MATH*2270 Test 2: Key Concepts and Problem-Solving Strategies
School
University of Guelph**We aren't endorsed by this school
Course
MATH 2270
Subject
Mathematics
Date
Dec 12, 2024
Pages
14
Uploaded by ChancellorSeaLion4237
Test 2 MATH*2270, Fall 2024 November 16", 2024 from 3:00 — 4:15 PM Name g@]w@fofl& Student Number Finod This assessment will be marked out of 45, but there are 47 marks available. There are 5 questions in Part A worth 5 marks, 9 questions in Part B worth 18 marks, and 3 questions in Part C worth 24 marks. A calculator and the provided HanDEbook can be used while writing this assessment, but no other printed or virtual material or electronic devices may be used. Keep your eyes on your own assessment and do not communicate with other students during the assessment. Academic misconduct will be treated seriously. Ensure that your answers are written dark enough for the scanner to pick up what you have written. Nobody can leave the room during the first 30 minutes or last 10 minutes of the assessment. This simplifies signing everybody in and the collection of tests. Unless it is in the last 10 minutes please raise your hand and a TA or myself will collect your assessment. If you need to use the washroom or need assistance during the assessment please raise your hand and a TA or myself will attend to you. You will be asked to leave your phone at your desk. There is one extra page at the end of this assessment for rough work or extra space. If you use it for your solutions, please indicate this clearly in the question so we know where to look. Do not detach these pages or remove the staple from the test. This may result in errors during scanning. Take a deep breath and believe in yourself. Good luck :)
MATH*2270 Test 2: 3:00 PM Version Part A: Quick Answer (5 Marks) Read these questions with care. Please place your final solution in the boxes provided on the right side of the page. Each question is worth 1 mark. Only the final solution will be graded for these questions. No partial credit will be awarded. . . : N d?y dy . : Al. Given the differential equation R T e + 4y = cos(z) provide the expression T for L such that L[y] = cos(z) is equivalent to this differential equation. If no L exists write DNE. [ =D 2xTF+D +4 A2. Convert the differential equation D(D? + 4)y = 0 into ‘prime’ notation. _ N, . =Dy Wy L,f +Lh1:0 A3. Given the functions y; = sec(z) and yy = sec(—z) provide two values ¢; # 0 and ¢ # 0 such that ¢y, 4+ caoyo = 0. If no such values exist write DNE. C l + Ca — =0 S0 s(-% Aone ofF [Mauad- Cosix) = cosl=» " i C| = :L sz"‘L Cy =~ CQ_ A4. Calculate £{5 + 7} if it converges. Otherwise, write DNE. St o S A5. Calculate £{t™*} if it converges. Otherwise, write DNE. af;ivfi%w’f”-’ ONE Page 1 of 11
MATH*2270 Test 2: 3:00 PM Version Part B: Short Answer (18 Marks) Read these questions with care. Please place your final solution in the boxes provided on the right side of the page. Each question is worth 2 marks. If you are correct, only the final solution will be graded. If you are incorrect, you may receive one mark for a single minor error and/or demonstrating the correct process. Bl If y; = V4 -1z, y» = In(z + 4), and the Wronskian W (y;,y,) evaluated at zo = 0 is non-zero then on what interval of z will the Wronskian W (y,y2) be guaranteed to be non-zero? If no interval is guaranteed or you don’t have enough information to answer this write DNE. But I drein of \.(, ¢ \,(,_: Ju-x —>' Y-x20 WC‘1H‘4‘>=CJ9+(JE\.—>< I n(X+Y) ald-x el Wz X Inlx+4) = X¢Y -0 R CET S lfié’iflL Y >x K+ 4 2{H-% PR o S XE (4,4]) XE4. 1t =4 _— -4« xey o X E (’H;H) Must both be round # B2. Given that y; = e** and y, = e 2® find Wy, y2). - d@'}' g = G'Z* = -—Z_Q?(’(f_ Zgzfi/g’—’zi ~2* 2% e ~28 i Y -4 Y Page 2 of 11
