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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 xcoordinates 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 steadystate 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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 Winter '09
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