Sample.TermTest2S08.Sol

Sample.TermTest2S08.Sol - UNIVERSITY OF WATWLERLOO TERM...

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Unformatted text preview: UNIVERSITY OF WATWLERLOO TERM TES’lf # 2 “ SPRING TERM“?0’GB COURSE COURSE TITLE SECTION(S) HELD WITH OOURSE(S) SECTION(S) OF HELD WITH OOURSE(S) DATE OF EXAM TIME PERIOD DURATION OF EXAM NUMBER OF EXAM PAGES (including this cover sheet) INSTRUCTOR EXAM TYPE ADDITIONAL MATERIALS ALLOWED AM 250 INTRODUCTION TO DIFFEREN- TIAL EQUATIONS 001 N/A N/A July 14th7 2008 4:30 — 5:30 pm. 60 minutes 5 pages D. Harmsworth Closed book Scientific Calculator Name (print) I“:— ID N Umber \N‘Nw~_um.m.w_MWWWMH—«WWW"""" W Signature Instructions: 1. Print your name and ID Number on this page and sign it. 2. Answer the questions in the space pro— vided. Continue on the back of the page if necessary. Show all your work. 3. The last page may be removed and used for rough work. 4. Your grade will be influenced by how clearly you express your ideas and by how well you organize your solutions. MARKING SCHEME 3 H i __1 Total 1 45 L AM 250 ~ Term Test #2 Page 2 of 5 Spring Term 2008 1. Use the characteristic equations and the method of undetermined coefficients to solve the following: a) y” +2y’+ 103: 0 CH} ‘/ Char. ELLA: mil ’r am i' \O :0 {9 W fifim : «:3; C9 The gmm “data is \3: ét [C\ Cosg‘t 4' Ca 9‘" gt] (53> En] / b>y”+6y’+9y=2t; y(0)=0, y’(0)=0. a W3?“ = , \e m: -3 («Maxi CD -> ”We ComeemequD Radio's 75 M ‘5“: Q6“ + Cate'yc. @ Erfleewmdw wfiw)t3 3f A£+g $9 50 ‘3‘} : A \ am} (3‘9”: 0 ' édiaéi‘wd‘ns SW33. M3“ *k" DE ‘3‘“3 CA 3' (“J0 4 mg i at/ f ”x 3f «32> So C‘A: 9» f' N CAmg=o > .3 J1 “Bate, A:% lad (5'1 EA: (9‘7 ,/) _- ,, N ~3t - 2:1“ The, Saga son \3 '3: Q8 ¥ Cafe“ 1, %L’ 1917’ ® - - at: g /, NM; 3’ = ~3c‘e3t + c3e3t~3C£66 1” q Q) 90 “We {JVA COfiXéfions reel/MIR, “\‘kx\’ y - _ fl ’ C‘ 50 j 1/ amk o: —3c(+£a+% \x9 2‘ / , H‘ , , -2 , 2 K “Cane, p' 57, ml (.93 3Q (1 ~ 01) j‘ AM 250 - Term Test #2 Page 3 of 5 Spring Term 2008 L53: 2. a) Evaluate £—1{32 + :5 _ 5} flu«WWWWW.WWW“ :9 / ,3... _ 2.. L 3 .B. + 7?: {9 games ' {,esfiigex) 5+5 s~l :27 3‘: MM) + blavs} \; (:5) s-A 9m \~ 6% 5» E'VCS/ K 5;—S" awe) ~32~GA 9: A: E Wkflfcxafe, i4 5’ .. E C“ i513 i E:(§“3:; (5th t :5: "SE \ \ + (in) Q 8 «it 7; C \r/ b) Evaluate L” {m} ~53 a: a? s Q) ‘/ £628)?” ”‘3 @ ~3’C -‘ s ‘5 W ziw<2<3+m~3g __€ igr'f% «AME—W ‘ s W 5 9“ i Kara“) H 7 _. "3{ L [,2 Sat Cosmle‘mc \\ - e mat—E: we , /3~ * (a a s \ CE) \\ E g” (”gamma ”3;: E {:95} e) Write the following function in terms of Heaviside functions and find its Laplace trans- / form: f, for 0<t<1 f({;)—{ 1, for 1<t<2 3§é\ for / are W Huh: ’ (5’8 , 1: [Harman] 4 {Mr-xirmmfl 5r Mei): Ewe) E» New“) r (5f) M4.) 3) = we». (HlHlt—U» (time-2) t me ~ (Jr—w \«m- (trim—r) NH) «2 re ~ \‘- «r [Rafi “£431 W53) AM 250 - Term Test #2 Page 4 of 5 Spring Term 2008 Lg] 3. (a) Use Laplace transforms to solve the IVP «s 2:” + 455 + 4x = 2e-2t, 35(0) : 0, we) = 2. SQXBF gawk 1Y0) é H[9X§g‘j ~ 1(0)] + H H3) “2 5;; g) 3: (ffist‘fl “m z 2 + 533 Q) ::=? ><(s}‘: €§5 t ,52‘. (it? fists); K93? v ~=a nitr: Qtégttrffégt, (é? Cg} (b) Evaluate f0 00 236“” sin 3tdt. HINT: Compare this expression to the definition of the Laplace transform. iflhfl=fe*flas So Kigksxs'éttlt = B 5%(fsh3tflk AM 250 - Term Test #2 Page 5 of 5 Spring Term 2008 (This page is for rough work) ...
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