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# mt2_219 - M E T U Department of'lMathomatics Introduciion...

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Unformatted text preview: _ M E T U _ Department of'lMathomatics Introduciion to Diﬁ'érential Equations Midterm Exam II Code ' -MATH 219‘ Last Name ; ' "Acad Yeéir 2008‘ 09 _ ' . Semester "Fall - Name ‘ . 31311618“ No - Instructm 1 -~ ‘ " Department: Date 20.12.2008 Time =.13 30 Duration Question 1 (2 + s + 10 m 20) Given the equatiOn 13(13):; = go), Where L(D) has the characteristic polynomial L(r) m r20" m UK? + 1)2 -!—' 25], and 9(3)::32— ——8+«em +4me" J’sinSm. (a) What is the number of arbitrary 001131;th expected in the general solution? Why? ”:5 ;- Numéwx afaréﬂﬁmrj Com/{5 '2' I"? (b) Find the complementary soiution of the differential equation. I" :0 ___, ﬂ H2 1—07.. Oiﬂwns : / Q)5.:—/+5¢ 2r "2' .. _ . ‘ gccmzqycczz+cge +6 (C4CoséxfC5-Vﬂ5?!) (c) Find the SIMPLEST FORM of a particulai' solution without calculating the actual values of parameters that may appear in it. - 3674:) sak‘sﬂ‘e/s‘véAQVh-om. 006” /_l :0)¥:30 WW9 L co): 0 3*(0 /)[(07‘f)7‘25'] Alena? 0L Ana/1.29M ww'on' Sahsﬁaa A[(0)Z.(0);::0 waM’) ‘ ‘ w ' 2 3 4' 7‘ W V 9" 2r; 5rn53< 21¢:- ms5‘x [(1%2z329x38)16)6Hnéz’ﬁCCDJ-s )va J 29C 5‘} ' “ ~5‘K3xeroi 2c 7’ZQAeﬁw-t‘ DMZ-99:6 {7551' 7494?”: M (31/53/5qu!“ (m #6631),w€oé4(21f2 ﬁe SrmﬂQé/Af. gCOr-m 0F %7(2))(' p . ‘ cmIAXZ-fAzz 37‘4374 4+4 3613:! A 9;) A I 7 -+x€ux(/+5~sxﬂn\$x+z4§a355xp+ +22“? (AJMMﬁAAéB 66.35%) Question 2 (5 + 10 + 5 2 20) Consider the equation x239” «Zary’ +2y m m3em. (a) Proposing a trial solution of the type as", where r is a constant, solve the corre— sponding HOMOGENEOUS equation. :- ;zx’" [r(kr/)-2r+2]x=0 =P(f“—/)(r‘—2)20 / hi 2%- ” M, a: { yfrerzz F212 X15293}: 2 (b) Find a particuiar solution by the method of the variation of parameters. propose >1}; (2):. M(x)9(+C/2C2)sz QI/(pA-W "amxch miésér‘éﬁ ¢J%WEHM Ljeewr f @261. 742% (1sz wzqz~““r 9c @ (A, 1‘22 (12 127(63 —-—=r “(4/2- 946% a: *M/2 firings/7e: m(;~(-—()<’9r-—e d,<x)\$(/~P€?€ ’ 7‘ <1! 6 a?” cl; 1 e -wmwmap 2 2)? 30 7c 2 3f at. 2P6”): 7c((»27€ +76 6? 3X6" (c) Find the solution which satisﬁes the conditions y(——1) = y’(—1) m 0. ‘5‘ (y): y; owe—”7P <20 2 Q? 7‘C2'K2'f‘26x Question 3 (5 + 10 + 5 r: 20) We want to construct series solutions of (:32 + 9)y” + 4my' + 29,: 0 about the point at \$0 = 0. (a) First of all, Without ﬁnding such a series, estimate its radius of convergence. (b) Then ﬁnd both two independent solutions in series farms. (c) Finally, if possibie, express your solutions in Part (b) in terms of elementary functions fmx (a) 5/2766 K0530 (5 an OQO/NAA’YPOcn/é ’33 3::3 {EM fermion has) (3 SQHQA 1.99:2” ”in"? F4: , QF 44.? (per—Man u3\\ ﬂcx)3n 2:5?»7?‘ 3f7j<3 (/9) Suﬂgwﬂn (D; 3’cm (w XII-“O no :9 4° )9 2 4 anh X114 Zth’k _. Z: nan—04,12: 7’- BZﬁCan> any; + h: o H: 14:10 “:29 ﬂ ‘4( ”f4n%2)c2 ]%:O or 2: [3(n+/)(n+z)67nf2 r) ... h n-- no ‘ M Wﬁw , 9L4» Maw/rmce Leia/44m? 2 f .. F0 1’ w: Chm " *3" 0% 2 e 2 , . ._ ”L C: Q2 2"“35'570 2 l 03—- 5 f {)2 "C. C? :f~~ Q( I H ”I“ f " = '“ 3 j P 3 I ( k .. \ f 3 4 1 2 _m Q C(Zé t: (”é") QC} I 02é+( ( 5) r ab 50%;”, m m 21!: / 4° 25? 4 Z "24' 2% 247 :25 +raoZ(*W)+QIX [3") yo”: 202,“? +l¢ o 241;] lczo 3 Ace kno 2 M f .. ﬂy: ‘L__ f (x): M (C) X‘”’* Z ( 3“) H M??? or g, 263-9 LL50 .....- .3 m It 2 ‘3‘” y (9: r #3... y w :29”) h Mae? 2 2w * s f 47 Question 4 (15 + 5: 20) Using Laplace transform methods: (a) Solve the initial value problem y"—4y'+5y=6(t~—vr), gm) =1, y’(0)=2, ‘ .-where 6(t) denotes Dirac ﬁfnmjon. £7”: (4)} 2 C/MO) ”£ng L¢7;5":;,j 17%!“ 27)}: “(7" S ~f‘P3 (5 i4§+5‘)Y(5)-:: 5—2+ ﬂ 2,5 0‘ 2 0&1‘. SIT j(é):€ COSZf'VL 2(é‘f?) / oi 62:; i 7-“ [iffnéjhj.f( -«-—p i r) (5,2) +/ 21.4 \$«2 —- S C" (05%, 5: £{stéjnf—mfl ”‘3" If f (SHZJ?+/ (‘3) Determine _f(t) satisfying the equation f ﬂu) sin(t — u)du = 1 -~ cos t. , ‘ fzwae'izrs/mvé / Cost? Wafffc'a}[{£m£j Lifmjw—u / .’ ' =P£ffzm§3~l~ 5-H :5c'3‘?+/J ...
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mt2_219 - M E T U Department of'lMathomatics Introduciion...

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