Math33BExam2Soln

Math33BExam2Soln - MATH33B Exam 2 Name: February 26, 2007...

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Unformatted text preview: MATH33B Exam 2 Name: February 26, 2007 Section: For full credit, show all work. 1 (12 points) For each of the following ordinary differential equations, determine its order, whether or not the equation is linear, and whether the eqrration is homogeneous or inhomoge~ neous. _ 1 (a) )7, +3012 = 0 35" Lye/5‘3: {65/ m {£3 oﬁmég (b) y’” +xy’ + (1203217)}: m x3 E: {"55 e NW a S: r g In: Km (C) )7” ﬂ ‘“ Sin ()5 +37) :5“ are: er" } ywé‘arwr e; r j, 2;;_‘:~}e‘,.‘€‘>a :35“; (d) x2)?" + xy' + 2y = sinx f} ‘9‘ ‘53 !L‘é“ Em my." Q Show oﬁeo ft 9min 2 (10 points) Show that if the functions 311 and y; are lirrearly indepenrient solutions of y” + p(x)y’ ~+ q(x)y m 0, then the two functions y; +312 and y1 — yg also form a fundamental set of solutions. 5.5% mng it) K§}r'\z§2 ﬁvga =- fg‘vﬁggkgcr airngg mfg git}; *3 ﬁve a "WW, 3 (26 points) Determine the genera! solution of the initial value probkem: m2. 0, ﬂit/2) I 0, ﬂit/2) y” - 2y’ + 5y 23% . WW “Rah 1”; g, Q g A” “We. Q k g iw m \m M; @ A” am js§2\ a aw E My; é if Q i a?“ g E 1: W . Q x a M: WM a w a ,, ﬂ 3% I E , m m C i w % m j a x f 1m? , 4w é mm; mm 1 A2 4.. E , x M x3; 3 a 9%, mm J g V; f 3% {fax urﬁswax ={;.i\ . WWW £6 £6 « ﬂ Méé w Xvi z 2 x Mm an a Am Q Viz XWE; 33;; 3%“: ﬂag m 4 (26 points) Given that (1 +x) and 8" are solutions of the homogeneous equation corresponding to xy" m (1 +x)y’ + y m x263, x > 0, ﬁnd a particular solution of the differential equation. X. . M C? 2"?! ?‘~ 3“ a; \5 w; {Egg—XXX x; l ﬁx v5 1 ,3” , i ; w woo m a: “k 33?; K K “)5. X' if 2% §“%E§x%xgtwﬁga~¥xﬁwﬁ\$x£ i ,5 X “"3 A a an § ’3 \5’ .2: _ 4% an” ‘ WW .-‘R E 2%? {Kg E 2 “a W} ‘32": "3% {3” f“ ‘3 7‘ 2s X: z .x I xii} 1;: 2‘ A?“ j [ﬁxé :2 f! X\a§ 1: W “X a? )1“ a); «u 3’ \H gﬂ 5:: Kg. .xﬂ n g >< % ,ﬁ "1% Vg‘; fixﬁwkxgﬂﬁx N5: Wig/52W“ 34*” j 1:3 r , “*5 3g “an: Y é‘V‘J’ <3: {fix Cit/<03; “we 3! x5” xiii 33" Rafa: a‘ﬁw i a. fig; ’ Hg we??? "1 f, 1 “gri/ ﬁx‘ xx w mm— £~ 3/2 \g 3’ “ E K :1“: géswa‘éﬁéz j? “gag” f: i: “3 «x v— : ‘72:: i “Lg “Hf ”” “,4” w» “m of; 36 3%.? 0%“ 22"" 5 {26 points) With no damping, a spiral spring is stretched .98 m by a mass that weighs 1 kg. The damping constant ,u = 7kg/s. If the mass is subjected to an external fqree of the form 4:105 2:, ﬁnd and identify the transient and steady state response of the system. How many times does the solution cross the time axis as t a 00? .. WW 3 v z . NE E ééfiéﬁ} K” as ii} ...
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This note was uploaded on 04/02/2008 for the course MATH 33B taught by Professor Staff during the Winter '07 term at UCLA.

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Math33BExam2Soln - MATH33B Exam 2 Name: February 26, 2007...

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