mid2sols - 2nd Midterm exam. Math 2403 K4— K5—. Fall...

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Unformatted text preview: 2nd Midterm exam. Math 2403 K4— K5—. Fall 2007, Oct—25. Professor: Luz Waney Vela-Arevalo. Show all work! 1. Consider the differential equation y(3) — 3y’ + 2y 2 0. I (3) Verify that 311(3) = 6—23: is a solution. (b) Find three linearly independent solutions. (c) Use the Wronskian to show that the solutions are linearly independent. 2? ’1): 6 _)_x 2 ,2,‘ CD = _ __ V f dlx u: _ "7- 7‘ 3‘“.— l 2 2 ~ 6 ‘1’ e "i e Z 6") \ji’: “261x / 3K’54e / 3'- ’" $36 a? :9 53+ 3 15points ~ . ""1 m 5 19) he"; is areal, Wm {2(1): M+2Xf~2r +1) .. r flariz .o - . JR 3 W vzsi ‘ (gait, ’flvr‘afl. \nflwmgflénl‘ ‘50\ I“ M jt '—‘ ézx / 31»: €41 / -3'5 2 ’7‘ ex 0; , - 1 ’2X 3." 7C6? 2;: x —-)< . ) WfWk‘M ' E29?“ 93‘ mote? : é” (a (1))“ C (6 ("’5"ng N (‘flvqhgn : 4 6.4.2: e,>c (x k2)67< +r><e>< e”: ('61)) 7— Héx Jr‘E-“G‘X 2. Find the form of the paiticular solution yp(x) of the differential equation but do not atteInpt to solve for the constants: (D2 — 4D+ 5)3(D — 3)3y = 37(8352 + 33: + Des": -i- 336% + (1 + 5:1382m 81111:. ' - , 4iJ££-*2D‘_ ii to): (rumogoaf .5500”? “MT”; 1590mm W~ y: “Mi, 2+E,2+i, a-C,1~t,Q—i, 3,3,?)- 3 7 1X 31000 :- ‘7<3(A4 ®¢+C12+DWS) fly 4" (Elinve' + 3w r" w ' 3 N+EX)€ZX 7i +7((A+gx)e <3an + x0) (>63 - 3. Consider a spring with constant k _= 34, attached to a mass m = 2. The mass is also subject to an ‘ external force of 20 sin t. (a) Write the model describing the position of the mass a:(t) as a function of time, neglecting fn'ction forces. Find the general solution of your model. (b) Consider now a friction coefficient 0 = 16. Write the model and find the general solution for the position of the mass. (c) What can you say about the long term behavior in each case? 25 points a i u p :QOS/Qnt t ) ‘2“ +54% (“Half-ti? 29> “if” “I'd W'X :1- lOSi/n‘t ‘ 7i€€€71 0th Jr Cflzfi‘rmifitti. WP; ACéDi—i— libs/indie We ’Awt’tbmc MEI/Amt rfisfimi _ t r Hfi'W'ttlfava't—Hefiw MM imp: “Hwy out Jr C M > 19.. :3 =79 we.th . z:th We meet Ago) 5:16 ‘6’ ‘8 s) 2%“ + ma +54%: 20 Wt- r (JCT): r1+9mt7r20 ~==> int—41L 0‘1 W"? ‘E’x‘ +i??¢ : 10W; we , Wth): Cie cart—t C16 Six/nit Agm X9 Awb)E/£nt ;A+86+11A3mt+L/5rgf\tm $ . F'L 4- @W93‘FLCflBt*fi§/m% d‘f .. a ficfln‘ (other jealokfm ‘. 9(9”+37<if+1?7‘9 3 ( Salt/Eng ‘. A s 1 b: L ‘ C) FWH‘ (Goa: Suction hm :wo osciUkaHm- $56M Cause: W (isidbihm (lbw: at? (Nd/nSl'P/d'tv anal mtg W particular Idem!“ fee/’5“ L 4. Use the method of variation of parameters to find the particular solution of the differential equation -—3:E y”+y’— 6y : —3e p(r):y3+r,g: ((+5BKT’13 <73 {2“321 29‘ 15points m - “ax e I kfic" five. 4' C?» 2} ‘ LA 2 Mn 3?: UH 637 r. {AL-e _. To Punt-i t1? 2,13 ’3‘ {Quiz}- ’ W I axe-“O % Uz(5€ ):"5€ ?— 5 LU '6 4- Hi '6 ‘_ a ’57 ’6 _5)‘5a urtrageeyut’me“ :05 m (’56 )— e t 5 Pam 3 4:75 "(Wm uxrjiséx:%x Mal Masg'ig—e chciéee, (-375 . 47c -si< w a. :3 4“ 3,” Ling/nae. 4,3?6 e “'5 x8 +3156 5. Consider the romance of Remeo and Juliet described by the following system: R’ salami} (WWW ), R05) is the love/hate of Romeo at time t, J (t) is the lovelhate of Juliet at time t. a measures their cautiousriess (both avoid getting too close,) and b meaSures their respOnse (both enthusiastically advance to each other.) Analyze how the solution depends on a and :5, drawing the phase portrait in each case, and describe when the couple will live happirly everafter. ' _ b 3 .. ,_ “’— - I a fi d , )[43 A) b 20pmnts _, m.- t.“ l i ” SWF’M :maib y) W, a a ‘ ' 9— r x) ‘1 l 2 i l E ' 2. Mb b QM 71¢~a+f91 9*"?‘1I1<b “b Arflf : lob y) LIL: a Sol; g);CiG)C (Wat .. .. game cam ‘- fiF‘m )2 #040 “- \ Mimi/t 6. Find the real general solution of the linear system 2:1 raw —3$1+3}2 ‘ 3 \ $2 h “hi—$2) ‘ k: (:2 #1) I ISPoints ptm2(»3—>s>t~4~w 9— : 22+ 4’) +5 cs a: #2:: . ' u f ,HL' 1 (avg/meals“: (490%; *"K '" 1:” A?’1’{ I {1-)}: I («-2 ‘l—tl ' 40-: is ( l Ul’UJE z: ' j ’2 ‘H CMQ jolal‘fam (emgfifix) '- g kid) 8 ) {lit (-WE*W ) ...
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mid2sols - 2nd Midterm exam. Math 2403 K4— K5—. Fall...

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