AK-MATH225F09-WS8

# AK-MATH225F09-WS8 - MATH225 Fall 2009 Name...

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Unformatted text preview: MATH225 Fall 2009 Name: Worksheet’S (3.6, 4.1) Section: 30 lﬁlonﬁ For full credit, you must show all work and box answers. 1. Find the solutions of the given second~order equations or initial-value problems. Use the method from section 3.6. (a) y” + 5y’ + 6y = 0, 11(0) = 2, y’(0) — 3 yic’r—E/ y] :f‘C/rt 7”: (“72"“):— yéa’): /<‘+/<'Z. =2 / 7 [(0):VZZ/VB’ZL:3 PL+SP+C9=0 ILL: Zv/L, ’ZL,-BCZ—K,)=3 (r+z/){F+3)‘O Li=z~a=~7 141%?“ (C) y” + 43/ ~ 0%,7 W) = 0, MW = 1 7 a y’lt} : "2!. sum; 1' ZZLCczsz’zi) (“NH-i=0 7&0); K50 7 [0)~ ZLL: !) (.‘L\"Li fizz." 2:47 C t); ‘5)" (2'3) C 250\$(ZJ:> +£9mé21> yét); lid, C05 ZZf) r/4L‘3'0LZi) (d) 16y”—8y’ “459:0 C rt: 6; T1: +30: éEéBI‘t 7 "6’ 3F: \. x ) .(ﬁr‘z \$r+IL{§;O I LCOSZ36)+’[SII)CB‘C) J v t m p : ‘5 I; “a” ‘ mo \ 7w) L/é. 2“}05 (3t) +Lzeﬁs";n 6.3%,) 32, F : (a : Fm— M—ww-W*W~mm~-«v~---~-vWm. ,V n / Bzx \ [—lszcm — ”'l-TtBL 2 2 Consider the harmonic oscillator with the second—order equation (ft—til- + 4434.511: 0. dt (a) Find the general solution of the second—order equation that models the motion of the oscillator. Use the method from section 3.6. Pi: , ' _. _, \ \. 7:61,— C : 6L z+c)1\$:éz:le: L't—{CQSbHS/ni) --WM_ ....... ..... w... mm... anknh‘ 1+qmv52-o t ”u 13 +45 "got; -th iyLIFKe/(OS \$2 I r‘ I; Z WWWWWMMM~NMHM mmmmmmmmmmmm ”Hill -, "\i‘ :— "' "(it »C r— ‘1' (b) Classify the oscillator. Unaic. FaiKM/Dﬁai (c ) Find the particular solution with the initial condition y(0)— — 0 and 11(0): 1. _ 2:6 2-5 1715):- VC‘t): 3/46.; Last —/4 alt—5:", '“ZLZ/C; glob-f/ézg; (cast 7L0):L':O l7’(a)—~Z,/ 2/14 H47.” ’ (d) Write the ﬁrst— order system that corresponds to the second—order differential equation using 11— - -- - a) 3-, -7 age—3: 57 Liz-5 Sﬁ'v )Find the eigenvalues Do not ﬁnd the general solution. ”4le + L! t 3‘": A 1&1 t rm 2 '2: A (f) Sketch the pése portrait afassify the or1gin. \ V /\ : “2:1: L iégifai ”bl/7L ) 7 :11? [a I)? heads/g) (:f; «it; > 0 {or érq Vader :ZVS) \j/ VC’C’é’J 1'": Cut) 3. Given the second—order differential equation ayll+byl +Cy : f(t) For the given f (t) below give the form of yp(t). f“) yp(t) EX:1 A (a) 3t” 24%» B W“ 4tz+lewe +C (C) est 0< éé‘b (d) “We” MW e): '7 (e) t6“ [#ch B)!1 b m 9t3+t+362t Amps-H3 tact +0 + 04 5t '5-[: -4: (g)563‘+e" 0/ c + (56, 4. Find the solutions of the given second—order equations or initial~va1ue problems. (a) yu+y'—2y=2t, y(0)=0, y’(0):r‘%: 7”: “7"”"L'7V27ﬁo/7h‘é 7,, 217/; ’+;,2’/27,o:?