ma366final10-1Soln

ma366final10-1Soln - (1) The following vectors X1 and Y1...

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Unformatted text preview: (1) The following vectors X1 and Y1 are eigenvectors for a certain 3 X 3 matrix A corresponding to the eigenvalues 2 —7,' and —4 respectively. Find the general solution to the system X ’ : AX 5 pts. in real form. No complex numbers allowed! i+1 1 $1=': i ], ylzi: 3 ~21 _1 Wm 99: vol/um l7: *4»: toe-Mame ‘: l ‘ Wt): est»: 64th] t {NYCX LOW T'Lvl‘W/ a Wel/WNU- ‘8 \ eh“ = 31W?“ 2 {flu/oats + Mamet»)? : exém; W cm) " ‘~ » all he tthMXMMB $0: 82w“: ELEM" Wm 2 [6 ngtw {6:005 —€2b(m§tvié{mt)91) film: wt—tgmxfim : was =— gwm mt «Flew: ‘ : (mt m) + “mt 9W) LOWS? twat : Wm {QM : Wm Md“ 1 ‘ l ‘ ‘ (fistwg \ (fit-wt) + 4, (Wit » 9W3 {fiat-Eng \ P Fl? é¥~bfixfi @216 C WV + It mt : ezt gm; + 4 5*” C69? -—2<;Mt + i film) 3 (2) The Characteristic polynomial for the following matrix is 19(7“) 2 —(7" —— 4)2(T — 5) and the vector Y1 is an eigenvector corre— sponding to 7" = 5. Find the general solution to the system X, = AX. V21; 10 pts. \: 30\ get 03" weak 33’§3 %: A'Ari ‘3 “Ll '2 D] V: [4+ “’2 O ‘2 0]1[—12+% 4 4+}: [-4 4V4 “'0 .3 0 mg 3 b we “’3 e film 6 t 4:» b 5 {3 “2’ 1 ill? «ma-:0» o6?- /l~ aitl«s[ll+tl?l «m l U I“ X\ XJfi: eA‘VX‘: \ :: €4k<j+£g3g)x‘: €4JC<1+B£>ELE we): BM; éwm I 2 eat mm: eflnxfll tow %= a: v3 W 6 El 5 pts. (3) Given that X1, X2 and X3 are eigenvectors for the following matrix, find the general solution to X ’ : AX. Hint: To find the eigenvalue, compute AXi. A: —2 —10 5 X1: —1 X2: 2 X3: 3 —6 —l2 5 —2 2 5 X“ 2 (7 fig \ 242M: 2 y \ it: 92 g -1 i: +2 4]: MM 4, ~12: S “2/ ”l7+\'?x'l0 “'4’ W?" 2Y2 2 0 ""l r1+11fil> 4 ml ‘3“ ~l Mir 92 «i0 é [ Z X: 2. JDMD‘X: [*8 .= ‘4 21” IKE/1:?) Mimi “l, 41 s l gezwo “3 Z ‘2 g —} ‘\ AXE: [*2 “‘0 X] [ 3 \(7 42 S S 80: m: 152’“ , y: (H : fiwmomw \f; gym Mew exit) « JV = Cl Clfiwatl: ij’ée L 5 (4) The following matrix A has characteristic polynomial 19(7") 2 —(r — 5)3. 10 pts. (a) Find e“. (b) Find the general solution to X’ : AX. 521 A: 050 015 E SW Pm: -» (mfg? mm @06me NE: (went Alana-WW ‘ E3X=0~ So women 9%: civil“) Q, (5) A certain EL); 6 matrix A has characteristic polynomial p(r) : —(r-2)2(r—5)4. Let/X be a generalized eigenvector for A correspondinmz S/Give a formula for etAX that does not require summing an/infinite series. Your formula should use as few matrix products as possible relative to the given 5 pts. information. tum. is Amt} . amiwmmt we W thto (6) Find all singular points for the following differential eQuation and state which are regular. Don’t forget to justify your 6 pts. answers! 562(2x — 5)3y” + 533(233 — 5)yl + (x — 1)y : O \. flag) . x4 m Wt X1(2\<~r§);: 0 2? X20 w x: 4: s—a giWfi‘AW My X20 )8! mag? \1“ + XC‘XZCQWS}? \I’ + (X4) \{;1) fix” W) in) we r W » \/ .le Xsox‘eox W‘éw «r Vow, WVV’A'W) Sit/Mal \ae swift, Wise , Chute Vloifizmv 37> #0 , So xsolrgawqwffiw 3s s l2><s>tixl<2e>3~w bxs)jy’ +@,\’i':o V “‘33s?” i”) W3 i \ i \ \ V in. g‘ Myer) [M V i F \ \lwiismwgiiw giggmio "tows/ii 29:” so will Mama?“ Owl Wm, 9M2 (x): MWVSBV/lflg):g) (231.9% 30 Xzz‘i lé Wb'i CA (“git/vim $3M?! :9 i Manama” + n50 (7) Substitute y : 220:0 anmn into the differential equation 4y/l+.(x2+4)yl and simplify until you obtain an expression of the form 00 oo ZR” + '32:” + 2%" = 0 n=? '7 n2? 71:. g Where the exponent of :5 in each sum is n and the question i l 00 marks are explicit expressions. (You do not need to use exactly 3 summation signs.) Do not simplify further! 10 pts. mg #1 : DO hawxlfirl t.