AK-MATH225F09-WS12

AK-MATH225F09-WS12 - MATH225, Fall 2009 Name: \ Worksheet...

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Unformatted text preview: MATH225, Fall 2009 Name: \ Worksheet 12 (5.1, 5.2) Section: fidlubo/D For full credit, you must show all work and box answers. 1. Consider the predator-prey system: dm 3 1 __ ' a? = ‘2“ a” “£0929 dy _ 1 1 _; __ L 2 i ‘L : a? * 41—2331)?” 7 é 7 A"? 90») (a) Which variable represents the predator and which represents the prey? X'- F’uefi’“ / "ate—e»ch ea», 0%)») Posi'f'i'vg <7_ ' PF¢7 / Fn'ééracit‘i‘on €,¢r~ml (- X>) flvkémsxvg (b) Find all equilibrium points. AX .— ' A“ t _L _-l_ :: 'Zt' TX'i'L/X) O 9")”é73 Z’Xy o J: v + F0 ; ‘ k" 3:0 \ X( a 7 M ' a) ,_L>:43 er / '3‘t730 fio/ (1' H'— zs I Z I 3. )=5: 3(1—33‘2.>‘:O 4?: % Xf‘f Liv-mmwjacobian matrix, 0135qu an equilibrium pOints' at: a“; =34 :7 ’5 x 9x 7 7’7 I 37— L) [,1 ,3) TS q Ll/5)4 _L7’ L 3'4 +é:o A; L~H Z dd (*9 EL 3:0 M t w) Y z «e «M Amen—em we“: We: Aim (d) For each equilibrium point, sketch the phase portrait of the system near that point. Sketchze'ach of the graphs separately. Continue work on the back of this page and/ or attach a separate page. 5% 4 7L1chC»Zw<al. 7r} ’ 7 , 7 , - 47,,.,7__x7,_=_o~, WW,,,$Qj_gfipfl__.4:9, . a *— w 7 WWW” H W" 7 L9 { —.> ,zj,>=,_z,;%7g)+ Dom: mféflg Ma) W "0 6""? ’wT'ts m5: 6 '9 W “4:: {tfi'k ) WA Fifi; 1 O A \/J 1 Z .. ; — 7 5 v m" m ' 0L$ {- ‘> N AVVMMWWMM~Nn 2. Consider a pendulum made of a rigid rod with a ball at one end. The second—order differential equation which models the damped pendulum is: d20 b d0 g —— —-— — ' 0 = 0 dt2 +mdt+ 15mm where 0(t) is the angle at time, t, measured in a counterclockwise direction from the downward vertical. The parameter 9 is gravity, l is the length of the pendulum, b is the damping coefficient, and m is the mass of the ball (we neglect the mass of the rod). For simplicity, let I = m = 1, b = 2, and g m 9.8, thus the equation becomes: d20 d6 , d0 (a) Let U = a and convert the second—order differential equation to a first—order system. M vzaa alt;- 01 53% = 426: : -0ll$’san-Zg—i—‘ a“: : ~fiflsin9 *ZV (b) Find the equilibrium points of the system for 0 S 0 < 27r. 0‘9 :0 st 4V —v~——azs57ae~2¢=-c> IE: ‘36 m \ ‘ vac? Viol ’fifislnemp $706.10 9”»Q,T (c) Using the J acobian matrix, classify the equilibrium points from part Tigw; <0 ) viz) ‘1 8’0: 50— ylo/OB: £1018 '451/4 fl. «fig v’L):/4 We 1 ., w I Act (“"6 "Zr/Q KO Act (tag «2%) :0 (—Al («2%) +01 3:0 5%) 52—26 *‘7 $~=o AZJrZ/k +fi3:0 AZ+Z&-°HE~=O A: -2: , -21. 11774???) 2. A“ z A; "1+m>o [LCD/C?) T5 SplHKl Slfit A: '21: 2 Hal”? / AL: —}~WLO Mm , -V,._w__._ 7,, (d) What would happen to the motion of the pendulum with the initial condition 0(0) 2 g and 0(0) = 0? (Will the pendulum oscillate? If so, will the amplitude of the oscillations change?) In qurxsd’ Portrc‘jfé/ “poio’é'l‘bq LO: 5/9/24 60 boar/ls tLW, £20: “Braid/‘1 foi‘n‘b «if ’67‘L1(_ origin, m This Wins «the. Pwaolw oscilwcm wail: t . édfirffgiafii far a me Q l g '5': 0 c, 2“: 3. Consider the same predator~prey system from problem 1: if” _ _§ +1 dt _ 4‘t 4” @_111 dt‘y 6y 2‘” (a) On the entire any—plane, sketch the m— and y—nullclines and indicate the direction of the vector field along each nullcline. (c) What is the behavior of the solution with initial condition Y(0) = (2, 1)? (Use your additional information from problem 1.) TM, sold foxy 007“ S/D/\r\c\i flak/Carats fl»: EFat(/,3), ...
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This note was uploaded on 02/08/2011 for the course MATH 225 taught by Professor Staff during the Fall '08 term at Mines.

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AK-MATH225F09-WS12 - MATH225, Fall 2009 Name: \ Worksheet...

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