~f L , i ’50 r M m I r +r Z 7P'4ng7f/"4/yp’20 Asa \ (rm/:UF'" (2 + 4 “ZlﬂmH-B) =2rt~+0 r? ’ (2% L +; «2,4 t +1416) :2ﬁ0 7“:/<‘C +/ be '24:; A,Z/5;O .L , ., - —z :o/ =/ L‘é) 3Z.C/Z{Tfl"/Z.Z/C/tvﬁfé H»; .2: I Z , . . 3 ' ‘2 7m: was; Zip/Zack! 7'” .W -mw‘ _ v z [(0): ’ZZI7L/Z '- :’ L'Zt ‘1:- L )* ZI'HQ 55' a / 7 a [t -‘"J C- + 4:”, 7 0 LL:‘ZI.'K’//LL:‘ “U41+i.ZI:Z .L 7 ) 2‘ C 2. (b) ﬂ+2y=5€4t “3 /L|:"7E_' / K12'Z— ___________________ m “WWW dt2 [— b “ﬂwﬂﬂ‘WM—ﬂwp _, L 73: ’ ” +2 :0 7K6 ' ”+2, :gc’ 7k, 7h 7;, , 7/“ W by if, rZ+Z,>O 1&0 t7 '="'[email protected]) 2,; «Z 7? / f F \ // “'th rzzf? L 7/ =lé046> ‘ ‘ vL} -qb “‘4 I: M‘s 6136:4051 gaﬁsml/Z‘O l @045“ +29% :56 5 \ New-*6 99., 7h: K‘Co\$[/2\,’ﬁ) +ZLSJOKPLi) S -L,~L_./ F5, c y,,+3y,=3 NM Lt) r/L, 4050215) ﬁggjﬁtﬂgéﬁ ~ I I_ — t I l 7* 7h’*3i'~’o/ 7“"? W i 7,9 ”32/ >3 5. Find the solutions of the given second-order equations or initial-value problems. (a) y” — 22/ — 3y 2 3te2t rt 7:5 7}»; 7h//#Z/7/,/ﬂ37h;O/7h:c 7? ')F” J yf 1%),» 3&6 7"er «3 e0 7”???)3: L FJB>ZF+|> :0 7f 0L +ZMFH5)‘; r :3 J F: A t 710 I, ‘5 2,146»; 1L15- 148*; LIZ/H“ @525 \7’“ .2 14, 3t 5K7}; 9%}; MHZ/6119445) fVZEtg/Z +7::Z/7L6+ﬁ/¢J @55sz J5, ”=35? (H4 54 «350535 5% maimqgae)? :35th «3453 +524535>c 5355 W05“ *ML-E 243/3): 0 u _ g: “—227 ﬁzz't‘gc (b) 3/" —~ 21/ + 5y: 31t+e2t MW‘”“mw~~~w.~..,.%,.,__m m m :2) ; Mi,“ ,, , // 7h ' 7h “2% +575 0 75,: JJ’ 7F . 7,9 «2;, 45;, 35m F vZF +e~o 7 A~5+g+o< Zt‘ -Zl/LI-Zovth' I: at alt F ’” Z. *— Z C 7: ,4 + ZKC / 7f” :L f”: I 1‘24 a 4+ ‘5 it ”E 51152:» 5 2J5 We} LC Maj + {ﬂy/55%] (i ’ : 66/, ~ €t+élt :: tz COSZZ/f) +ESI\4[Z:é>> S vgc 1E_).C5,§z t +l’Z/9 15/; ; ‘5 t K &J_ 315+: ’ +/ , OKZ‘Z 7% /<‘ c: @421) (as \$1 > SOFA 5/7" =3 mags—o 5 5 . N : 5 A; 23‘ ‘% +5550 / if) r/L, c cpsZZt) +Zcha‘n [25) J g 6;?— +§£5L54 2.5 B’% 5 2‘9 S C 7 : 3‘945 Q I 217 7’ s 7:: 4— 35 ...
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