k λ + =, which has roos = k λ= λ = k + k 4j Whn k <, h roos ar compl and h soluions oscilla Whn k h soluions mana from an unsabl nod Hnc, h bifurcaion valus ar k =±! Inrsing Rlaionships 6 = a b c d Th characrisic quaion is a fa f a f λ ( a + d) λ + ( ad bc)= = λ r λ r = λ r + r λ + r r = If h characrisic roos ar r and r, w facor h quadraic on h lf W s by quaing h cofficins ha (a) (b) TrA (h cofficin of λ) is always h ngaiv of h sum of h roos (i, TrA = r + r f) a A (h consan rm) is always h produc of h roos (i, A = rr )! Inrpring h Trac-Drminan Graph In hs problms w us h basic fac ha h ignvalus can b wrin in rms of h rac and drminan of A using h basic formula 7 A >, ( TrA) 4A > TrA± ( TrA) 4 A λ, λ = Using h basic formula, h ignvalus ar ral, unqual, and of h sam sign; hnc, h quilibrium poin (, ) is a nod Whhr i is an aracing or rplling nod dpnds on h rac

81 SECTI 65 Sabiliy and inar Classificaion 55 8 A < Using h basic formula, h drminan of A is ngaiv hn ( TrA) 4A > and TrA >, so h ignvalus mus b posiiv and hav opposi signs Hnc, h origin is a saddl poin and an unsabl quilibrium 9 TrA, ( TrA) 4A < Using h basic formula, h ignvalus ar compl wih a nonzro ral par Hnc, h origin is a spiral quilibrium poin Whhr i is an aracing or rplling spiral dpnds on whhr h rac is posiiv or ngaiv If i is ngaiv h origin is aracing, so ha i is a spiral sink If TrA is posiiv, h origin is rplling so ha i is a spiral sourc TrA =, A > Using h basic formula, h ignvalus ar purly compl Hnc, h origin is a cnr poin and nurally sabl ( TrA) 4A =, TrA Using h basic formula, (ral) nonzro ignvalus ar rpad Hnc, h origin is a dgnra or sar nod TrA > or A < Using h basic formula, if h rac is posiiv, hn ihr h roos ar compl wih posiiv par or h roos ar ral wih a las on posiiv roo In ihr cas h origin is an unsabl quilibrium poin In h cas whn d A <, hn, from h basic formula h roos ar ral and a las on roo is posiiv, again showing ha h origin is unsabl A > and TrA = Using h basic formula, h ignvalus ar purly imaginary Hnc, h origin is a cnr poin and nurally sabl 4 TrA < and A > Using h basic formula, h ignvalus ar ral and boh ngaiv Hnc, h origin is asympoically sabl