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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AK-MATH225F09-WS12 - MATH225 Fall 2009 Name Worksheet...

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