AK-MATH225F10-WS11

AK-MATH225F10-WS11 - MATH225 Fall 2010 Name...

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Unformatted text preview: MATH225 Fall 2010 Name: \ Worksheet,11 (10.1, 10.2) Section: $0,011 0/15 For full credit, you must show all work and box answers. 1. Find all the critical (equilibrium) points of the given systems. (a) ((3—9: = x+2y X+Z7 :0 0‘qu :X‘7;:[ ) ~ , .: Z 0 fl = 2M, M”? 289‘ ‘> ”’5' dt "HI/1,30 w lag: _ . l/ X:O 6:913 x=~o‘ 7U”) f3 2. Given the system — = $+2y -4 2 %—§=433+3y A;(LI ’53) 941: [1+3‘IL AOL?Firithejawaizmmn‘ A,=-1:(/1 A 2977:?) ' 112:5 ' (411mm ‘“ LIV/0 “(3.70 “LE/)(LOZO 27:“ é:>;: 5) (H Li ~Z.)[\f.)n 0) A}_L1/L~S:O (Zvl’iZVz ‘Hv,+ZVL=o LA.+"3(IV5):O Vi} “V”; we} 1:, AF"! 42:5 ('V\)/ v'm /V’ Z i} 2‘31 ”(PU/2:97! 1 ><Z~t>~ t/ 1 ma 5% 1) (b assi the on ma \equ1 1 Hum pomt 0,0 0 the system. A :"i 4 0 Z— A2,: l) (c ) Find the solution that satisfies the initial condition (x (0), 31(0)) vector. 7[0)>CZ~1)1’Caza):Z:) 3. Given the system (a) )Find the general solution. «40% Eek 113— 1:0 A7,: MW]: 7'25 (ye-01971) ~(S?/»5)=o (:55 E (v; AL+ZA+ ’70 *Sv.+S\/&fo (A +/) 7-5—0 ‘— vl gape-(CA ~t z 1 I ) Classify the critlca equili rlum po A. A 2’24 40 ' @aafic 5 —>(c ) Find the solution that satisfies the initia{condit16n ('07; (1 0)“ WM" 7 O 4. Given the system ‘IL I dX 1 1) d—t=(—2 —1)X A1/Z-I)A AI: Z— -—Z,I-A (a) Find the general solution. I [/4"A.1)V7-'= E; F“ 3:10 “it ' l V Adel Z " ('Ié #00379 (Mi) b M) 1, l)[~—Z):Q A1+I=O _ (\JL) v, +vzj~o AZ. :M‘" 0 V2,: (”l +2} ‘V// Zl'l+L)VI>/ V’_’/ V’ EHL APE.» 61$ (1)) : Cost» + is in E \ 6 V’ H30) :ZCOSJUL Lgmflt) [VH‘L “(@gbfl- [Cosf‘ [sinks/3‘9 Cesw'r Slot zz~cj smti +£(C05t ‘ 5W5) ”er WMWW mm» m». "WW-«MWMW mmmmijw __> east M713 / Xif) 7‘ C ‘CCo$+‘Sin‘f> + C2, (iat My (b) Classify the critical (equilibrium) point (0, 0) of the system. Arr; cmibl/ a>~o [E (c ) Find the solution that satisfies the initial condition X(0)— — (— 2, 2) )Report your solution as one real vector. 7(0) ZC (vi) “a! > Z 5) C,='Z “C(+C;>Z (/9241, O w... ~..n...m~~..__‘m~__. .M __..._.___........._......,...—————-—— 5. Classify the critical (equilibrium) point (0, 0) of the linear system by computing the trace 7' and the determinant A. Use this information to match the given linear systems to their phase portraits. dx 3 dx 3 1 day (a) E — —a: + 5y (b) d—t — —§x + 1y (c) E = —5m + 3y @ 7 d_y dy H | H + = —3ac+y H | I 8 + [\3 {d 1 dt 2 dt 5? % ...
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