tast Name First Name "."*"*(lN CAPITAI- LETTtrRS) MAT 485, HOMEW.OIIK 11 (DUE, TH. DEC. 8) PRACTICE PROBLEMS FOII FINAL EXAM Final Exanr witl be held on \\4rednesday, Deceluber 14, ?016, hom 10;tE tq 12:15 in MAXW'ELL AUDITORIU$'I PROBLEI4 1 (15 pts) Daiertrlae the general solution ol the equation (cosr)r' * (sir*)'y={cosf}?, Uo Use l.ariation of pan:amotors to obtain a pa;rticula.r' solutiorr, deiermile the geaeral solution and O' solve the ioit'ial I'alue probleni S(1) = 0 g'(1) : , a) (5pte). Dctcrmineaba.risofsolutionst{)thehon:ogencousequationg" o,:l r Ur-[ - 2A'*:ry: 0' t Write dorvn the gorreral solutiol to the horaogenecus equatiolr ,: "t b) (r pts). l, I Corupute Wronskian l, [ l'[ II W:det I Lt c) (b pts). Find the variable coefficients Up:utvr + ul : u1 : ', I 'l ?r1 (1) aacl us = q2 (t) I I LJ I la t: for the particuiar solution u2A2: [LJ --'-- l') Hence ur l{ence ,,:l vt d) (b pts), *lrn thu initia-l : Ir rr=[ I trVrite the general soiution !/c arld detorrninc valrro probieur'y,ur, ; u(1) :0 , u'(t) : 0. Jt Ir tlie coflsta,nt ctrofficierrts cl i. e2 to a.= 9rvp :t L Ir P}:|OBI,EM 4 (15 pts). ItiDd a basis of the spare of selutiqns to tbe follo'wing s.ysiern '&t:2x-U-Lz gt=2x-g-ze z'u2lr:11 -22 a) (5 pts).Fiud the elgenvalueo "y=)r1:i2:l r, = b) (8 pts). Find the lineally f [ Jr iudope:rdeni eigenvectors: : -*' l't' -;1,] l;' ] tl] ? -:liil",=liil' lz-l,r L; -2 I [", -1 -1-)r -,1^,iL;: c) (2pts), Write the getoral solutioa [ lol , t-:lI Lt L3l a vector forro l*",e, ,'i I Lt L( )l l;[:] I = c,e( ,,|- L;t,)l il = )lJ I + c"e( ,rL il, al )_t PROBLEM 5 (20 pts)Colsider the follou,ing norrlinear autonsnrou$ systeil *'=*(p2 -l) ) {:f@,il g':'r*u2-4 { -stx,il ) a) (2 pts). Shovrthnt (t:,y)= (0,2) , {t:,y): (0,-2), (*,c): (3,1) and (c,u): (3,-1) are the only cquilibrium b) (2 pts). Compute the Jacobian matrix a.t the general point rt-.^.\\: lf" lrl -t( : r\\:t.,y) lg,, is) -Lf J c) (a ptu). J(0,2) pointri. (r,g) )( )( )l )l ? Elaluate tlie Jacobia.n ilr&irix at the equilibriurr-r poirrt (c,g) = (0,2). : !( M( ) tI L )l'l . and find its eigenvalues: .\ :( ) ,i2: ( )?. Clussify thc Iocal tr&jectories at this equiiibriurn point as: (underline the correci answer) r Attractivc notc, stable r Repelling note, unstable r Sadclle, unstable r Ceflter, siable for lirearized $ystelnr unr:ertaiu for nonlincnr systc[r d) ( pts). Evalu*te r(0,-2) thc.Jacobiau rnatrix tt :Li ltir tt tlreequilibriuur point (*,y): (0,-2), ll,ardfinditstseigenvalues: )) ir:( ),Az:( )? Ciassify the local trajcctories at this cquilibliurn point as: (urrcleriine the correct :rnswer) . Attrlc'tive note. stable . Ropelling trote, urstable r Saddle, lrustable r Ceuter. stable for linea:rized sy'stem. uncertain for nonlinear system e) (a pts). J(3,1) Evaluate thc..Jncobiarr rnatrix itt tlie equilibriurn poini : [( Cln^ssify ) ( L( )( ) I ) I (,:r, V) : (3, 1) , l,andfinditseigenvalues: lr--( ),)z: ( )? the local trirjectories at this eqrrilil:riuur poiut ers: stab.lei....... (underline thc r:olrect answer) r Attractivc notc, stable r Ilcpellitrg rrotc, unstnbicr o Saddle, unstablc r Center, stable for ]inearized f) (a pts), r(s,-l)' systeur, urrccrtain for nonlirrear systent ix at the equilil:riuin point (r,y) Evaluate the Jacobian :ItL( ]( ) ( )l )l , : and find its eigenvahies: (3, -1) h: ( , ) , )z: ( ) Classify ttrc local trajcctories nt this equilibrium point as: stirble....... (underliue thc correct ans*'er) r Attractive notc, stable r Bepelling note, unstable e Saddle, rnstable r Couter, stable for linea,rized systcrrl. uncertail for nonlinear systen ? I PR.CELEIVI6 (15 pts). Coneider a nochoruogcueou* frrs oreler erluatioa: qt * * - e.*talr(**1) = J(t) $