Hw8Solscan

Hw8Solscan - MM 94’ 9 we new L. are (-Hy. Hui war to...

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Unformatted text preview: MM 94’ 9 we new L. are (-Hy. Hui war to fis‘fm ‘l’iAQSIL first)“. page”) ’i‘he University of British Columbia Final lflxamination — December 2009 Mathematics 265 Section 101 Closed book examination Time: 2.5 hours Last Name: m_____________ First: W_ Signature ‘ WM Student Number WW . m 2510 Noit- OT (Jim; will u” S A " l I tru 'tit 11“ M r ' pLCId ns . c 3 s. a Hwio. w Be sure that this examination has 13 pages. Write your name on top of each page. EYou are allowed to bring into the exam one 8% X 11 formula sheet filied on both sides. No calculators or any other aids are allowedj ~ in case of an exam disruption such as a fire alarm, leave the exam papers in the room and exit quickly and quietly to a pro—designated location. Rules governing examinations 0 Each candidate must he preparer] to produce, upon request, a UBCcard for identii'ication. - Candidates are not permitted to ask questions oi'ti'ie invigilators, except in cases of supposed errors or z—u‘nbiguities in examination questions. 0 No candidate shall be permitted to enter the examination room after the expiration oi" oi'ie—haii' hour from the scheduled starting time, or to leave during the first half hour of the examination. 0 Candidates suspectc-zd of any of the following, or similar, dishon~ est practices shall he i1‘l'l]]'i(-§(llai:(~‘.ly dismissed from the examination and shall be liable to disciplimn‘y action: (a) having at the place of writing any books, papers or n'ieniorai'ida, calculators= con'iputers, sound or image play» ers/recordcrs/transmitten-s {inchiding telephones), or other mem— ory aid devices. other than those authorized by the examiners; (h) spcakii'ig or communicating with other candidates; and (c) purposely exposing written papers to the View of other can- didates or imaging devices. The plea of accident or iorgetfulness shall not he received. 0 Candidates must not destroy or mutilate any examination mate— rial; must hand in all exan'iii'iation papers; and must not take any examination n’nitt—irial iron] the examination room Without pel‘misw sion of the invigilator. o Jai'ididates must follow any additional examination rules or di» rections coinniunicated by the instructor or invigiiator. _J Page 1 of 13 pages December 2009 Math 265 Nmne: Page 3 of 13 pages [10] 1. 801w: the initial value problem: 11 ((-23: smack,” —1 —- (63" (30811:)y, with y(1rr/2) emu/g (Cpc5{n x>3£+(e1Cb$%jry 3 "i U'oor CoSDL a, : —~....L. exam; ihljv.fiofv 1 Mm w :wS as!“ win-w 14: sa'mc. ch41 :- c0314”— : (A a su'nx 5m 9: £3+C§fixgl ’ W E! ESthx 3.: m .e'x‘ o’x Strut 3kg):- e‘WL m: "L... a"? TC] Siam 9:45. "-1: I . [e'ni fig] (’3. 