SampleFinalS08 - UNIVERSITY OF WATERLOO FINAL EXAMINATION...

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Unformatted text preview: UNIVERSITY OF WATERLOO FINAL EXAMINATION SPRING TERM 2008 Student Name (Print Legibly) Signature Student ID Number COURSE NUMBER AMATH 250 COURSE TITLE Introduction to Differential Equations COURSE SECTION(S) 001 DATE OF EXAM Tuesday, August 12, 2008 TIME PERIOD 12:30 am. - 3:00 pm. DURATION OF EXAM 2.5 hours NUMBER OF EXAM PAGES 11 INSTRUCTOR David Harmsworth (001) EXAM TYPE Closed Book ADDITIONAL MATERIALS ALLOWED None Notes: Marking Scheme: 1. Please be sure to identify your instructor above. 2. Fill in your name, ID number and sign the paper. 3. Answer all questions in the space provided. Continue on the back of the preceding page if necessary. Show ALL your work. 3. Check that the examination has 12 pages. 4. Your grade will be in- fluenced by how clearly you express your ideas, and how well you organize your solutions. -- -- -- AMATH 250 - Final Exam. Spring Term 2008 Page 2 0f 11 d [7] 1. a) Solve the IVP .1: 21—: = 2y + 932, y(1) = 2. d [10] b) Find the general solution to the DE _y = m(l — y), and sketch the family d3: of solutions (you should be able to locate the x-coordinates of any extrema or inflection points). AMATH 250 - Final Exam. Spring Term 2008 Page 3 of 11 a [4] 2. 3.) Find the general solution to the DE 3/” + y’ + 4y = 0. [10] b) Solve the IVP y” + 33/ + 2y = 4t2, y(0) = O, y’(0) — O. AMATH 250 - Final Exam. Spring Term 2008 Page 4 of 11 R [10] 3. The speed (V) of a wave in water may be related to the wavelength (A), the density of the water (p), the depth of the water (/1), and the acceleration due to gravity (9) What can we determine about the relationship from dimensional considerations alone (ie. from Buckingliam’s Pi Theorem)? AMATH 250 ~ Final Exam. S pring Term 2008 Page 5 of 11 4‘ Evaluate the following: l6l 3‘) £4 {(3 +813);:2)3} [3] c) c {(3t+1)H(t — 3)} AMATH 250 - Final Exam. Spring Term 2008 Page 6 of 11 [2] 5. a) Express the function f (t) in terms of the Heaviside function, where __ 1, for 0 S t < l fa) -{ —1, for at 2 1 [8] b) Solve the WP y’+y = f (t), y(0) = 0, using Laplace transforms, where f(t) is the function in part (3). Provide a. rough sketch of your solution. AMATH 250 - Final Exam. Spring Term 2008 Page 7 of 11 [9] 6. a) Solve the system I’ = 33 +11, 3/ = 4.1: — 2y using the eigenvalue method. [5] b) Sketch the phase portrait for the system above. Include the vertical and horizontal isoclines. AMATH 250 - F inal Exam. Spring Term 2008 Page 8 of 11 a _1 _ [10] 7. Find the general solution to the system with x’ = [ 1 :11 Jx. AMATH 250 — Final Exam. Spring Term 2008 Page 9 of 11 8. We’ve seen in this course that if a mass m were to be attached to a spring with spring constant k, and we were to consider the motion to be opposed by damping rather than friction, then the displacement of the mass from equilibrium would satisfy the DE mm” + 73' + km = 0 in the absence of any external forces. [3] a) Find the dimensions of the damping coefficient 7. [9] b) Suppose the numerical values of m, 'y, and k are 1, 1, and 4 (in their appropriate units), and suppose an external force of cos Qt Newtons is to be applied to the mass. We anticipate that the mass should eventually settle into an oscillatory motion of the same angular frequency 9; find the amplitude of these oscillations (as a function of Q). Show that we will observe larger amplitude oscillations if Q = i than we would if (2 were small. x/i Note: You’re being asked to find the amplitude of the steady-state solution to the equation .72” + a;’ + 4:1: = cos Qt. You may do this by considering the complex equation z” + z’ + 4.2 = ei 9‘ if you feel comfortable with that approach. AMATH 250 ~ Final Exam. Spring Term 2008 Page 10 of 11 Formulas (you may tear this page off and discard when the exam is over) AIV‘IATH 250 - Final Exam. Spring Term 2008 Page 11 of 11 For Rough Work Only... (you may tear this page off and discard when the exam is over) ...
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