f02-fin

# f02-fin - Math. 224. Final Exam FALL 2002, DEC. 11 David...

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Unformatted text preview: Math. 224. Final Exam FALL 2002, DEC. 11 David Gurarie Name 1. Modeling: set up equations, classify them7 indicate solution method (a) Set up a model of 3 migrating populations (x,y, 2), so that 20% of 1: migrates to y and another 25% to z, of y 15% migrates to at and 25% to z, and of z 10% migrates to :13 and 5% to y. Besides as grows at annual rate a, y — at rate ﬂ, and z decays at rate 7. Write the matrix of the system, and explain how to solve it. 3:, : (obi/81x +53 +31% oc—uis' .Is” ,4. ’) 2i 3 ~27: +(F”°£’)3 “05% A: .2 [5”.‘9’ .05? .. : £253; + . 095:3; —(¥+.l5\)2 ’2 S.“ “2%.. {64%; 2 é? ELAQ/anA/nﬁs A (b) Set up a differential equation model for temperature distribution u in a uniform rod 0 < a: < 1 heated by a stationary source f = 1, with boundary conditions u (0) = A (ﬁxed temperature), and um = 0 (insulated). Extra credit for solution. ‘ —» :; i 0 ‘5 X < i. ufx? 0(ny ‘52 ) g (B (O W4 : , a, 3 n . a _ 9 l ’0 J' x i ' ""‘ ‘ “W m . _, .M H 7 , 7 W7“. ., V _,..__ #1 "7 i / gghiﬁ’w : (LL, :: gr. 4-» (3/ u;- (‘2 X £ i.” m.-_..w..... Wan {D W \ if..- an _______ Mama ,4 (c) Set up a model of ﬁnancing loan at interest a, and payment rate p. Sketch solu— % tions, explain the minimal payment rate7 total payment and the overpay factor. I ((1) Write the Duﬂing type oscillator with force f = as — 3:3 v .2, as 2—nd order equation and DE system. Formulate the energy conservation law (compute potential U Sketch phase—plane and equilibria, indicate their types(Hint: use critical points of 2. Linear oscillators. a Write eneral solution of forced oscillator: 2'3 + 22': + 4x = 1 + e“, and sketch g phase—plane trajectories and solution curves. Explain what happens to a: (t), as {L' *9 OO ,_ i t t. i NM _‘ I: \y "o " E) ‘l‘ '1" Q {’9 we“? "t: +58an ( . ‘w/ A. ‘ if . Mn] (b) Consider periodically forced oscillator; + 4x = sin wt. Write general solution and lVP-solution for so (0) = 33’ (O) = 0 (Hint: sin wt = Im (4%)). lb : A2 4. (f (c) Explain the modulation and resonance phenomena (determine the resonant fre— quency w), plot them. Find the beat period for w = 1.9 cw w 1/- - ms 3+ g; 3: ﬁ .9: [H = WWW—WWW.“ : 4'“ ‘Zszgaf’i m3 i 1.: 60;?- = ,0: (d) Extra: Do part (b) with damping coefﬁcient d = 2. Find the response solution, plot its amplitude as function of w, ﬁnd its maximal value and the corresponding frequency: , ,. z] M_________,._a.. j o o a 47-3 h=* ___‘ ._‘5___~_1_ M_L I Z (s+a)(32+b2) L12+b2 (3—H: 52+b2)’ (s+a)(32+b2) __ 0.2+b2 (52+b2 s+a) (a) Solve ﬁrst order equation: 3/ + y = cos (2t), y (0) = A. Sketch solution. 3 S A‘ (3“)X“) ‘-’ M4 \$ st>- “W‘” M 52” “ “(31+q36w) 5w " J- ) + 49: ft! S+ 1’ \‘W/t/ \_ 5,. am ,, (b) Solve the oscillator problem: 3/” + 4y = e‘t, y (0) = 1, y’ (0) = O, by the Laplace transform method. Sketch solution and explain its behavior at large t. 4 E; __ ”' "" + T— -— (5+6}(51+‘{) SW 3‘ “9" mm- 2:. > saw > ((2) Write and plot the fundamental solutions K (t) of DE 3;” cases: a = —2,0, 1. Write Laplace solution y (t) l 3) as convolution integral (Hint: K = £‘1[W13) ) . ,1" _ '4 :D {.3— l I” g _________ ‘ 4 :7)?