0 j :- Ea)!» I’)ece1‘1‘1ber 2009 Math 265 Nauru): Page 4 of 13 pages ['10] 2. Find 3,1} solutions of the (ilifl’erential equation ($+1Wfi+%$+1kflH=0 ufilv =— ("‘\(L‘_) +C. u. z: .1- + C. ‘ CDC+D 2 Nadia— December 2009 math 265 Name: Page 5 of 13 pages [:10] 3. Comider the (lifl’(n'(21n;ia.l equation :9“ +210?! ~|~ le = 0 (*) where p and q are centimious [inletimis for all If. (a) Can 39(1) «2:. smug) be a solution on an interval containing '15 m 0 0f the differential carnation >1: '1? Exalain rein" answer. 1 l .3 C? EEC-)2 50% EU) «rim gear. 1‘53 (mafia) Una) :- 2 (“55th 5' t'Z-E. 5'3“l (9)) = 20>st — HtlsfhctL) 6% 'tczo 3(E)=0 ( {OH =-‘ D 3"C{=)= 1 (“Tux l7“ :0 .50 910‘ Hm} f; t‘O’ICMSISf_ (\‘5, Plan-3 lane-k ink ODE) (b) Calculate the \N’l'm'lskjan of" 'l; and t2. .i.‘ t t; r. ZtL’t‘st?’ khaki“ ‘ at (0) Can 1’, and t2 both be solutions of the same differential equation Explain clearly. 0+“,- ar't {30 Wmnsl’M‘AM W510! ,0 '{Lu‘S Cam-49+ be“ :0 114% w am an», mam w MM 12:0 - as» was em wmluyaj. gown» "i’iresra cw, lye-ii}. sgfims oF- (*). Ha, “Linnea a 0+ Pf-t)+ 7601: :0 m {15.x 5d“ a 2 + .z-tff-t) + tiffh‘,1 3C3 C13 (Us; rte): ram new Sufism C9.) 7‘» rd— Q+ “25716) + flail :0 2—- 2 11:00 = 0 1.27612): [ 4-D 7({)3’E‘_1 1; {so} itsz mm (mi a He mffi‘a‘wf qcfl 1mg Mai-ms all" i=0. 9,, waqu be MMJWJECILM UM ghwua/{S CLO l\KClVCl€, 4:30} Ca, 910 $0 and», Iain/\ch \‘kciud.’ IJGCGI'Ilbcr 2009 Math 265 Name: Page 6 0f 13 pages [15] 4. Use, the nmthod 01' undei‘crn'm'led coefficients to find the genera} soluijon of y” -+- y m cos '6. $51M {‘6 MM- 89% 5 21"4’3 ‘0 wtvkcffia N‘s—t r 5. 1}. L z CLC05t+ (22.9244: Joe.” 1615/ Pam/£2. srL at? MAL“ PAL, _ goon/69¢; “fl, 5; iCACDQ t”? @Slylt) _Qfi . "(gm = t (—Asmt + Boast) +1 [firestw {35m} Yp"(k) z: {:("Acorf *’ 139“th + “( ”A9"'£*B‘°5't) '1' ( - Asr-ai + {host} “Manhasxnfl - MSW + 13m“? Yp"(k) + YPUC) '1' Cést Eh (AVE-kBsfi-‘ky- sthf+23cosfl + {:(W 9.3.0 .3th «LPrco H30 23%! 8mg.“ Y? Ci“) *3 2;: 'b Start M sag“ I _l._,__ “ We): ctcosm tics-“*7 ’* J79“: December 2009 Math 265 Name: __ ______ Page 8 of 13 pages [20] 5. Consider the initial value problem 1U” 3211’ 2y 9(1), 9(0) m 1» EMU) m 1 Where A I 3H: :: 1 " U1 (£1); 9“) {3 if: 1 Lg} \ (a) 30mpute the 11214312100 ti‘a.1’18f01"1’11 £{g{t)}. (b) Use the 11'1ethed of Laplace transforms to solve the initial value problem. (51mg) «sum 1%)) ~30 F55) ‘7’”) “H” a “z 5073 1%) (5‘~Bs+z)~s-. + 3 =~ 23%;} Fe) 2 5.4. + $.33} Lea-394?.) [52-354‘2.) “5 :we, 2 M) + (523(50' S {34054 ~ - 3 ._.. “SJ 4%)” 9“ WM 1': .—--—-v- + \ (1"?— M956“? (5' ‘0 5 (5 "0(5'2' (:01.qu ,L _. -5 Penal +Gifl£~+hs¢v‘ 9’ I" 5—1 94 5-\ 5 J " '- (Hna 2. J- ‘- J- v-l e UhL5l‘V‘GL 3 i " é“ +ii'§}L "' (2-% 517-; +2.54.) Elma at» q '- 2‘t I Mémsk Fh\ 30ml 305))“ 3+ 21:8, ..—- (J£_a*+:az#c>} » w v . «QED 5: éfwew) "" u,L-t)[__u_eflt~t* J“ gut-fl L W WWWMHWZf \ n we! 4“} 'f p-CL. PML‘J — I: m S 5-1 3’2. 1 : ALS'MS’LB ‘1‘ BSC5“Z)*C5C5"9 5’50‘ 1—» Afifihzb A13: s—m l ~ 3 We) (5:1 3&2. [2 (1242-0 05%; December 2009 Math 265 Name: Page 10 of 13 pages [20] 6. Consider the following 2 X 2 matrix with real coefficients 31 Am(() a)’ and consider the system of differential equations “rhere ::::.