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07m2-solution-yellow - 5A Midterm 344 2 Spring 2007...

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Unformatted text preview: 5A Midterm 344: 2 Spring 2007 May 25, 2087 1115131116101”: G1 W'ei /3 / u e VMIFW Name Discussion time (circle one): T821111; T413111; T513111; T613111; T713111; RBam; R4p111; R5pm; Bfipm; Your Scores: 1.‘ 2‘ 3‘. 4. 1‘ .. Your Total Continue on the back of the page if you need more space. Good luck! 1. (12 pts} Find all the eigenvalues and an eigenvector of the smaliest eigenvaiue [w-A z ; A~*Al=: 2. ~/\ 3 2.. 3--)\ JEN/1-1 1) : (HUME—17) “2(w2z\-!/’+ 2/6124) .2011) (1+1) (1—111 (HAM) + HHS) Z 3 .3 "a «Z 3 ‘3 "M9 0' Z 1 2. (33 pts) Wxite the differential equation ‘7 rc"+2:c’+2m-a:'m0 into a system of first order differential equations" Then determine the equilibrium solutions of this system and sketch the horizontal and vextical nullclines and the directional field of the system, determine if the equilibrium soiutien is stable or unstable. Cw 39x", idem XXI—:9 3. (12 pts) For the linear transformation T : 1R3 ~+ R3, T(v) m Av, whey-e A is given by 1 2 1 A: 3 4 1 O 1 1 find the iiemel, image, nullity and rank ‘l a ( / a / ("MIL Illa-'4 "Mi AIM/NJ) ( 3 9L [)4 (0 *2 "1 0 l I c1 ' I 2 { / 0 -—l I l _..5 (/I"Vs :— O (a f l “"9 0 f / 0 0 0 U 0 C) L/L+V$ O MIA—Cg, tzbk M35 L/L:—w$ ___ ' r m We _, f /- 5 My M W _ I SO [MM/A7 35‘ fl; DfWWI 5/91} 7/;[41 WWI/(I : ga/ 4. (13 pts) F0: the system of differential} equatiozis 10—1 X’=020X 101 (a) Find the geneial solutiozin (‘0) Find the solution satisfiring the initial conditions Km) = < (5*) [inf %w»(’ 1%: FnijM/a/i'xd/J : I—‘C‘JH \________/ ...
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