m257_316finaldec07_Solutions

# m257_316finaldec07_Solutions - 1. Consider the differential...

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Unformatted text preview: 1. Consider the differential equation (a: 3w2y” + Sxy’ + (x -— 1)y = O (1) (a) Classify the points 0 g at < 00 as ordinary points, regular singular points, or irregular singular points. 00 (b) Find two values of r such that there are solutions of the form y(x) = Z anm"+'". 71:0 (c) Use the series expansion in (b) to determine two independent solutions of (1). You only need to calculate the ﬁrst three non-zero terms in each case. (d) Determine the radius of convergence of the series in (c). [20 marks] " t X _, 5 ,0 (a) X=O A? A Iféﬂzﬂ SM/i/xz/IK 7/ f/WEA’ Q49»; X35}; ~§«( WA”; #14 Wo/IVZS 0< X< a6 AK! 01270-0sz Pan-r73. '2 ,— (A) 7/{1A/2/C/4zr62 /J.‘ ~r(r’/)+§r~§’=o 0K37—3’W5ﬂ’” O 3r242r,/=(3r-—/)(r+/)':o 1‘: U3 9 r=~/ (C) g: 2/ an xm’”, I g, mm + 40x7” nfo" n+r+/ m=n+l Z: Xm+r Xi : a” X : m=/ 4”)" r :0 5 9o M7" '7' xéy‘sf” aﬂmﬂxmrg " z ammwﬁx + QDTX “:0 7‘ 0" mw” r' x279 2 an(n+r)(ﬂ+p’>x s" 2 am/m+r)[m+r/I)x Harm/M n:o mm ADO ‘ mﬂ’ ' 0? Ag 3 [0m [3{m+r)(m-rﬂl).,£ 5%+F)— /]+dm,,]X +dof37’6504—5/L/Jyr msl ‘ 56204’7/M5 C017; 70 0: do [3r2+27”/] = 4; {Sr—4) [7"+/)-:0 z 4) ‘l/a #5189”! [3:0 =77 0M: “am—4 {m+r>(3(m+r)+2)—/ =—/: am raga...” , Wm” (m’l){3m’l)'l “4(37'7'4) a _ “40 ,+00 092: "62/ :“io Q3——"a'2. =+_._.—Q——Q M340 2,92) 4 3(5) (0 -I 7— 3 ‘ z X X -’ X +X, _— .. 17/00 00 {/+ Za 60 ) a 7’=/ 3 a —— ~am—l __: ’ m—l .— " W1“) [5 m m+//3)[3m¢3)—/ (3m+/)(m+/)—I m[3m+4) a : “a0 :‘Qo Q 5"az_.,:: 7‘30— / M3744) “’7 2 ,2{/0 Mo 2 r.‘ I {/2 (x) : Q0 x’é (/ — >97 + 7C/A/m“ > [4) Silt/[t A)” am "‘4’?" ‘l :2 0 war Iia/zzr 0; Cameémg- in”. ’ (m+r)(3(m+rl+z)—i a: ﬂoﬂ/Ja/IS fspd. 2. Find the solution to the following initial boundary value problem for the heat equation: at = um+u, 0<:c<1, t>0 ux(0, t) = Oand ux(1,t) = 1 u(:v,0) = c0s(21rx), 0 < a: < 1 1 Hint:/cos(mrx) cos(x)dx= (—1)n+1 sin(1) (7rn)2 - 1 0 [15 marks] ,4 00/ 2:0,: I Jaway you/raw: WZx) 0: VQX ¢r w =7> W: Amsx + gS/ll-X WK = ~ A‘s/Ax +6’a5x —— I x4440); 6:0 WK(/):.—A5‘£n//) :/ 47A = ml) fo): - (05K Sinfl)‘ ZN” amt): woo + fo/x) Hg = [v32 V).£ == /W+V)XK+[W7‘V> s f%+m +{\/,(X+ v)=¢v{=\9x—v 0.: “KM/75):: Make V>([0)f) :Tp VX[0)2§>=:O ﬂ HAW): M)+vx//,ﬂ z? \Qz/Jﬂso COHZFXD : Lax/o) :wzx) + v09 0) £0 v(x,o): Cats/22729 —W[x) 5635/2170???) 1’! A/ﬂw 5377/1”! Vﬂ/Mzﬁ Fold Viz/2‘) :2 X60 7(2‘) 773‘) ﬁx) . , 2' '5' 7:[/»A2)7” =77 7m: c ’ A) v \ ' X\\+)\z.X—;O X: A605AX +65/VMK X : //l’/\Sm)‘>(+6>) (Ad! X\(0)=O=X\0) 1V0): 3A:o :7 6:0 (Mi/i=0 X\(/):’x4/\Sl'n/\:o 5v Arlsz V):0)/),,. X”: 05mm) £— ‘X’ p/lzhe ‘ v06): A06 +Z/{nef ’4 (osmiﬂ .. ) ns, 06 605(277x7-Atosx 5 VLv/o): 260 + f '4’, (£50171) 5mm “3/ ml ’ 524; {-l) f A": 2, j [605(21725) +605Kj605[”79‘)dX.—: £[I 7MP“, 0’ Sin/I) 5' I a 77 (S d=£5kzﬁl+ﬂl :oZ; A0 " 2‘ 2! (05(2 2‘) 71' gay?” X [ mzr 5M0) 0 Z 00 NH _ [W2 75., 47";— .2 20979): —Co5K + 02873— +2ﬁtf 5”) 8 p"), ) fashﬁx) 748 605(2/71) —-—\—- 5 ’2’ S/nfl) g, M ’65") / 3. The displacement u(x, t) of a string of length 1 satisﬁes the wave equation utt = c211”. The string is set in motion from its equilibrium position u = 0 with an initial velocity g(x) while both the ends of the string are held ﬁxed. (a) Write down the initial boundary value problem satisﬁed by the displacement u(x, t) of the string, and determine the solution to this equation. (b) Find u(1:,t) when g(a:) is deﬁned as follows: ()_ 1 if0<x<§ 933— 0 if§3z<1 [15marks] (a) a“ = Czl/xx gc.