# 219M1 - 1V1 E T U Department of Mathematics Math 219...

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Unformatted text preview: 1V1 E T U Department of Mathematics Math 219 Differentiai Equations Exam 1 19.11.2011 Last Name 1 Signature 1 Section Name 1 Student No! Duration I 90 minutes 4 QUESTIONS ON 4 PAGES TOTAL 90 POINTS SHOW YOUR WORK (10+10 pts) 1. Consider the following iﬁitiai value problem: , 2\$+2 4 m m—MW 4mm. y+\$2+2m~15y m2+2x~15’ 1’” (a) Find the intervai on which the givea initial value problem has unique solution. Explain! I K OHOL'HCDVN is ck linear“ equﬁkﬁeﬂ- TMQ (“othQMOﬂh "The. cia‘ 1K+2 _ 200+!) L4 1 L aha >12+ZX~\5HM LX*53£X"3> ) E+ZXM5 {x-xv5)(><-3) gonﬁﬂuooﬁs on 44%; Irﬂ—er‘v‘cus (w. col—*5), (15,3) ancé (5)00.) - “Theo Tnixﬁoﬁ Pon+ x93; ' {5'} {‘n joke; ;ﬂ+ar\r6\\ CgimD' \$0 “H/‘ﬂ QQMQEKOH has; LLJ’NIO‘UQ \$0Mwa on we; :nmmu came) _ (1)) Find the unique solution. ‘ ilk—+2 a.“ K j x?- +2X~N5 :m'ﬁﬂwcdﬁmns {ad‘or‘ HLK§ 1:“ a “5:3 «— \$+1x~15 ‘ r 2‘ _~ ~ Hu\—¥'§‘P\b 5041A s)?ng me/x x +2-X ‘5‘ Cxl+lxw15>a)+@~x~v2>j __—_ 1.1+ 1 . J < Lx+lx—\5)é > :—H C>¢Q+21x4553 :1 W24 y: A: C, When young) 3:40: L154— EWIEDHZ) : W\QD+CL iogwurx x>gn 2 (20 pts) 2. For the differential equation (256% — 31269) dm + \$9er + y) 65y = 0, 2: > 0, y > 0, Show that ,u 2 ﬁg is an integrating factor and soive the equation, _ ‘ j +0 Qri~ ( ' - \$- eke—15> - 3.1.. W 3 M 0H1 9x3- xomk \ M £3 5 ‘:J . ‘ a b 2.... 54' a ‘ a3 5 e ) e: LH’Eﬁ) 3(1Km Lj >:___.__L<ea +361 3% L X's—— ><:2,— )8“ E5 :3 :21 , e f i a a > 9 Q CW5) ) W m Q L +3) 3; 3 (Dix w L - >4 w m 2,. 9:9 _ >4. ‘3 X X; LA _ I Q '% %mﬁeﬂuqﬁmx bammﬁﬁ aMm+ w V1 0 nﬂ M " ﬂ“ " thh muiﬁfiiaci 3’33 \$0 )2; 12:» an \m'i—Qal 3 "Fa (Hm: r": +uncﬁ¥i©m LX335 H 9%” K 294 (Salim) QM eta >¢~ :3 b 2 \$3 ‘ MC Jrei, “‘1‘ A f B m CLVN\$L \f\( i3 Cﬁﬁﬁfb n 60 [A L3 W0 . "a .3943 éolgi'HOﬂ C4 *kﬂ CAT—G‘Lc‘lr‘ﬂm4-fqi QMQQO\‘*W\©tq r?