Midterm2-math108-Williams-solutions

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

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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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## This note was uploaded on 09/29/2010 for the course MATH 108 taught by Professor Trangenstein during the Spring '07 term at Duke.

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

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