math2930_2011fa_week07_PS07Sol_Zehnder(1)

math2930_2011fa_week07_PS07Sol_Zehnder(1) -...

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Unformatted text preview: gt (HIM-H )% -=t>r-—-\$ Yl: (a 2 Y1: 756 it th". #7 VP: Ailey-i. 7f“: iAf-E ‘29 «Ai'e 7F": ?A€’f‘ .. 2A\$€ ’t «b24755 it 416214: ’16 #15” N 7?» '3. 2 . _‘ __ ‘ - - 2. ._-r- a ’- l- A; [L- if- +12 4- M7114 I 2e“? 3,5 m— ._..__—— 2. The characteristic equation for the homogeneous problem is r2 + 277 + 5 : 0, with complex roots 7" : 71 :i: 2i. Hence ycﬁf) 2 c1 e—t cos 2t + ege“f sin 21:. Since the function g(t) : 3 Sin 5215 is not proportional to the solutions of the homogeneous equation, set Y 2 Aces 213 + B sin 2t. Substitution into the given ODE, and come paring the coefﬁcients, results in the system of equations B t 4A : 3 and A + 48 : 0. Hence Y I 7% cos 212 + % sin 215. The general solution is y(t) : yaw) + Y. 10. From Problem 9, yaw is known. Since cos mat is a solution of the homoge— neous problem, set Y : At cos wot + Bt sin we 13. Substitution into the given ODE and comparing the coefﬁcients results in A = 0 and B = ﬁ . Hence the general solution is y(t) 2 c1 cos wot + Cg sin Lug t + gig sin wot . 9. The characteristic equation for the homogeneous problem is r2 -I— L03 : O, with complex roots r 2 iwgi. Hence yam : c1 cos mo t + C2 sin mo 25. Since in aé 0J0 , set Y = A cos wt + B sin wt. Substitution into the ODE and comparing the co- efﬁcients results in the system of equations (Lag _ w2)A = I and (tag — w2)B = 0. Hence Y : m—u- cos wt. The general solution is ﬁt) : 310(3) + Y. 14. The characteristic equation for the homogeneous problem is 7‘2 + 4 = 0, With roots 7" : i 22'. Hence ycbf) = (:1 cos 215 + Cg sin 2t. Set Y1 2 A+ Bt + 0152. Come paring the coefﬁcients of the respective terms, we ﬁnd that A : —1/8, B 2 0, C : 1/4. Now set Y2 : D st, and obtain D : 3/5 . Hence the general solution is = cl cos 2t+cQsin 2t * l/8+t2/4+Bet/5. Invoking the initial conditions, we require that 19/40 + (:1 : 0 and 3/5 + 202 : 2. Hence c1 : ‘19/40 and (:2 = 7/10. mm. W “0%) (m mas- o z u m 3’ 5.1mm am N/m .‘DF" “(5)10; pct/G) 5 /0Cv~h(3 5 aim/J WILL” “FELLJO M 3 Ade LJQ‘Z “I‘D/J‘sﬂmwz‘: 7 ~ 7 w _ - - 2 r r 5%.: *3/45 mztf/jéé‘ﬂ Ma): 0:4. at I: 51%, £43 wad _ _¢<’/a_/_= 23% 3:} Va; {007/ ' (14/175. 00 ?’/ﬁ"l¢¢‘/i) (m) A 232/57 U.) 4:65:27 “(1&5 77' {I Tﬂ/r'Z/Z7ZJ€<J4 3’7? I 0 29M m‘ﬁgasf‘wzo V L L = {H W” S Q1: /o“(;,/M5 fo 3 0 7 f I : @I/ejéym/ﬂscc Ma: 4C 2 , as” [/01 Q : A (a; gas. '5 7d 551'”: Log?" __ _ ,é 1 (ﬂora, :74 f 7 (W) ‘ /0 _ .603 (WM 1‘) 0? J i "WU/435111051; *idvﬂ <13 W“; (970') = Odo/5? 5&7 ]T.(a} Here m:1, 7:025, w§=2, F022. . « » Hence Lassa} : Z2: eos(wt — J), 3 where A = 1/(2—w2)2+w2/16 =41xf64—63w2+16w4 ,and toIiJIKZE‘TZ). 9 (b) The amplitude is Ampfiiude ‘ 18.(a) The homogeneous solution is 71605} = A cos t + B sin t. Based on the method .g of undetermined coeﬂicients, the particular solution is Hence the general solution of the ODE is Mt) : uc(2f),+ U Invoking the initial conditions, we ﬁnd that A : 3/ (W2 — 1) and B : 0. Hence the response is 3 a“) I l—w2 [cos wt ~ cos t]. 3 8 lg “0 +éctf+LL Caz/3,. €4.3qu 3 a) Sﬁcdj ﬂake. R V3. w (A: 5“pr £3) L.) w E 5' H3 6 Male /2/ ,7 f32 37 mg“ x47 ‘7 7249—} /% Lo 314-3 ’39? U 2359 2/0 /2, 21,9} 2 5’5" / 3 2 60 /¢ 2‘99?! #5" “:75” M, ,ér / 7 .53 H? .445” /7 ,35’ 2 .35 C. ) «CE, W! Ar VIA-,1 44 0min Am, (may 5744/?) - ' 3- 'I j ’4. m ‘ a J- '1- [K- #3 +1. .22: M WW we“ I! / 1- r f 6f 7‘35»: 15% ~ca I! E 5 .4 3 2 1. DD muaiéq 10/7/11 4:50 PM C:\Documents a...\hw3.m 1 of 1 %example for Math 2930 Fall 2011 %numerical integration of 2nd order equation %u"+.2u'+u+.2*u“3=oos(wt), u(O)=O, u'(O)=O %from 0 to 100, with plotting steps of 0.1 time! units u0=0; VO=—0, Y0=[u0 v0]; %initial conditions Tspan=[0:.1:100]; %times for plotting [T,Y]=ode23('sample3',Tspan,YO); figure(1) plot(T,Y(:,l)) xlabel('t') ylabel('u') R=0.5*(max(Y(800:1000,1))—min(Y(800:1000,1))) 10/7/11 4:50 PM C:\Documen...\sam91e3.m 1 of 1 function dy=sample3(t,y) w=1.; dy = zeros(2,1); % a column vector dY(l)=y(2); dy(2)=cos(W*t)—y(1)-.2*y(2)-.2*y(1).“3; m mus—gt _ V 7‘ ’ 21:; 27! _;__2_77'_____ __ Mm “Mm—#mmm‘M—twaw 0 Wm V971. ___ __._____WLEL§.M_MMV%6 W Y mm ' WWﬁ—éﬁoiijé’mmﬂ 5mL¢+ @36‘i'w_u_é)__mﬁﬁﬁw "1 V SM pd; ~ (9 ' ____ _______ __________ Wﬁriirdﬂw _,______MWﬁﬁwwjﬁwwwjﬁégm "fame 31“ ' A/3 4» v‘: Q Na‘fc +143} (/95 90:: smﬂjW/Qﬁ“ “(277)7me , ...
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