245ch7testsoln - + l'c-) nt { o' 6s t'rt - b...

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Fall 2007 ileth 245 Chapter 7 Test (7 December 2fi)7) Name: S&rta ltkch- Classif the ^=[1 l) Prob. l. Consider the system r'= Ax where the matrix A is given by fixed point x = 0 of the system as to its stability and type. fr"A +At l,,3*vir]A a( A, dc(A-rD= l''." tl= e-t'- 4=o '/ I 4 t-rl \ J fL rr +t-1= [1 1r -3 = (r-s)ff+l) =a l =_,b+3 TC, Sx.A F;.t L oa^ F*uE WAL. Prob. 2. Consider a second-order system r'= Ar where the matrix A =(qt 4z) n* a pair of \4r' uzz ) complex conjugate eigenvalues r = d i.iat . Theeigenvectors of the system are found to be of the form 6ttr =(a+ib,c)'and 5t:r =(a-ib,c)t'. Findthereal-valuedfunctions u(r) and v(t) suchthat the general solution of the system can be expressed in the real-valued form r(l) = C,u(l) + Crv(t) . C Ccart &sart -b E^n C dost rL 9al^ro b tt,lrt)= V'ct) = { n)_ ,al'rutt _ cr/ =e( \ = e*f [" \ (:') (*"t*isd-t) -0"*) / a+tb\ ra/+it^r)f s.t ( n /e' = e \r-J (L 6sc^rt +lasaiot +tbAsuL +, Jf ( d-e^;a'f +b os-h) \ C %irq.,g /
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Unformatted text preview: + l'c-) nt { o' 6s t'rt - b sur'<,rb\ e ( c cos t-^rt ) {t I as*uL rb e6,,,f \ I C $,i. c.rL "l Prob. 3. Find the general solution x(r) = C,x(t)1t; + CrxQ'1t7 of the second-order system x' : Ar where the matrix A is given u, a = .tl \l | ) FiA.& *Wvdl',",t A, a,*(a- rr) = t;" ,:i\= (3-r)0-r) +1 = ?2'4ft 3+! = Q-- 2f =o ift+rrm b&e., \n!.,rQor I-= 2rL. T,;A#' {r;t &*d". ^ Q'Lf%t%), (';' ,:lX$]) =' + fi'- f i'=o f* tfu l*^;,arp^Ue s{,,fn 3. bq* f* ,a.ft*, x2f?eor* \tl TR* #",= gr(dr* *i*)_r['el J Srlb{i,|.rA d"b +h obe. rf"Ie*** (F"Ia[)f t= (l l)r.i to r fi l,)r{ %!t,-" rupfuFar*t,-LiJ Ato'= (? 1)r', + (', .l[il* + r? (1) il eorz,2 =(1 ln)q + (^,il(e) = fl ) .^tt {a^l f.r r,l,-}l cr! ,tS, y, ^1.= I t9** o)w,ct ?=k rhr''^ 2n= k-l ,m tC* +t' S,'rt^a\adugr,r'wutr r1ul, ?= (,:,) = f, fl - (i)-lT;. W Safr-kln ra r Krt) = c, (1).'o- c.i (i )tf t* r, fl)u"L ff)dtl = 4 (f)dtn c,Ul)te'6- fi) t'l ,...
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This note was uploaded on 09/04/2008 for the course MATH 245 taught by Professor Alexander during the Spring '07 term at USC.

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245ch7testsoln - + l'c-) nt { o' 6s t'rt - b...

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