Midterm2

Midterm2 - Alex Boismrt Math 3313 Winter ’03 8:00 AM...

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Unformatted text preview: Alex Boismrt Math 3313., Winter ’03 February 29, 2008, 8:00 AM Midterm 2 Name: __ _ Student ID: \/ I? Signature: __ Check your section: _ 1a (Tu) _ 1b (Th) TA: Neel Timviluarnala _ 1c _ 1d (Th) TA: Eric Radke This is a closed-book exam. Do not use notes, books, papers, or electronic devices of any kind. Do all work on the sheets provided. Do not use your own paper or blue books. If you need more space for your solution, use the back of each page; you may request extra paper. Be sure to state clearly if you are continuing on a different page and label the problems well. Do all 5 problems. For full credit, you must Show all your work. Do not. worry about oversimpljfying your answers. Please clearly indicate your final answer, for example by putting a box around it. Problem Out of Points __ l 9 2 9 3 i 4 8 5 12 (1) (9 points) For each of parts (a)-(e), determine whether or not the statement is true or false. If true, Show why. If false, explain why not. (a) (3 points) There exists a differential equation of the form it)" + MW + G‘th = 0 such that y1(t) = «3t and yflt) 2 t2 are two solutions on the interval (—10, 10). (kids/0w OM; (M exei; Mr LOFDQLZ‘QA WEIde "Ewkcaflflia ecouoi in O on Heath ef‘ MAM ahead Ottere‘ ta wreak-35w e‘iél’c +14%- Ee‘fa-Q eczw‘: 0 at boil M We: (b) (3 points) If A and v are defined as we we) then v is an eigenvector of A with associated eigenvalue 2. “Ma—To awarigmtiomwn (Av: LE)(?>:(J:3:&(;):;M (c) (3 points) The differential equation giving the following direction field is autonomous. .* § Myrrh . Ehxsssaviiniflanflfifii . . . ..i.........-..- ..- l... V v u u - m \.L§l§n§i?5$.\5355w_159. » . _ . J _ _ . . anzktmvtvinl‘viasfivtkgl . _ . _ “a... .;-,.u\a\1\hviir%niéifitnvtk\ . . . . . $.¥;~2¥ ma;%finéfinti¥annx: L1Ffmfflwmxukkwx?? ".azszzzs1.4»‘ufie4€z‘ .. v o .~¢»Q» A . c . x ; a o A t 9 v w 9 \ w \ v ; 9 ow "Jx\»92 “HA; QVdeflngo/x W (SLAVE 1\/\ VQ. NW (2) (9 points) In this problem we consider the differential equation :12’ = x3 — 43:. (a) (6 points) Draw a phase line for this autonomous differential equation. Classify each equilibrium point as either unstable or asymptotically stable. £00: xii—letfl'fifl: XCX~2Y><+2> (b) (3 points) Sketch some solution curves in the tm—plane, indicating the equiiibrium solutions. ‘// ///' \\\ @\ a ,j (3) (12 points) A 2 kg mass, when attached to a certain spring, stretches the spring 20 cm. Assume this system is unforced. (a) (3 points) Calculate the spring. constant. for this Spring. F =mgzl<x so seem-02 :ekzss g (b) (3 points) Find the damping constant ,u for which there is critical damping. cggnal-j/{lalf4 n r” ,fi r x a + %% “40‘:ij C/rl'i‘C—GM/j MW \\ LAMA C's—1403. s”? 90% @1390 C: fléifiz¥rCflo E. (c) (6 poirits) Find the solution to this system given the initial conditions = 1 In? y’(U) = 2 m/s. mm mm :0 g; a :1 l ymwcrno ==~>Q+¥330 =7 A“? (4) [8 points) Find a particular solution to the differential equation 1;” + 93; = 'sec(3t) (Hint: Recall that ftan(t)dt =1n|sec(t)| +0) LL12. we umbu'hm“ {39‘ PWW- L6 {1+CTSSO T» gifawgfl, Kjfi 004316 \81) ‘2: VI gtvfij‘flf Z '2 Vt \gp’z 3v. cesc 34:)“3V1‘5M (3%§+V,%M (31c) wit 6&4) Se} v, gm (am-x4 3930 17mm 5 “2 " C1le [30 “QVLCDS(3B 4 BVKLoiBQ ‘3V2,’ 311/189 max 3.9% 613?: gwmgbfl -— ’5v7jem (319. T WE‘ka "VD flue : W 6% 3% £05( 39 *gvzf 9M 3%} 2 ged 31:3 \JJ EBA/«(356$ + \l-zf CDSC3E\ DO (5] (12 points) (a) (4 points) Prove that the imaginary part of the solution of 2” +2" + z = to“ is a solution of y” + y' + y = tsinbt). (Hint: let z(t) = $(t) + iy(t)) (b) (8 points) Use this idea to find a particular solution of y” + y' + y = tsin‘(t). (Hint: solve z” + z' + z- : is“ using the method of undetermined coefficients.) let-9405+ =3 9W9 : {statute ere: (tHUeH' g0 w \mr «Ca a 90mm ya omega/x (we) a“: 2:. ( AHB) 63+ P3: 6% 111‘“ {EHM 2”: Eva) HEWNHEHM = ecjkGM +1A1¥m 2-5 2”+ 13% :6“th Hui—{35rka gape AU E3 r‘flm‘s 1E A7? abwfl "ZEN: Wee, Mink, 96:4, DiliJréHPrzo. kaw Pvt-m”) Elma: 7 32> EMT-Gilt +U-Z‘cfié‘h‘ £34k mmgiwu 905+ , : (Lt-(4023 + 0%;me :: (60%]: +1059 +5 COW) 0053C +8”) I I . by SULLW I 2: HE+L)QcE>1CW\nJC 0‘ 0‘ WARM $0 Am \filqv‘yfjclcgflng ...
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This note was uploaded on 04/02/2008 for the course MATH 33B taught by Professor Staff during the Winter '07 term at UCLA.

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Midterm2 - Alex Boismrt Math 3313 Winter ’03 8:00 AM...

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