MATH*2270 B3. Provide the general solution to (D + 9)(D — 3)(D + 2)y = 0. (/LPMMHI-HI'C < [r+9(r-3(r+2) =0 o) z=9 r=< r=-2 Test 2: 3:00 PM Version 22X : ~9x _ . B \,f: Ce 4-0283)( + Cze . . d’y | dy B4. Provide the general solution to —= + 2—= + 26y = 0. dz? dz CMV&W({?’-/C C r?i2e +2L:0 7 f;ez;dhfiJMNfir B pYQ) :—__2_"—@‘1" o8 2 oy € (oo (50 Co siN(5)) [ cos(3t) o5 ( S -_— - E5 ¢Z4(3)* Bb5. Calculate £ { } if it converges. Otherwise, write DNE. ] __L—_ S P 85 52'4—(1, Page 3 of 11
MATH*2270 B6. Calculate £{(e"t)"} if it converges. Otherwise, write DNE. - 734 Test 2: 3:00 PM Version 41 (s-4)5 B7. Calculate L{f"(¢)} where f(t) = *+ 2 if it converges. There should be no £ symbols in your final answer. Otherwise, write DNE. fit4) =2t " = 2 24 - o) - §'(0) A 5N ST Lidr2d - s fld ~ Fro Py a = 97-(_8_!?-4_&5_>_5(2_)—-0 OR S N 1ifuR= 1523 & 2 a a o . _ R st (Grg)rse Saptas = § o 8 I S A L B B8. Calculate £ {m} SCS“’Z.) < SF 2 = - j_ _ j__ HSCM" % _ _E__, :L{ ‘ { S sz $:0 72 51, T L Szl =5 ._&- - -—L’ -2 : -2t -2t Lo -4 s yl]-e Page 4 of 11
MATH*2270 Test 2: 3:00 PM Version 2 BY. Caleulate L7 ———_ &, ; 9. Calculate £ {s2+23+5} 52425+ L = (57‘-;25"' ,>’I+§_ T bl - 2 foet sin (agEt) Page 5 of 11
MATH*2270 Test 2: 3:00 PM Version Part C: Long Answer (24 Marks) For each of the following, please write your solution in the space provided, showing all your work. Partial marks may be given in certain cases, so please attempt all questions! d* d C1. (8 marks) Given a differential equation a—g - 2bd—y + b%y = 0 where b € R. 5 Z (a) (2 marks) Determine all the roots of the characteristic equation associated with this differential equation. If there are any repeated roots be sure to clearly indicate i ( havacteristic r¢_2brtb’= C-?’ (r_ b)z:O r:bl twice. (b) (6 marks) If one of the solutions to thls differential equation is i = € prove that the second solution must be y, = ze®. Plugging these in and showing the left-hand side equals the right-hand side is not a proof, and doing this will yield you zero marks. We have one solution, use fedwction o€ order! @ y. v»(. @D - X \12u _ \}.\lebx P bv.(ebx i bv,‘ebx N bl\/_eb - e = Ve r2bye™ + bve e \ it % b ' (f\\’ri“” S:? 2 bye + v ) - ol vieb* + bve?) + bi(ve™) = 0 e (’,\I{-bovsco\ofl\fr\l'! > _ ) A Teon] o [ ape 2] + v g2t o) < 0 \" eb’( + bve” N\ bx @ \r” gbx -O & Direct :L'V\J't‘tjlovl-:on.> \_fq_ _ Cxeb% . (D\e;v__, ' MW VI\ o Aineorly ind. o U= Sodx:(’, Lo from\{ \f’\f\\ " Al’rerm}n‘ve \ = SCAX =Cx+D. Shown @ \41 . (ceD)e” on on e 14 o CO(’"\’ $¥op 7 il a(\ts x m Page 6 of 11
MATH*2270 Test 2: 3:00 PM Version d? d C2. (8 marks) Solve d—% 4+ 2% — 3y = e % + . There is extra space on the next page if z you need it to answer this question. Use vafejy 31 Yndetermined Coetficienks! ' @@C[nomc’refl'sh‘c E?: r2+2r-2:=0 A = C\e'3x e (r+3)(c-) =0 D> ¢:-3 =4 .@Awwfle \1[’; Aie 2 + p2x + Ps @ Q-BX oceurs in koth \'fn and L{P \1P: AIXQ—BX ¥ Aax 4 Az Y= fe™™ - 2Axe™ + fiz s, Be2ST by X Yo' - 3P’ "3F\(€-3:,<+ qpxe " R . PSS D ope™ s apme™) > Alhe™- 3PpE™ +h) -3(hne ™ hux A | 1 . ! —3% Collect coe FEicienks = F Yoarious Fuctionrs - . X xe > L9 - oA - 3]+ [-6R < 2R ] 3 +x[-3p]+ [aps-3R)=6¥ ' DNS. Generole e%_w.‘ons bosed on e Lredki @ -3X 0= 0O Xe (ongts ARz-3Qz = © = JPhz=3P: %—Qz = As | =Ly = = =2, 2(3) e - [Ae= | Condinued below. Page 7 of 11