“ SQS >¢+Mmcq )¢ (10+10+10 pts) 3. (a) Find the general solution of the differential eqnation y” ~ 3y] _ = 0' (ll/x QWQQ+Q.V:§§Y{C- 610‘ Lieu—Fran: V‘Z—m E F‘wa :2. CF—H) Crwj-l): 19* f 3"?" .,\ mi: awe ﬂamerdel ﬁaluﬂ’laﬂi jC—t): Lie, nuczde (b) Find a suitable form for the particular solution of the differential equatiozi y” —— 3y] — 4y = 5 —§- te”t + 2sint. (DO NOT EVALUATE THE COEFFICZENTS) jcf>l5+££%~*2_5‘fn~t W 36%) man; an 04 [+3 clafllvdcllvai (Loam lane unﬁt-Pram as) an linear“ ﬂ ._ i . “"4? #6: .u @mbimOCﬁOr‘x 01C ill, “EC”— , a I 3‘51an C4333th- I—JJC "5 ‘~ 1 Qt 0c wdmﬁis Silage; @ ['3 m EOILL+1©H o—P «Ll/«e; M «A. \Lt . 1 -~ s h; aqmaJc—‘QOFN I NQM EMOLJ)\CL +vﬂlcweém \$4 l—t; cam l-EQ, fairijqwi‘s , L wet: ‘ we N ‘ '56» be: Utﬁ :; Ari—Etc; +<A:e erleh‘ETTECwOfg‘E _ (c) Use the method of variation of parameters to ﬁnd a particular solution of the differential equation 3;” w 339’ —— 4y 2 s3? dc - WV 4: _t' 9.4;: ms / LL+€L \f/ Udl V‘t’e’ \/ we ‘ 2+4: I m-Jc- are at were “(2—0) :5? she LLA—Lle. V ~43 L133, If “we / Ll r _,I. I i :3? ca, tom—me; +169, waive \/ ’f i / 3e w ~—- ' 9% :e n We“ : aka waif»; Mtle—H veequ (BFBQ (A #126. V—eLlaQ H ' Eﬁ wt / HA”: 3+: _ S ﬁfv 1) me. Lb +Lie, V .... a, o.) t “+3 l “We ) \(l__ i ejﬁ , 5] “at \r: "fr—"(2. _ JC / 55*" / - ‘Fb Mine ' 3i: 3%: 3+, 4 (20 pts) 4. Find the power series around 329 m 0 of two linearly independent solutions (that 155 solutions which form a fundamentai set) of (9 — \$2)y” + 6y = 0. w ‘ GO “*5 f 09 h“?— _:c\ Q ar_ :nOKm‘M. 39;: ﬂmvoahﬁ Satan” h ) “(1:4 / ﬂzb m :3 (0Q in“ a -—Z , W— E ﬁmLm—-x>o\n5¢m m 2 mCm-1>CKp>4 "’r 1:: n "‘2" mz?’ 2m” 00 a Z: Bonwab (xx-4m} Qnﬁlx’m _‘ 2 ﬂux—4) ﬂ “2*:0 W3?» 00 hi?) 9:) r\ Z (- 00‘ Xn . I I ‘ 54- *— n n— n ‘ 2 E) Lm'ﬁ‘l) Ln+\\ Qhﬁl XF’V \ﬁ‘LZ» msz, 00 ﬂ ﬁzz, 00 ‘ ﬂ Q ‘ -;C3Q2ﬁr0\o)+é2(30\3ﬁ~0m)x + ? 3C‘n—‘Y‘33LY‘T\) h‘ﬁrl m mi; ﬁnL.h—~a3‘9\b+é>qm1x:mo CL, oils—"3.3:. a3;m—_ML_ O‘n Chm-v1), w. 3 Ln-WZBLﬁWﬂ) W 3 Ln~+szn+13 M LanD) Lnﬁ-Zr) Ohm : William—— G‘h SLerz) LrvHB 3 v 2* (3490:.— xwﬁx "‘“ “H M" ‘ mi - .110 h / h WEB H) 0‘5 \$0 > 013mm“ ) moo: W W5 2 h— m wig/223113 = in <3 (Ln-\) “9—— 339an) 3 {Ln—3 (ma 3 (236%an in ~— 5 lﬂﬂ} M _ OKO 3 a m A 3" <3) LZAA~4> f) Clnﬁ) @3ch u) (BM-3') — ﬁ ‘ Um 5) go 3 n qo w W 3 (Zn—aha) '_ 5"“ A r-3— 5“ a — CZWWEDLZ““3>CDH~\> ...
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219M1 - 1V1 E T U Department of Mathematics Math 219...

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