final_sol - S Math 250 Fall 2007 Final Exam"l tou.9 NAME ID...

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S "l,-+tou.9 V. C= Math 250 Fall 2007 Final Exam NAME: ID No: SECTION: This exam contains 16 qrrestions on 15 pages(including this title page). This exam is worth a total of 150 points. The exam is broken into two parts. There are eight rnultiple chriice questions, cach worth 5 poirrts, and 8 partial credit problems. To receive full credit for a partial credit problem all work must be sliown. Wherr irr doubt, fill in the details. No notes, books or calculators may be used during the exam. Piea,se. Box Your Final Answer (when possible). 6:
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Multiple Choice Section 7. (5 'poi'nts) The autonomous equation y' : -U2 -A +2 has the following equilibrium solutions: ( ("DE - -2 is an unstatiie solution anclg : 7 is an asymptotlcally stable sollltton. (b) y - -2 is an asymptotically stable solution and y : 1 is an unstable: solution. (") E : '-2 is an unstablesoiutiorr atrd g : 1 is a semistatilc solution. (d) g : -2 is a scrnistable solution and ?J : ). is a,n asymptotically stable solution. 2. (5 poi,nts) Orr which of the following intervals is the unique soiutir.rri of the initial value problern zt-.--"1 (f - 1)',/'- (ltf)y : tani. Qtf l : i \\_ : t\ \^=*a .-.^s\,ok{- l= - t \ \.1 { t^t t!- !) q4t 5\'.. \ certain to exist'/ (a) (0, 1) (b) (1, 1) (") (t, f ) ,/?- (t(a))r*, +) \>c; r"+L ) t h^\< x, b.\q (I_D +Eer-\v, --> \z> L -T t\ J. \-.
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3. (5 poi.nts) Which of the following differentiai equations has g t : 3e2t u,nd Az : Se-at as rwo sorutions,/ tl""C"cUri:Llc po.L.^.-"lJ (r-:.)(r*q\_o (u) 3g" s-Sg' + lt : 0 1 (b))s" -2a' - 8s : 0 (.) a" - 69' + d,i/ : 0 (d)y"-2A'r8y:0 :fttr.-6 r=\t.L :- () o o --\ 4. (5 poirt,ts) Firrci a suitable form of a particular '!/" *2'g'*2Y: l(sini + 1). The A, B,C, D,E, F below are constants. \ \ (a) Atsinf * Bf cost+Ct+ D \ \ (b) (At + B)(C sinl * Dcos t + E) \ ^l (91r, * B)sin t + (('t -r D)cos t + Et + F' (d) (At + B)tsint* (Ct+D)tcost+ Et+ F i \ .
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