MATH*2270 Test 2: 3:00 PM Version This space s left blank to continue your solution to Question C2 if needed. — [} ::__L —3x’—\— ’2 “qp er 3>< 3 @ \,1 \%Jr\{? ng;?i The test continues on the next page. Page 8 of 11
MATH*2270 Test 2: 3:00 PM Version ) C3. (8 marks) Solve 12 + 49y = sec®*(7z). There is extra space on the next page if you need it to answer this question. X | o pneed 4o wse variodion of Pwneéca! z 4+ 44q RO i © \{ = €105 (F2) & €2 5Tn(2x) ~°=-49 =it @ i) L> ¥ = ECOS(?A’ S\h(qfid} @ o= Yt e V) cos(F2) +\28in (30 @ \NL\1||\4L§: dé‘t’( - T cost (=2 + Fsintlm) SN cosS(F) SN (%x) ~Jams(20) Feos(#) Ti T cos™ (0 +5inHFN) = :’_ W\L\fu\\fa) = ded L 0 smcm\ - —sinlI) sec’ (Fx) ) s’ (F) Feosl V\,lL\fu\(i) = dek c0s (74 ) 3 = D 7] costix) () = Falinie) cec®(#x @ LY W o - s:'n(l)iseé(?x) A Alkernakive i X ’% V2' = We = gee2 (30 Uz fant We - gec® (34 W == A - Feocl" S Lo (30 sec(AVdx AX ! S 9 L (A e Tadud F 21 @ V), = -X SN S@cB(?x)C))( F = *;L Z,: + _ o sin(#) Jx (= osL#x HY L / ] /:l—— & 052 (X dvw o - Fern (FX) . X i émwz(x)‘} - :_\_g L duw L "du o gin(@RdX B3 us —+ —ar Lo(yy 2 A < T T SM du rv»dwqc“* I h. ! _ T‘,:' _(A:ii 4—/@/ (there le lbe MOfé?)A i | S = Cfg cost( ) ’;L—sccz(,?x) =\ condinued }Oelow . 9L Page 9 of 11
MATH*2270 Test 2: 3:00 PM Version This space 1is left blank to continue your solution to Question C8 if needed Ve = S Sec*(FaAx g =2 " 2 = m_\/ S_l_":.l 20 (F1) A% :L 7 = S :l—gé’cz (,?“)d)( LH Tncluded in éan (20 £ Mk Ve = fi Lanlto LA e X _'/- S‘V](q‘ ) = V\\{.+V'z\11 @ \{F = 0’& Sec (FIx) cos(Fx) + 59 tan(#x) SINCHX ‘{p (s Hne. & [Foee g~ ————— — -~ 1 = = 2 (3x) We found V1= ;‘—{i&iclt?{j\) OR Vy = Iy kantli : = int (3 & = =1 SIn /J"[@ Hm{ e o 48 cos*(®) s = |— cos (F2 \(\\ e 173 thiswll B ) \ Mlb(lj‘fd/ ( \D\/’ %—g— coSL(-?fl) Cos*(Fn) in me +C. (Dns\'av\}‘ -V (e (3 - \B 5 fi& - t e Hf)e\/ m the same = % -\ coc*(9%) ‘1& Good work, you completed the test! Thank you for always being the most exciting part of my week! I hope you have a good night tonight :) Your teacher, Michael. Page 10 of 11
MATH*2270 Test 2: 3:00 PM Version This page is intentionally left blank. If you use this page to continue a solution, be sure to make it clear that your solution continues on this page so we know to look. nothee w0 o fiaish preof in CLb: Yuz [Lx+D)e \1 (ren. Sol: =gl et 1" : X C|€bx ¥ C,_(Cx*D)@b ce®® 1 catxe® v cape” L = C|€b’<+Cz'Debx + czcxe”x = (Q*’sz)cbx b Lol s \/‘V‘_\) S &, Ce Yo = K | Page 11 of 11
g SOIVI'V\fi CZ via varishion of params. @ = From ofner seludion: \{h= C.Q-SX FCae” TRl %e-sx, e”s, | @ A’55L(Wl@ L’(P: \/|\1' 1_\,“11 ‘1‘): V'Iefax*\ficx © W= det| e®C o*| = 7%, g¢2" -33’3,( 8" _ L’e,zx W= deb] 0 er| = - e €—3?<+x @)\ = _ Q-zx = xcx -3 X Wo = det e O ‘ - Q—3K(e—3x+ )() -3X -3X -3¢ ™7 e T e_(,x . xe-zx \ ~aX =rdlilir o e X i & @Vl e =g o= KE V2's We . e6”+x@‘ w -2K L]l@ N L’e"z.’( & 4-—L —_LX xesxd)( U':;L V= ée3x A SR SOl R ‘Xe’ oL?(}
- 1] gl = — [] +|- m' &= = -‘- N | K nN 1 &+ o — 0 1 X . K ~————— e Al H | . el e = '_‘qeqx—XQx-g_’_X |6 4 4 ONP e +\&)@M /47’( /’hdsé ong Wer's ‘)/}lfi Same? (from +he uy)dp#ermmed coeFeveiowts on PO Fand ) - » -4x =X -k X - [l - xe?® | 3K} o3 e xe _g____>e_ \f‘) ( X = -l» 7, >Q + n /"l m = =l xe” -3X -y | k| 1z 36 1 -3X e | o S =l L T (2 H)x = —,‘_Xe’sxo -Hy _ & H 12 56 o Y T A =\ =——Xe - L x = =2 Y 3 9