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AK-MATH225F09-WS9

AK-MATH225F09-WS9 - MATH225 Fall 2009 Name g I ‘5...

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Unformatted text preview: MATH225, Fall 2009 Name: g I ‘5' Worksheet 9 (4.2, 4.3) Section: 0 o l 005 For full credit7 you must show all work and box answers. 1. Find the solutions of the given second—order equations or initial-value problems. (a)y” —3y’ —4y=281n(t) 7“ 7473);» 7% O 755 77c175'3yc5! y’c— 954° r1'%p ‘7ng :ZCosf4-LZSMWS W ”’:—’.,:t/ I ‘ ' ”H'Vk”#7h*>/7H”3 \\\\ éGfPTSWD Fl + ‘:O 4’ [(3% CoS'iL“ p iSM-i) - W (Z:" /F'iL :rnv—ﬁyp “C if K \ C :é : Cosh- +15!“ t 7Ct): K. Cosf+ Kzsio‘ﬁ 7hli: :L‘ €056? 4/Léiof -, 3 sin L21) 7c . 7c” CED =505KZQ+TQQEEO 7(0): K, : i In». . N 7c%2tl/ 7c 'ZL°<£ Ztl/ 7g’:‘L‘iV<cw 7 H) r/CZ-sich "gLSIoCZO é—Zt‘LMCUZL :€Z‘t( 7 [1(1) ‘ Kl COSi‘ “ECGS(2'Q '34:} x:‘% J 7L: M at Viéwjptﬂk‘MZHD ' 3605620+MM 2. Find the solution of the given second—order equation. dgy (it? dt 71;. 7h”-5;h'v‘4;hio/7h=crt rz'w3ovqco (P—H)[r+|)=o PLH/ F:—\ W: /L;CL‘ H4} (8 M“ «3 d, — — 3_y — 4y : —8€t 008(275) 1m) 70.. >1011137o/ *Hyg: _ xét 6.21:1 _: —<6ﬂté¢f>[2t)+ ‘L (’S/C'bsxmzzt>> :_3€f)+21)t ELK) 2 , . \+2' 9:74éé’+2>t / 70 TZI+ZL>°<C( L)'t 3. Another physical example of a second—order linear differential equation with constant coefﬁcients is as a model of the ﬂow of electric current in an RLC circuit which contains an inductor, a capacitor, a resistor, and a voltage source. Let 'v(t) be the voltage across the capacitor and V(t) be the input voltage. The circuit has parameters L > 0, R > 0, and C > O which represent inductance, resistance, and capacitance. From the theory of electric circuits, we know that v05) satisﬁes (1% d1? LC 0— _ (it? +R dt +12— V (t ) LetIZ=1R=5><103 C=0.25><10 6,andlr’(t):12. (a) Find the general solution for 170). U) (0ng Vfav” 4— L5X103)ZD,ZS X/O‘h/ J, \/ : [Z V” 4 5CD ’+ i — V: )Z ' . OOV azgxloé 0.29/05 v” +5ooo¢’+ A»,oooooox/= Lascooomoo / 1 / w, ; x/h” +Sooov,+ Amooapomﬂ)! Vﬁcrt VF: VP 1+ vapﬁmvf F2 +sooor+ HOOODOO:O . =930®°°°° [QVSOOO i lzswoooo Aleooaood . VF: ﬂ‘ / VP]: 0/ «4/90 2 HWAZLRW - ’ ,————’~———_ -50<>o 23000 I": 5000‘" QOOGOOQ __ Z. HA=L§2//<}:IZ Z ’lDoo _ , -‘~lo ; rpween~wwiwww,_+Aa”e‘72W%?%EF 0. 1 I \ (b) Find the solution that satisﬁes the initial conditions 17(0) — 1 and — 00 fl.‘ i=0 v',[t)‘=—,’lbc>o L; c I o MOoo/(Z 'dtt’ooot v40) : Await/z; we) > — IOOOKrHOooML-‘o VHZ hi - vi! raw/.4, : HWQVQ be: :41 a : J. K, = "LUL; L / Z 3 )L, > ~ﬂ {Vuw~%Ai”“ie‘ff3+Z (c) What is the limiting voltage as t #- 007 (The steady— state solution. ) 4. Consider the forced but undamped harmonic oscillator with mass 1 kg and a spring Whose constant is 1 N/m. (a) \Vrite the second—order differential equation for y(t), the position of the mass at time, t, if the external force, f(t)=3cos(wt). M ”+JD7/VLZ7I'FL’L) / MZ-l/ [9:0/ £2, W 1 (b) Find the solution such that y(0)— — y’( ~ 0 and w M — What phenomenon occurs at this value of w? 7%}: 3Co5(%>::y[o> =27(o) O :2 W 711” also / 7% 7c - 7o +2: e 1 .1 111 rl+l=o / 14:1) / r>ii 3%)ﬂ5370121 ‘ \ \ z ram . @heszé:cos€+15/0Jc 7 ;0<€% ’-D<1 39 ft) Q /7o "=56 75: /4IQCDEHQ'+/4 Sift we: // . (X z , [-5 70 : TTCL it);LposijZer'tb-FVL/lops (a) _%é—El+o<é-§L; égL yéo); Ll+L1~O/ 1“ Li -; '01 :3 / 0(>L.[ i _: l vi. : I c I‘ll :1" Fa; at 65ng H iiﬁl Fr}. of éafiol 059 5,7. L1) ’ Z / ~00 ’l (d) For w from part (b) how many rapid oscillations are there per beat? (\0, pkfiVl OSC, Fer“ béxf; \$__...IZ——— :Es (e) Find the solution such that y 0 — I 0 = 0 and w— — .31. What phenomenon occurs at this value of w? szb=i I choﬂct'WCd cg 1,5‘: “"5 7//+713695‘6/)7(O’7I(O)’0 70: éhé c 711 /<. Cost+4€lnt ( PM (1)) r'siwtlestmmt) ~ 21315th + i (2% test 7C”, 7QI/+;c:3él:3é:i;t+L33/qé {2’0“}?le 70: “text: 7 / _ 05:25 4' ‘><.L Gt: \7 :/<' Cosf'l/(L5m'bl—ét5h) L ’ ~ '/ °( ti t i. \7/0‘):Za _O : L - 7c ti Z a ‘,’”“D< C t! ~ tL‘ 7 C't) : /L7/C0Cj3 4-5,n‘t-/:f®5 Zdta v¥w41ﬁc , L ' 'Co 'KLJC) ...
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