trialexam2_sol - CANDIDATES MUST NOT INTERNAL STUDENTS ONLY...

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Unformatted text preview: CANDIDATES MUST NOT INTERNAL STUDENTS ONLY THE UNIVERSITY OF QUEENSLAND REMOVE THIS PAPER mom THE EXAMINATION ROOM Zeroth Semester Examination, Ides March, 709 AUC I MATH2010 ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Time: ONE Hour for working Ten minutes for perusal before examination begins Check that this examination paper has 12 printed pages! CREDIT WILL BE GIVEN ONLY FOR WORK WRITTEN ON THIS EXAMINATION PAPER! This paper is worth 65% of the total assessment for MATH2010. Each question is worth 27 marks. You can score at most 65 marks for the whole paper. Answer as many questions as time permits. Pocket calculators without ASCII capabilities may be used. TRIAL EXAMINATION 2 FAMILY NAME (PRINT): GIVEN NAMES (PRINT): SIGNATURE: EXAMINER’S USE ONLY QUESTION MARK I QUESTION MARK COPYRIGHT RESERVED 2 MATH2010 —— ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Zeroth Semester Examination, Ides March, 709 AUC (continued) @ Q1. (a) Find the general eolution of the system [Iva/(Ni We“ -:~>~ ~v -9 f I”) 3 6+‘)()\~s) 4'3' ‘6 =) 124F4- :0 fl = t2; E’ut'ot’ofl'v , fl: )1, [—I-zt .—| ‘1) Q) s t—ZC V - ° =) (-Ivum -v‘ ~° $4 9; was)“ You +(7'Z‘)‘7”’° ’ Question 1(a) continued on next page. TURN OVER 3 MATH2010 — ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Zeroth Semester Examination, Ides March, 709 AUC (continued) (A) Q1. (a) Working spaceonly , I 25f __ “r. M” 506 M chf 2 m Question 1 continued on next page. TURN OVER 4 MATH201O — ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Zeroth Semester Examination, Ides March, 709 AUC (continued) M g : l4 (-ainc) )Cc‘“ 15* 5““16’) 4. 5 (($26) ) (will: -Csmczlg) ‘ @+g)azszt +C (A (3)9sz ~ (@W)[’ «516 +2sm1€j+664 K)[-1usll:.—s~\2£ (35' Li? gas It ) .— C(~c¢5Lt +1.90A3k) +3 (-2:s2l: ‘5'»:fo (c) Identify the type and stability of the equilibrium (critical) point at the origin. Q1. (b) Write the general solution in real form. \ I)- ~H—t ‘.:o (:‘MAD i; (vs): -— (~‘)S‘ :4- (= “(i-m) é/Zwrlgcf‘nm (“A (/3 flaw—.— (W m” rkJ/k o Wmctw) Question 1 continued on next page. TURN OVER 5 MATH2010 —— ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Zeroth Semester Examination, Ides March, 709 AUC (continued) 1. ((1) Determine the slopes of trajectories where they cross the lines yg # y1, y2 = —y1, 312 = —5y1 and yl = 0. Determine the directions of trajectories where they cross the line y1 = 0. Use your results to help sketch some trajectories. flew. i912: MSS‘JL ‘09)» ‘W/“C Jfl“ A" q i . , xv; 1 \ Question 2 see next page. j? "f" TURN OVER 3 \~‘ Q ‘. , - i M ‘J 6 MATH2010 — ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Zeroth Semester Examination, Ides March, 709 AUC (continued) Q2. (a) Rewrite _ y"(t) - 29 (t) + W)2 + W)3 = 0 as a system of two coupled ordinary differential equations, and locate the equi- librium points of the system. Question 2 continued on next page. TURN OVER 7 MATH2010 —— ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Zeroth Semester Examination, Ides March, 709 AUC (continued) Q2. (b) Determine the type and stability of the equilibrium points by linearization. 5" i5 0 l7 ag,‘ ) by» gtzé 9F —z-2y.~§;,1/ E:i 3;. (Q13?) ‘ A M Question 2 continued on next page. TURN OVER 8 MATH2010 — ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Zeroth Semester Examination, Ides March, 709 AUC (continued) @ Q2. (c) Use the Second Shifting Theorem (see Table) to find the Inverse Laplace Trans- form of 6—35 F(s) == s3 ' Sketch the resulting inverse function, f (t) 7:33-02 «£— [/45 fivm.,1v7’ 26% 2 5-? on 5w: ' f4) =- 0 E < 3 2L“ 4)t e 77 H liganmdq) Question 3 see next page. TURN OVER 9 MATH2010 — ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Zeroth Semester Examination, Ides March, 709 AUC (continued) @ Q3. (a) Using the Table provided, (i) Representing the hyperbolic functions in terms of exponential functions, and applying the First Shifting Theorem (see Table), show that 32+2 £(coshtsint) = 54 +4 cowl {[C-tfikt’): __.Z_..’ (3’4”)?! J5 o/(O‘MW‘); $255337? *’ cm“ I s L (st/f3” —/ W171 " ((5-01: wa‘w 3 _.: J, 29L++ ” (§E29+2)(5“I~L9+2) 31+ 7' SW. 4- 7 Question 3 continued on next page. TURN OVER 10 MATH2010 —-— ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Zeroth Semester Examination, Ides March, 709 AUC (continued) @ Q3. (a) (ii) Use the Convolution Theorem (see Table) with F(s) = G(s) = 1/(5 + 4) to show that the Inverse LapIace Transform of 1/(s + 4)2 is f(t)=te"'“ '-I / (#5; —-—- $4.9 ~ e 0/4/5911? ‘ {fle'wé’ta’r ’ 6-4% ’J‘t’it Question 3 continued on next page; :53 TURN OVER 11 MATH2010 — ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS Zeroth Semester Examination, Ides March, 709 AUC (continued) . Q3. (b) Use the method of Laplace Transforms to solve the equation y”(t) + 3y’(t) - 4W) = ~56“ with Initial Conditions y(0) = 1, y’(0) = ~8. 0A} 0/942): )7? fl» fKtyUZy’a?) z [ S 2X?) .- 53/19.? ’79] at»; [5 )7!) 7/0)] "(f 7/!) (SI/«35> fly—s +f—x “’5 375:; ' (3 440(90/7, signal; war In 3225— 2,5 wry, 7/: 52—3—25 (5‘-r)(s”P9L)" ; A + g , .9; v— a. . $‘~r nest (”'6‘ 5) {1- 925: 'A/fo’A— Efr—-!IS‘I—$r)+ C(I‘v’) s___=_.\ .' 525.: 25A —-é> A .7 ~t¢ 3=- y —S'= “5C __—_-; Czl GSA—:4 5": ~ I = A +8 ~57 B -= 2 S. 2’“): :5 J, €32 + ($39“? M (w awash Table of Laplace Transforms see next page. TURN OVER 5/57: - €f+z€¥ "“9 ...
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