-.I (I; I ) ' (132(1) (21) Write the determinant 3/1 All in factored form and then find the eigenvalues /\1 and A2 of the n’min‘ix A in towns of o. X’(t) : AX(t), (1) (b) In the case where a. 3, find the eigenvectors corresponding to A} and A2. (0) If a 5/; 3, find two Solutions X{])(t) and Xmfl), in terms of a, so that {X(1), Xm} forms a fundzn‘nental set of solutions for (d) Deduce the general solution of ( 1) in the case a 74- 3. (e) W hat ifs/Eire tho oigerwalnds) of A in the case where a, = 3? Find, in this case, funda— 1‘1’1ental solutions XWU) and X (2)05), and then the general solution of Gan) (Ehamvdms! MA'rt) :0 (MC (2* ‘ 3:0 == (B-r)C¢“V‘l Orr mputgomlut Law. eiaimuzvlubovc. r33 , Q: 0- }. , At) @0451 SWWL W‘iw‘K :5 d? can—ad) WC Viv-M \AaUL 'Urk . 03 J LUVI'H'IM film-3 GMK—UHa) 0» WWW (M 13.33:) =(3) J Gwyn + m, =23 w 155‘ +Lw}’°\!=(3:r) ’U"; ‘5" “'3h" amniotic rut-'5 rs 916313): ,_.| m -3+a.. “'3 = "A at, 1 31:. "A <0 M we 2 l N.“ be °~ 'Fuuclamaifl S-J °P SHAME as 1% as 0.453 EDGCBH‘flJer 2009 Math 265 Name: Page 11 0f 13 pages Extra space (if needed) b “1“ 1‘5 +14%. SEAR): COEC‘t) + CL $62113) qu‘cké 0463 .. «A x“): 61(3)?" + egg)” 5(a) I; 01:3 #100.“ aw(‘-3, om 5AM) ":Cf g. (0963‘? /Fcrma. M 95C“ ZR) 3. (g)te3’c+ ©5634; WEB) can“ m ‘3 Sega-8c: +hgaeiM—rxs-c3 w "6 = w Kiwfwaiow (33: (g A :7 (it: 1’ attaining LSW‘MP’ elf. 7H! fL=[;)te%t+ fine: 90©:(:) (a) TM” (C? December 2009 Math 265 Neu‘ne: Page 12 of 13 pages [15} 7. Consider the (iiil‘erential equation y” 4y’ + 13y z: 0_ (a) 71ia1‘1si'orm the above equation into a system of first order differential equations, and write it in matrix for-111 X’ :: AX (1')) Find two reulwalued solutimis Xma‘) and X (gift) that form a fundamental set of solutions to the systei'i'i X’ AX from part What is the general solution of the system? (if) Deeeribe the lJC-Bl'lEWiOill" of the solutions as t -~-> 00. we, grew Udaufs +6 fun Moi lune. Nok Hm Luci: avAw ODE _ ~ _ .4- MQS eiAaV- r65“ rzmqr+l5=0 and (+9. citywv'alws 5141‘ F - 2"“3‘1 (Cody rind/w“) Ti‘ chsf’wa'L‘E Main'x Err $.7th”- uu 0% have same claw. fi L. —q .1: "Trace, (MN ‘6 I: [3 : M04) . Any (Wok w‘a-lw'x rein So would wwlf 5‘3 (23d! M ” .1 ($0.:— 1M Cfiflm W 13-2 i am! sewn?» noei épl‘ 6. $13 4““ cl "“ . 34- *3 VI): (0) {£=U" Eigenvalufi. a a'r8(r\rp a) 43.: “Ia-“hr M d ._ Mvt3l UL: “L2: "3‘ A I 6" r l 1 __ 1 1,: L 0 E a: i b t 8H: 3 Q“: ((o${3‘fl 5" LSM (7.31).) in Mualw‘l 9° S A at A New \ .1: LCngt+&3'-h3_t Mg: a“ fee“: Btu-QSun 31c) ) 11%} e ( o 5 D 3* Sim = clam +avcei, 2,41. 0‘5 "(3’37 Do ELI-(1') Mai cud. bb'Hn fi‘fiFdeJWJIY :‘KCMS‘I‘? 05(141(R+1“M9. (3.: Cyclic) uflf'in dwfiehloiA-J ijw _ ...
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Hw8Solscan - MM 94’ 9 we new L. are (-Hy. Hui war to...

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