MATH2010_2007_sol. - MATH2010 — ANALYSIS OF ORDINARY...

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Unformatted text preview: MATH2010 — ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS First Semester Examination, June, 2007 (continued) Q1. (a) Sketch the trajectories of the following system in the phase plane7 indicating the direction of flow7 and classify the type and stability of the critical point at the origin. <5:>=(335><::> State the equation of any straight line trajectories. Determine the equation of any lines on which the slope of the trajectories is infinite and indicate your results on your sketch. (15 marks) €19,2nvalum (ll-=3. 7‘: ’l5 —‘> 50.6616, unstablz. €19an Ms (A (a?) Question 1 continued on next page. ’ TURN OVER MATHZOlO —— ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS First Semester Examination, June, 2007 (continued) Q1. Find all the critical points for the following nonlinear system. (111 =( —y1+y1y2 > Q2 3y2 — Zyiyz - 1/3 Then use Linearization to find the type and stability of all the critical points. Chficdpt -3,(i—52)=0 AND32(3(1-83§1:5)-52)=O :9 fir-:0 631-70 ==3 (Olb) 350 a 3'Qgrjfo =3> (0.3) #—-—-:_==——Il' 32: I b32—‘O Noisdlui'lml 32:la3‘25t‘32:0 => Di, = ("#551 55‘ 7/239. 3— 231-232 = " 0 7H -\ 2v 3 =3 Suclclle unerablaa D‘D [0'03 ( 0 3 (or ad #340) I,” : 9. Q Quiz-g =5 Sucldle unstable Dfla‘gx (-6 a» (den-6w) o I er<1= *i <0 ( (is-l: 2 “thief-46d =l‘340 :25 M Eocw) arsfnraL Question 1 continued on next page. TURN OVER MATH2010 — ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS First Semester Examination, June, 2007 (continued) Q2. (a) If F(s) is the Laplace Transform of f(t), prove carefully that df (t) dt The Laplace Transform of is 3F(s) — f (0) Then use Laplace Transforms, the above result and the second shifting theorem to solve for y(t) if ' %%Z=f(t)={$ withy(0)=4. 0?. «I. (16marks) PM = [9570(tldJr M 3+ ”WL/Z’T‘fil‘ f 54% c” an“ “1* a“ “Z a d " —s+ ,0 oo _S:l+ Au =‘SC .— er] — {Wu-5e ) v =43 usmS intafl mlim by la m3 = :> 5W5) ~3lo) : LUCE-Bl) “ mm st) -— LIJMH) ~35 _:> 5Yl5) -= 4 + g. I 5 ~33 z; : .31 ‘I" g; 4 + (25—3)“ (7! -3) :9 jfi‘) = flak?) Question 2 continued on next page. TURN OVER POM/J rang) +5 [e (peace = we) ~ft'0) . -ks flflfl'kM/l—é} - 6 Fl? MATH2010 — ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS First Semester Examination, June, 2007 (continued) Q2. (b) Use Laplace TYansforms to solve the following initial value problem. d2y dy I t . . d’y d—tg— — 23t- +y — f(t) —- e sm(2t) W1th y(0) — 0 and — 0 W5 : L / W) ant/716th? lap/06¢ (16 marks) L67! ) ,] 7—WWM y [flu/[Q e7” 52Y/5)-5//0j—//0) wQKSY/s) 7/00 7‘ >75) " F/S) W as): mm) = 2 (5-0 1+ 4 Question 2 continue: on next page. t TURN OVER Mac/c j : g + 5,16“ -_q érw24+sm2%) .2 2 4; Emmet +4cw2é) % 7'2? i7 = 6% £675 -z+z>’§*/cwM—4J awe-2+9) .0 I. t f ” e +t€ — ‘7 “‘2 : ...
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This note was uploaded on 04/08/2010 for the course ENGINEERIN MATH2010 taught by Professor Varies during the Spring '06 term at 카이스트, 한국과학기술원.

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MATH2010_2007_sol. - MATH2010 — ANALYSIS OF ORDINARY...

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