math256_april2003 - Be sure that this examination has 8...

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Unformatted text preview: Be sure that this examination has 8 pages including this cover‘ The University of British Columbia Sessional Examinations - April 2003 ’ Mathematics 256 Difi’erential Equations Closed book examination ' Time; 2% hours Print Name ___.__._.._.___.___ Signature '3 Student Number____-___‘__;____. Instructor’s Name Section Number ‘ .UNlVERSlTY OF BRITISH COLUMBlA . . S Counselling Servrce _ Student Development & Services 1200-1874 East Mall 1 Vancouver, 8.0. V6T 12 THE Special Instructions: Calculators may NOT be used. An 8.5” x ll” information sheet is allowed. Show your work in the spaces provided. - Rules governing examinations 1. Each candidate should be prepared to produce his ilbrary/AMS card upon request. 2. Read and observe the following rules: . I No candidate shall be permitted to enter the examination room after the expiration of one half hour. or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. CAUTION - Candidates guilty of any of the following or similar practices shall be immediately Missed from the examination and shall be liable to disciplinary action. (5) Making use of any books, papers or memoranda, other than those authorized by the examiners. ('3) Speaking qr communicating with other candidates- (C) Purposely exposing written papers to the view of other candidates. The plea of accident or 1°rrgetrulness shall not be received. '3' sm°klng is not permitted during examinations. April 2003 Mathematics 256 Name [8] 1. Solve the following initial value problems. ‘ a) 29’ = (t +1)e”, 31(0) = 0 b) y’ = -2y + e‘”, 29(0) = 1 ,c) 2:] = --2m - y y' = :2: — 2y, 3(0) = 1, y(0) = 1. d) y” +2y’ HI = 0, y(0) = 1, y’(0) = o. Page 2 of 8 pages Continued on page 3 April 2003 Mathematics 256 Name —____._._____._ Page ‘3 of 8 pages [7] 2, Water containing a salt concentration of 0.01 g/L is introduced into a beaker of volume 3 L at the rate-of 0.1 L/s, and the well-stirred mixture is removed at the rate of 0.05 L/s. Initially, the beaker contains 2 L of pure water. a) Let A(t) denote the amount of salt in the beaker at time t. Find the initial value problem satisfied by A(t). b) Solve this initial value problem to find A(t). c) How much salt is in the beaker at the moment it overflows? Continued on page 4 m 3- April 2003 Mathematics 256 Name Page ‘4 of 8 pages The solution y(t) of a certain initial value problem is approximated using the Improved Euler method. With a step size of h = 3 / 4, the approximate value for y(1) is found to be 1.20. With a step size h = 1/4, the approximate value for y(1) is found to be 1.40. 3,) Using Richardson extrapolation, find an improved approximation to y(1). b) Estimate how small it should be so that the approximation to y(1) is correct to Within 10-3. . . - Continued on page 5 April 2003 Mathematics 256 I Name __________________ Page‘s of 8 pages [7] 4. Consider the system 93/ = ‘2m + y + my yi=3-2$-2y+m2. 3) Verify that (a: = 1, y = 1) is a critical point of this system, b) Find the apprOJdmating linear system near this critical point. and use this to determine the type of the critical point (circle all that apply): node, saddle point, spiral point, centre, stable, asymptotically stable, unstable. c) Sketch some typical trajectories in the ray-plane near (an = 1, y = 1). Continued on page 6 Apm 2003 Mathematics 256 Name _.____'__._____ PageL6 of 8 pages [7] ' 5, The current I (t) through an RLC circuit is modeled by the initial value problem i [ll 21'] + + C I = 5mm), [(0) = I’(0) = 0. Here 0 > O is the capacitance. a) For which values of C does the transient response oscillate? b) Find the steady-state response when 0 = l. Continued on page 7 April 2003 Mathematics 256 Name ___—____._________ Page]: of 8 pages ‘ [7] 6, The temperature u(:1:,t) of a rod occupying the interval 0 S a: S 2 satisfies the heat equation I ut=um, 0<$<2, t>O. The rod is initiale held at temperature zero: u(:1:,0) 20, 0952'. At time t = 0 the ends are placed in contact With thermal reservoirs (and kept there) at temperatures 0 and 10: . u(0, t) = O, u'(2,t) = 10, t > 0. Find u(:z:, :5). Continued on page 8 April 2003 Mathematics 256 Name ____,_____.____ PageaS of 8 pages > [7] 7. The displacement u(a:, t) of a vibrating string satisfies the wave equation ufl=u¢m O<a:<1, tl>0,, and the boundary conditions u(0,t) = 0, u3(1,t) = 0, t > 0 (one end is fixed, the other is free). At t = 0, the string has a displacement of u(x,0) = sin(57rx/2), and has no initial velocity. Use the method of separation of variables to find u(m, t). ...
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This note was uploaded on 03/28/2012 for the course MATH 215 taught by Professor Keqinliu during the Spring '08 term at The University of British Columbia.

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math256_april2003 - Be sure that this examination has 8...

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