Math2271_1112_T2_Solns

# Math2271_1112_T2_Solns - ZZ WNK5 Math 2271 Differential...

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Unformatted text preview: ZZ. W‘NK5 ’ Math 2271 Differential Equations for Scientists and Engineers: Test 2 Wednesday February 29, 2012 10:30am to 11:20am Name: 60 L01 [bids Student Number: Instructions: Complete all 5 of the following problems in the Space provided. Notes and calculators are not permitted. All cell phones and pagers are to be turned off. Students must present valid university identiﬁcation cards. 1. Find the general solution of the ordinary differential equation 4— N “(L3 { y’”+8y”+28y’+ 32y = 0. meAor tam; (D3 Jr 8B1 + 18D 1‘ ’5ng —. o Clmrhcluighg Ecbln ‘_ x3 + 8): l’ 28A 4' 31 3 D U) A :' ’"Z l\$ a (00* \$53402, L- 7A“— ?pi-z)‘ + 225m) k 31 = o (FM/“f ll‘c eharaclnx‘sbL ﬁbucdt ma '. >\"’+ lo/\ + \(o MLI A3+5XL+18>A +31 —()\3’r 2x1) ® txz+1s>~+ 37. -Lb)\1—r 0.)) lb)‘ *31 —( mx+§13 0 \$ -, xg-LaxH 299431: LMZBLVHMm) —» 1200*3 a? m CEUAAMAX {L \LUVM '. xzjﬁt—ﬁ W Z. - wank—a 7. :“Bi/LH ’ We, 351wer Saﬂghm is 'h/wts: «2‘ «But, 1350 2 (3‘6 my Jr C; a [43qu 5 WHY!) - 2. Using the method of undetermined coefﬁcients ﬁnd the general solution of the differ- ential equation y” — 4y’ + 4y = (2x — 3)e2"‘. Hwaxmcwe a \ 12v“- vm‘rk L2 b’ADM [Ag-:0 Exgmvmﬁmm X54; + 4 : 04,31 3 0 we, hm 0‘ o‘wbkc (00* \$9 ‘Tka, “Omega/news 50.0416 LW :6 271, 1x, 33“” ‘T- C16 4- C7121, 6 ~—-—-— Nm w w w m m minivans sow 409” 0‘99th on "ML 9H3 5 our Aﬁuan. Ear Ms. Spemjl cm, M,st a pmii’cviar soaedv'm'. 32m, 3 + EN, 6 + 715:: ff“- 12.51, QPLXA : be f; 1271. but : mat/Hva + amend)?” ‘Z 23L ‘Puo: 27%} e u + 1/184, 6 \L SE ‘5 a: Li) : Am [3}; Db) 0w“ 4 2H Dawn) 6}!“ 4’ ZEKEI} ’r 23?) e,”L + E [ Lab 4- but/7’) Cm 72L 3 A [433+ ‘8» m) 6,2" + EMﬁHzJ’ +bze)€, —> 7"“ L3? 7‘ \A K4¢1‘t%bb+7x) + 3 [438+ m? «HOLY "‘ o?“ 01*») - ‘W 3wzflm3) + ‘Mﬁ + 4&316“ ='-‘ M WE») cu : (11/336123; ® 7”” 2Pr>~3 :3 14:1; 2 EB=Z a g; E. aus‘w‘w‘iwx Sojldwwi APLNA: L—ézlzi'éLZBE/Zw Cit/um} SaQAAw 36M: gum + mm) " 7” 2°“ - 3 lat, - Q6 ‘1 Claw; + l-§1>&t.éd, )6 3. Consider the ordinary differential equation 4 Marks -* y”’+9y”+27y’+27y=0 E = 0 It is known that one solution of the equation is 3/1 = e‘396 . Use the method of reduction of order to ﬁnd a second linearly independent solution. No marks will be given for solution methods other than reduction of order. ,3: Lt’r '31“— M 3" ® M‘ l l 431' —3bL 3 “312 — a}: M -?>u6 3 (want u ,3 l ,3 lu— ‘3l‘*l‘*%‘*)0 Lt lu‘~3u‘>e t : (MH- bul+qu)6~3& u x 43%, \ ‘ j; : “Blu’bu'+‘lu)t 4— Ullvbul +0” -3 -= (um—w my —2au)(, * WM“ L431 —_ Lui|_ quu‘flq ul ‘7?” u I 3} I ‘31' + all“ 40” +‘lu)+Z lu~3u>+Bu @ : Um 6—33., : O J. UM:0 :5 LL: Clxz+ CAM, + C3 4 marks « 4. Find the general solution of the ordinary differential equation x2y” + 4333/ + 2y = 0. “\L'WoA -\ : Ld’ u, a (,2 7’"; © ,2 . 9—3; 3 d = A an} M A13 At An, AN, 4% Alt) R -% -2d —1 1 41% ’l = 6 ‘1': _‘3 t 6, 0 Li -A > L A1 317' kdil I: Tim. one be.me Z?— -—Z-2 , C e K&~ﬂ>+4e%ciéﬂ+23:0 (122 5” Ag .. 47 —7 .3 + 3d; + 7, a o _____, 6‘? (ii 3 ® Ckaru’ﬁ ers’nL Eemhmz /\ +75>\+‘L ‘5 0 =3 LA+13LN+\)=O :5 )\:’\ —7, , a; , HQMCL 6:- 6] a + CLe/ 2.} : Bu‘r 1:6,; 97 245le o -‘ '1 3Lw3:cie“°° iclcﬂm : 3+ 6; ___® (MAL—v, [m m a» mag) WV *3: lm >5 5‘: Mi (9 vaﬁ “Mm, M M 056'- \W\Lw\—\3 + 3va +3FXDLW‘3 0 .___._® we, MVA acbvdma mkgmf; = o 1) tm+zymm:a (D SHALL W\= "\ mag MA:-"Z A“ {m9 M54“wo{' Wok I‘WL \$012M“? W ﬁn): .C_\_+ ® )0 21/ 5 WMF¥6= r 5. Using the method of variation of parameters, ﬁnd a particular solution of the differential equation y”—2y’+y=ezln\$; x>0. / (All/VAL Umbbl: ovwl M’ LﬁltviL N N WA.“ W 2 6/21., die/3L Lab 2/1., Zap Mm) at t 6/ “HQ.- bbe’ :0 a a Wl : (7 \$6M“ \.,. __. le/ZLQMIJ. 8&1},th (Hat) C? “W00 W, (DI/3*: ~ bl, by»qu > W1, L»)? ﬁWL lvﬁeﬁrw‘z ‘33 PTA/3‘5 {Luv}: “ REL thcdm ='\<&m~>tl¢ 3— WM = +93%» +33 : sgLLx—uzm) L? “r Mlt’w) 7" 31M); (ii. .\_ DQQVDL 4... 3 MCE‘L I: ><>QWL’3-’ a, 3 at [,vac’t 7 gnaw 6L Pmch soLﬁx/M Is mag): {lg/5L(I~Z)ml> +0L1¢iUn>L~\> 4 ‘: ﬁll/d} COL (lzuvxbL " 3) ———~® ...
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