— ’S—2S+L§<§ +ay’+y=f(t)in3 for arbitrary source f (t) (for all 4. First order DE. (a) Sketch phase-line, equilibria and typical solution curves for modiﬁed tion: y’ 2: y2 (2 — y). logistic equa— (b) Could you solve it analytically? Outline the method and ﬁnd implicit solution 1 * 4(-2+y) (Hint: 1 _ _1_ i 92(2—31) * 22/2 + 4y if” #1,. f w *1- ‘i‘ ‘2'“ :1 xi “‘ C ‘i‘ "t £3 4’ {2 m3 j WWI; LIN“ m Wﬂ m __“WWWMWH.WMMWMWWMMWWvMWMW (0) System (a) is harvested at a constant rate b. bifurcation value(s) 6, sketch the bifurcation d b below, at and above the bifurcation 6*. UK to survive ? Write the DE model, ﬁnd the iagram, typical solution curves for Which values I) would allow the species l 5 i F l ,2 (2, x A: 2,5"2 A ‘1_ 1 2— .. ﬂ , identify them. Choose one of e linearizations and approximate A l m (1 ~ Iv/a — 21/2) y(1* C17/2 ‘y/a) (a) Two plots below correspond to a = 1 and. 5. Competing species: { -2x_3 l _ .J 3 ﬁnd and classify equilibria, writ equilibrium them (a = 1 or 3) solutions for each 1 _ _ ll . _ q XYZ/Dzzzzwzzznwér 5,. «4‘4: {/KZKKKKKKKKKKKKQ.WH zz/uzgzggggr(ge,y.x \ -zzzzzzzxgzgege(e?nn \ {Kara/.{a/(K, : ,z/uzzz/zz («gr/.1 q grgzaz/(z «(rrly . \ . {azwzz(£z(x«zw‘._xv KKK/(1(z (913%. \$64» ‘VMVVQVMKLKWX,,FRNQA («nan zztrzwmz (,1. x . zzzzxggw/lka.) .a..a wzz¢m(xz,7vx\,.. RV » a a ,4. a a V; .,./ z a...» .. 4 . a a. a g - . * n r. 4 a a- n .rr;r;»v, ,\ [Ila/.zxrkie + xxzerraOvarrwzrzﬂ * -21. 131 y, .<\~:\.,H1.J.>,. 411.1 r.» z. z _, u zgxmmxtfkrﬂrrrvrrw, ﬂ «\zxwxv1rtm4‘TTHTTTWTTWrzw tailllf J».le .1..in «libiﬁu . . ﬂ .1 i _ \$25.? 5 Z Z m \ 1 er“??? i if ,. [KKK/3k). «(ELL “ h 5.; ii .i. i i f ¢zz/5_”:m:p: . n yep/P512??? Z f ( . \$ l¢¢¢2 J). x. x z. ¢ y£¢¢¥¢1¢¢x<9wy , , *iu. .lv/v.¢¢H/» I»! 1.1 7K, ‘ a - relive! [will ,. J «3 ,A *1“ lvlv.¢¢u¢lvt¢.¢..\k.,x, ; f; . ,+ . . \ , L _ T¢I¥¢ll 4.....Yk in. c: J , 13,, 15,, .a. , e . H x H i . , ¢¢,¢¢¢¢\k¢¢.§\y,»lrza# :4 . . \ V . 1....vlynllelwlrll i. >12) ; 2:»; 2., * 2 . . \ \ . . _ o @ , i L; .p 1.7 r v » r . ) .vavxlmivllvnl .. ,\ r H _ xx») IlwlvllTlev I. litigilr llllll.l’ u i . h m _.L AW .4“ z s s 4 2 1 B a 0 2. O n 1 2 1 1 x o u a o I r! r In» ‘ II. 0.! 2m Ammo Mix-awe” nutcme 1%“ r W ’L 1. 71/2: ) (b) Sketch trajectories and explain the coexistence of species bl ed on your phase—plot and equilibria types. (a) cam-\$4 Q‘i’ (c) Extra: Discuss the bifurcations as a changes from 1 to 3 ( ﬁnd the bifurcation value. based on phase-plots), 10 6. Dynamical systems. (a) Determine whether a given system is Hamiltonian, dissipative, gradient. Find its Hamiltonian, potential, or Lyapounov function. (1) E11) (111) 2 ,_. : 0 _‘> O‘VJ-c! q —- z +X 4.- 3 /3 v )0 6M 1 (Z 3 53 ax J i 2, j in l VF:D "—2) Lﬁm' " 31.. £7+£F.2X\‘ MQW’Q/M ' 4” a —- E q 2 , d 6, VP = (1:83) +5534“ 0 N (b) Identify the above systems with the plots. Mark equilibria on each plot and identify its types based on plots (Hint: for system (ii) see problem 1(d)). g lax ._ ,é’x +53, #0 ﬂ PM, 42025, -l ‘~ &“&(‘§‘F<>\$—(—(—(—(—(—‘-kk?kk(/" \sssssssc-t—Ll—eA—kkere 11 % 1 Z (0) Pick any one of the systems, ﬁnd its equilibria, write the linearized equations (Ja— ' cobian) for each equilibrium, compute the eigendata, and determine their types. Compare topart (Extra credit for additional systems) ,A a‘>_i;g (i7 _ ‘1 1. CO)“ f2. 4.? a _ Mgr w Kata/({LQ A’LL 23] Laval IJz.‘ ﬁg ((1) Extra: Show .hat system (iii) has a conserved integral (ﬁnd it), and explain how it relates to trajectories. l W ,,,,,,,,,,,,,, _-~wmm'”:‘m"r"'ﬁ“”‘= ““““““““““““““““““““ W (Roi) l T _. Happy holidays! 12 ...
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## This note was uploaded on 05/09/2008 for the course MATH 224 taught by Professor Hahn during the Spring '07 term at Case Western.

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f02-fin - Math. 224. Final Exam FALL 2002, DEC. 11 David...

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