# mid1sol2010 - Math 257/316, Midterm 1, Section 101/102 8...

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Unformatted text preview: Math 257/316, Midterm 1, Section 101/102 8 October 2010 Instructions. The exam lasts 55 minutes. Calculators not allowed. A formula sheet is attached. 1. Consider the ODE 23:23]” + (.2: -+- o)y’ A (:3 l 1);} 2-:- 0, in which (.1‘ is a constant. (a) Classify the point :1: r: 0 (as ordinary point, regular singular point1 or irregular singular point) depending on the value of o. [5 marks] (b) For a z: 0, ﬁnd two independent solutions (for a; > 0) in the form of series about 3: :2 0 (you need only write the first three non—zero terms in each solution). {20 marks] (m the we is him] " “90¢ PMW‘ M low-ow Wu- ‘ m L if} t) \ 23"}. in Hip ms. m..—~ 0 WW I W a... .. _ _ ., I _~ 1.: @698) 1:. wind w) DL‘O '5’ 0‘ (“ADULT {bi-"ﬁt l B um 35‘ (“MA "’- t‘r“ “LE-'4" 1) t “L ) (limit.— 3c~90 ’i 3; ~90 Z- Z- I >930 it‘s C\ |‘ ' a ‘I ‘7 nu-r '3 hm 3o!)th : lim + : 1 1K 1:0 l g Qreszur slmjdw— lpocnk tie->0 l 'M,""~)O 2,, 7.3L, ‘ J‘_ f” iWS‘N‘lL I ' % “i O l ”O a irrtéxﬁw- gibJWT ‘ M122 Lt.) th who/“2i 6‘? MrJL “3" “W4 “0 W :3 ‘"' Zﬁwitnﬂﬂové‘" O '3 Z)?“ H’BC“ WW") Guy” 'l' 3" Zhﬂquﬁvi" --- {pm/ﬂ i {9“}: 5'0 “'30 inlo M1310 l'l n Hf— “3 VH4” n of r1 a? “+4. '1: g ZCX‘VY’\(/V‘vf Ml“ any; 2 Lm‘hr“) i am}; "" 2 h“: 9 1|.) h2g> h’wo 0 Fa ‘n~\ 1‘ C” burr :9 QmﬁL- i” W h “9‘ w!“ r; {flint} .trml1W09L + gatikath MIC/Mei wi Eel“ " (Met 9"“ int-1‘ . malls. V131 \ '7. r a... a O ‘3 Zr-{I-rt\ rev—i f: 12.4” “‘2’ «vi 3 L'Lr i". “i X ‘5‘) (L:.—”’\rv{' a rﬁwreonuL relay-3,12, Gm. m (Li 1 Q“ i k {(MQLM-ti "l x ‘ Gm, tsﬂi : t" 1. Cl“ MW (in “Ole-‘63 “1 2. Consider the following heat equation problem with zero boundary conditions: at 6.122 ’ BC : '2L((}, 1.) r: U 2: u.(1r, 16) IC : Mac, 0) :2: I (:3) U<w<7r, t>0 (a) Apply the method of separation of variables, and ﬁnd the solution if f : 3Sl1‘l(2IL’) -—» sin(4—m). [15 marks] (13) Find the Fourier series of the 27r-periodic function with :2 3.1; on —-7r g a; 3 7r. {5 marks} (Hint: ff” :1: Si11(na;)ci3; I 21—;"(m1)n+1 n = 1, 2, 3, . . .. } (c) Use your answer to (Iii) to find the solution of the above heat equation problem with f(:i:) : 39;. {5 marks] I v/z’!(/ _ Tlg Xniww‘xtﬁ’x {Q} Obs-1L2)“; XL:QTL\~\| '30 KT E =9 "\$2., 1% _ you‘uhlun: X”: t >0; XL») I (ran; (Show) \~ (5 ﬂin’Lﬁcmh “\Miml Sam-i“) we, M‘XCC-‘hXtcﬁza 90‘2X5Cﬂzﬂ :9 Km: {TBS-“WM tom 1 KW) ’-= E’w-nlﬁﬂ © “wt 557i T" “W ' er 3‘: v3— h:\('li‘3]t‘{”m n- CA'S {IN Java-3 wt. Crmné‘l" ‘cx ling? k l I. . ( lioOQ-Ntﬁ‘b /C,Ur\cﬁnr\klw\fg 3‘ (Kl Khan ﬁh‘Li‘») i . v _ ,_ | B an; “at “Tn ~4sz mist T‘ a) “(to c Leer L, T S3 (*0ng 2 L11: . My“ — , . '3 ‘l’lnug. {ML L‘m Ct \ 510idx‘1l M C1} x 3 En [A 'JL:\. Qty gr OWUKG\$L\ “ 00 7 nor. 3 MAL (in) {g {:Lmi a: 069% ~:. g lashing); Mi lo 33 17 :”l l w}, WW laminar: t: c. o é ,8 kl" _ r5” (bk) hm : f’y 3r 5%Aﬂm {mg Fgﬁgsmgmﬁl MW (Am: realms) L‘s-Q) clam, m C) ' h=i ‘ , , b E W {3 ‘W 310:») Hw‘a‘l; t ~11” BM h-"vogﬂﬂ. k, W I {if ‘ , 2V ”' WP tam—ii L3.“ 3: "bimlﬁkﬁ Elm, 2 'i it? JG) *4 3 ‘3 Wit-3 “ ‘ T." ' 1“ (a) «m, QM; at. Q wits—riff"; w w CL“ “hm m3 “‘7” if) ...
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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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mid1sol2010 - Math 257/316, Midterm 1, Section 101/102 8...

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