‘u(o,t);'0 : ill/ﬂ) :zc.‘ ammso raga/Maya) [17' (70 (x) 5/ 77/! 0a.; {X/YWf/éA/OF mm PAY/a2 0?, X"? a . ) 77W / A 45 yy 27344405927; 6&4 AI. mm)- ZC Java/5) . 9Q - ' . izmM/ﬂvﬂv WM): sznCoﬂWﬂf)¢5n\$/n{nrcfﬂ5/n[nnz) Vlzl ‘ (1090) : % K145010271): 0 :37 Anso ms: , Imam/#5 3 2’ i” MW)(‘0\$@YCD") 5/404”) Vt:l _ gakl/iﬂX/O) :: g 5" (n7¢)5‘m(n72) I Ytsi . __, 2 , d .. n (FT-Z) £300 5m(nﬁ70 X U[X)£)r; 2 8n 5/‘nfnYZfBS/‘nfnﬁx) h=l (A) l/z \ \ z = 2 jsinln‘EDde-J j (55(nﬁx_)|o larazmh/vﬂv B“ (We) 0 ﬂ (mm) MT) , .2 {\«(rl /] ' (MTV 0 C I 2, % Ll—(AY‘J 5M (WYC‘WSMUIVX) u \$3-, ’ ‘ ﬁzCW—l n?— 4. Use separation of variables to solve the following boundary value problem for part of the annular region: 1 1 u'r'r + _u1‘ + jugs 7" T ug(r, 0) u(a, 9) 0, a<r<b, 0<6<7r/2 0 and Ug(r,7r/2) = 0 1 + cos(40) and u(b,7r/2) = 0 [15 marks] (M66) 1 RN) (9(8) 2 \\ “ (,Lzo T E\\+rR\-: "‘ (9 :1 A1 “a (“’3 C9 mums : \ U : rzR\+rlZ\-«)\ZIZ=O a 0 A20: LTRT‘Y: O g‘b €13 C/r R:- Chi—+1) ’— >\ )\ W! W” X2430 50 29M Km..- cr +‘Dr' 9\\+/\29=o esAcosAeuaslMé 9\:-A/\9)’n/l.a+gAeas/\g G\CDB :0: 9\CO\ :- _-; 0‘ :17 B:& / Ago @VV/zl wills/MEET: 0 MD = nfr V): on), .. z .—._... 51 00 z: LA0+°(o/lﬂr3 4" él CAnT)Y\+ “Y‘T’An‘j €05 O: MU>39§5 {60" “04131 ‘l' :6 AmbAn‘l'Kn KARE (OSAV‘E nsl "an, 00 air, 9): xoﬂf/b)+ Mm LVN, '7)“ T—M Cosh/H (>0 ‘ H.059): 1+ 05(49): {0/44/654— 5‘xﬂ[a"/\YLgZA"a,A“_J€05(2nﬁ9> 'l l O?) afA; '_ guza’AZ A . Lime) :: ﬂ 4 T ' gummy jam) A a—xz - gﬂz (la/‘2 5. Consider the following Sturm—Louville boundary value problem: y”+Ay, 0 < x<1 (2) y’(0) = 0 and 11(1) + 1/0) = 0 (a) Determine the form of the eigenfunctions and the equation satisﬁed by the eigenvalues for boundary value problem (2). (b) Show that there exists an inﬁnite sequence An of eigenvalues and estimate An for large values of n. (c) Now show how you would Lise the above eigenvalues and eigenfunctions to solve the following initial boundary value problem for the heat equation: ut = um, 0<m<1, t>0 ux(0,t) = 0and u(1,t) +uz(1,t) =0 u(ac,0) = f(m), 0 < m < 1 [20 marks] MIL) ﬁ“+/\€/=0 “NE/“L y‘wbo 4/0) #ngso {/(X): Amy“ 738/VX ﬂ‘st/asﬂo/x +3/4(&\$ X 1.127“ w/x/r) .—: XS/‘w 6. Solve the inhomogeneous heat conduction problem: u(0,t) u(x 1 W ,0) sin ) 3 2 0 and u\$(7r, t) = sint um + Ze—2t Sim—:3) +cccost, 0 < a: < 7r» t> 0 WAN; Al 5367/5/96 97%42’ [15 marks] /N/%0Mp&mgaa5 6c , 7%!» any) : w/x/t) 7A V/x/z‘) .. -' (5’ #14: W); *Vf— V'Xxv‘je 0: “(Q5): Vy/Off)-1L 3? V[0)3£)= 0 ﬂat: LIX ﬁat) :: 5/416 +\§[7/g)@ \Q/F/ﬂso o’zirzgx)+)((/€€ ...
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## This note was uploaded on 04/06/2012 for the course MATH 257 taught by Professor Peirce during the Fall '08 term at The University of British Columbia.

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m257_316finaldec07_Solutions - 1. Consider the differential...

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