“ 5‘0 it & Vt ’ W: g: WW 3 z mam 2 wow i “:0 “4:0 l><s 4 [>9 h {>0 V, i 4T ': :0 4“ Wk“ = %D 46W) QMHX 2 g GINAX ' e be (we y” s mm) om“ = (mth (Minnow WM l1: ’2 , flit»; A9: (m) (M) (n+sz (>0 / ‘ h ZMH>OMX “i” :4) QMX‘A :10 w t i ,E.“ t“! S (RDQMXK'F ‘V\=\ “Maw! l‘ § j“ N i. X7 wt (8) In attempting to solve a certain differential equation, we sub~ stituted y 2 23°20 0mm” into the differential equation and sim— 10 pts. plified, obtaining Z 3n(n—1)an9§@+z —3n(n*1)anz@+z ~3nangQ+Z ~an1@: 0, “'20 “=0 71:0 n=0 (a) Continue the solution process to obtain the recursion re— “ k 1 be lation. W 3 /X7 *1 ‘9 (I R? w‘ igfiwanmau g. ,., v M H . go %W\~D WK :: Emma) 0m) QMX : %(~2+?{@u) mx + 3(»\+2V1') OM i W; i 60 N n ‘ as Z, [momma w» mm Wm M awn“ x a o “90 ‘ 3 (Nil) (Mb aw? [’BVLGH‘) a: C :ifl?i«%fi%%:ti >flm"; 8W1“) QM / _'_ Dal/EH /// :39 OMQ 1W QIM/ (b) Find the first three non-zero terms of the power series expansion for the so ution yl satisfying y1(0) = 0, 341(0) 2 2 \l‘ = 00mm mum :9 W0»: 90:0 \ 1 / ' / fl! = OH + mew «ma +w refine): / r v // l/\:(> ~. Q2 * Jam 0 2 O // z : K t it i L m i Oh’ ‘ 3mm 0‘ ” or 9 (9) The following differential equation has a regular singularity at x a: = 0. 9 pts. \’ ‘ gwemim‘ 132(373 + 2x + 1)y” + x(4:c3 + x + 6)y’ + (a; + 6)y : o, r» > 4 (@6ka & (a) Give the approximating Euler equation. (aware ‘80 We fifty 3495mm mm :0 we have; 4306: x‘nw , $3M: 4X19+x+g I fix); We 1/ j :9 no»: l , age»: 6 , Woké AWWflWWwfl Ellie/c" W035“ kw», {um : $3 110) ”+ 961%le 7’ + Y\(o)\{:5 :13“) x‘ "+ WW w :o/ (b) Give t 1e indicial equation. Let we“ , Wm We: swamxl’th swim ~+ Meg; MEN tn t5 val”: o (YZK‘PFE) XV?“ :9 New) : 0 A— imdr/OA(€2 graham V Kimmy =0 3? Vivi” 97"! (C) Use Theorem 5.6. on p. 289 of the text to describe/the expected form of the solutions. Do not find the/(feffi- n: ’2‘ (2:); $%E\ cients of the series expansions! If, {in be J23 be /’ D9 {/1/ s 50 We: oxllwxwylm >515: lax“ : Mix)“; QM + >< EM” .4) K M51 be u w : (1(th w + m 3 (OxC/oleoe 0 Mi) 6 pts. 10 (10) You are given that = 9:2 is a solution of the following differential equation. Use the method of reduction of order to find a second independent solution. Other methods will not receive credit! 3:23;” — 3553/ + 4y 2 0. 91m: $00 = X1 Lz-N—A X1 ’l3 4‘5 so vi 2 ml"; 3 l; = WM €49: 6%le : X3 11 (11) Use the 0f the Laplace transform to compute the tr Laplace rm £( f) of the function f (t) defined below. Other methods Will not receive credit! OSt<3 4, 3315. LUV: €26 6% (U; “k K‘mzéefi (if 3 2 Mt W : L e atngng \ _.\_ (2”9t (D) + i Esq?” & >3 ,b “S g 4 pts. 4 pts. 4 pts. 12 (12) Find the inverse Laplace transform of 252 + 4 A F 2 . 2 M (5) 8(82 + 4) fi + =39 233.4: Mfiéfifigm g 3 [$33 4A+B§+ cs: {Afiéfi C‘SWA’ 4:445 A?! M “L g ' i2=A+E :b gem “:9 R9“ 3 "‘ $4 0 = (1 x17 x£ " ( )7 age) (13) Find the inverse Laplac r eform of —55 2 2 4 F(s) 3—8 8 + ). : 8(82 + U965) ( \+ (.7, » k) . k % Mesa?) 13 (14) Find the inverse Laplace transfom‘figof M ( #1 4 Ham 4 pts. C25 / rug. 1‘0 Lia“) I I : F<s>= e - “ $5 (5 + 2)5 we / km mew We mm: 6’; [Mm/ea: L~‘(¥LQ): gm : 215 Wt). (“794 eater 8 pts. 14 (15) Find the Laplace transform Y(s) of the solution y(t) to the following initial value problem in terms of a and I). Do not find y(t). All we want is Y(s)! y” + 23/ — 4y = g(t), 31(0) : a, y’(0) = b. 1 1“ Where QED? 0, ogt<3 3‘ u/~» 97(75): t 3<t ‘ ‘ mm 7 — flxwv~fp { :9 edit): WM q”: Sim» 37w) «— Ylo‘) r: S‘Ym— 06—10 V’s S‘fl£)~ YMI 5%»er \f: We) 3&7 \I”+2\f«fl : Sill” Q9” l> + 2‘3 KW)» 10 - 4V4) 2 {Slag—4,) V5) w Qs~20x—\o 03(40): Wet :— Lt’>>> Matt) +a Mali) / Ge}: HOW) : LB) agar» + Mal/tag) as ...
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ma366final10-1Soln - (1) The following vectors X1 and Y1...

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