m257_316finaldec07_Solutions

m257_316finaldec07_Solutions - 1. Consider the differential...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
This is the end of the preview. Sign up to access the rest of the document.

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)+)((/€€ ...
View Full Document

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.

Page1 / 6

m257_316finaldec07_Solutions - 1. Consider the differential...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online