Midterm2-math108-Williams-solutions

Midterm2-math108-Williams-solutions - Math 108 — Williams...

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Unformatted text preview: Math 108 — Williams -Midterm 2 .- 4/14/2010 60 Minutes; no notes, books, calculators; you may use the official formula sheets. Name: Section 03 / Section 04' To receive credit, you must show your work, explain your reasoning, and demonstrate the use of the specific method indicated, when one is indicated. Clarity will be considered in grading. “I have adhered to the Duke Community Standard in completing this examination.” Signature: M Total (out of 24): 1. (3 points) 3 Find the eigenfunction associated with eigenvalue £0 = 0, or prove that .10 = 0 is not an eigenvalue: y" + fly = 0, 0 < x < 1; 31(0) +y'(0) = O, y(1) = 0. )mO :Lé"s ' Wéoliw" tFCaX‘tC’L Li’ ‘0' 56.5 .. ..__ - me'm macro e er a: J raw-I) fl“) —;0 ® Wv awpunoh‘m g0: 9M '/ 2. (3 points)’ Suppose wl,w2,... is an orthonormal basis of real valued functions on the interval 0 < x < 1. If Zfizlcntpn converges to f (x) and 2:21 dawn converges to g(x), prove = Cldl + Czdz + ”°. 3 3 fire; Cn/Lk'] 2:20;; dH/k’ofl ° l' cm in» «m ediledele‘l—Malx o ; “tray; + aestf'amux + emit, m, a” azdlfd'xlzwm Wflim’fldxw fly MM moi [‘51)de Jig WWMW‘ if _ , . «r. at 0!, '\ ‘P r (:de / 3. (8 points) Consider the foilowing heat conduction problem: utzuxx-i-ZS, 0<x<1, t>0 ux(0, t) = 111(1, t) = 0 u(x,0) = cos rrx a. Sketch u(x,'I‘) for a large, fixed value T. Labei your sketch clearly. b. SOIve. { r MC753T) .10: & Bag (M, +0 7313’; ‘ 22:0 mo 61>" to ’22; mum) + 2“: chem V'V'" W n 1:- =€7 CnZIL‘Z—CJ jam-’0 flrgmméh - 0””0 Got/H42) @645 A. 5‘1 WW" Zing; 4’ kn “any dDMLx) :— O 1% bh‘C‘b) “1' Xnknbt‘) “(ano ink-hi Wi‘h‘m comm afifii-Md 1C; “(X103 =50 éTT‘aL . . '7 = t: 10.40) (Lu) mitt 5 is 40 echo - Emit)- “0 ‘ MR gm => \onm): ’erW" m baa-25:0 / Law) a0 =2; boLt3=25t aim-H .. _ f2 1‘: 1930:) +'.Tr"!on¢t) ~ 0 ‘ 0, Mm '45. ‘9 bi“): E}: T” r2; me Home?) =0 , bnmw => \ontt‘wb :fi; u(x,t) : 45k + (e' V 095111)} WW 2 4. (5 points) Consider the temperature of a wire that loses heat to the surrounding medium at every point along its length, 0 < x < 1. (In our usual rod problems, heat is lost only through the endpoints, x = 0 and x = 1.) The situation is modeled by ut+hu=uxm 0<x<1, t>0 u(0,t) = u(1,t) = 0 u(x, O) = f(x) where u(x,-t) represents the difference between the temperature of the surrounding medium and the temperature of the wire and h is a positive constant. Solve. Salad/will Vdfiitl‘oLQ/p e:th uCX.t\= XUOTCL') 3 XTJ ThXT'I-‘w x"-r xv m TELZ a 51:-) T x X"+>X=0 , T’ + (Minna W366, Mohxako WW 'tkis $1. perm Lgéeldxx mngon a)“ 5 éifl‘I—X-nx) Tn: An = (Hm; SO UUHtBZ' gen Xan) Tract) :— Nmo choose Cw $0 “1”” “wimzw’ch '39”; ‘ ~ '2 Cu Sin Mix = , lie, (in: ZKO‘F‘CX'33l/1VITTX A‘XJ ’Fl 5. (5 points) The vibrations of a string under the influence of gravity are modeled by Yr: =a2yxx—g, 0<x<L, t>0 y(x.0) = mm) = o where y(x,t) is the diSplacement from equilibrium and g is a positive constant. The solution can be decomposed as y(x,t) = 3150:) + ym(x,t)_. where ys is the stationary (steady-state) solution. . What ODE and BCs are satisfied by ys ? . Find ys. What PDE, BCs, and ICs are satisfied by ym? . Find the solution y(x,t). . Find expressions for the minimum and maximum displacement positions of the string. rung-acre: u_........__..— a; bjs 6+CLHO‘AM3 fl) $534150” r fifiézdzflsn—CS 155(9):|fi$(-L) =OE b. (3;):GMQ?» 3 %5CXB;AXZ+By1-C,. 115(0):,0 9 6:0, 93;“ ’D -- ‘35“)=Ax"--ALX ‘i‘W’O we} HL‘LMBL‘Oc? EVAL— uécxi:2Ax—AL a ()_ 3 z aim-4A -—-W— 916:3}; so “is x "20:10 ‘in mam (j: 335th : (LjfiH’iii‘oet : a1('$‘=+"§m)xx’a =3 lémhk 7- 93% aéizzy' chmi‘” “3 PDE iit’émet: 15M“ l 55(°3+%m(°¢3 ‘0 ,’ VéstL} +CAMCL1‘E) so ,9 ms: infinite, 1;)": Limit-3t?) $05 C; (blank) 6411. v: gm, imam vmiablag an VCMtDw KOO—(CB- Vtt’ = avg-V90: H 11— _ VCDJt)=\/¢Lt)=0 3‘1— ‘M’ rajl’ilzX’H.” 60h”; V090): “‘5 )2 “Lva DEX—T GG'T 7’ 3‘ 2330‘ -Lx) X1; Vt (290) = a fix :0 §a‘%m¥rv= Xrfi’bffi mx WWW-r0 A“ {M 2- [— Th‘W-X'l" ‘0 , A 70 2W4; Wd @0533“,- .7?t) 9 august-it + clsghfif): t (’CISMA‘F‘S t +Czfia'5‘.flfi‘h') TL’O) =0 59 C1=D THC?) = Load?“ 17 (1 ’ ‘ m : L )- F’le can ‘50 VCXMD): EEECKP"*L><) 6,1 = "gt-1351.): ‘1) 6m 1111)! " /(§m “if; 52‘an- r x M ) WWW... @931 #35 + m : r53 00 WMMMWW ..._. 2 n 7 7 Wad" "’40 + i an 5M"? 006 mm: E _ m - III-=1 1—,; 6?. Tim; M 03543;ng— ogo-[Ham MW -1 l. qu‘cwz Q‘xac‘ 5?: - 'rhUt Mix-1’63 3A M 5*— min-EMwM mm. cosmhrrt: -1 L J Le, LOW Emma: fl 7. . _. 20LLCX an‘) 1- 2%“3 L36) :E r L‘ICKfl'E 1A 05901 YMer wmw DOSaflb :5 , i-Er; “3% L. gum -r galmlww‘) +%L(xz_, “)1? 550‘) ...
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Midterm2-math108-Williams-solutions - Math 108 — Williams...

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