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AMATH 250 MIDTERM 01, FALL 2010 VERSION B Student Name (Print Legibly) (FAMILY NAME) Signature (GIVEN NAME) Student ID Number Tutorial Section (or tutorial start time) COURSE NUMBER
COURSE TITLE COURSE SECTION(s)
DATE OF EXAM
DURATION OF EXAM
NUMBER OF EXAM PAGES ' 50 minutes AMATH 250
Introduction to Differential Equations 001
Monday, October 18, 2010 5 (5 single—sided sheets) INSTRUCTOR G. Mayer
EXAM TYPE Closed Book
ADDITIONAL MATERIALS ALLOWED NONE
Instructions Marking Scheme
1. Write your name, signature, ID number and tutd .
rial section (or tutorial start time) on the cover. QueStlon Mark Ont Of
2. Answer all questions in the spaces provided. Ask cover 1
a proctor for extra blank pages if necessary. Show
all your work for full marks. 1 11
3. Check that the examination has 5' Single—sided 2 8
pages.
4. Your grade will be inﬂuenced by how Clearly you 3 10
express your ideas, and how well you organize
your solutions. Total 30 AMATH 250 — Midterm Test 01 Fall 2010 Page 2 of 5 [5] 1. a) For each DE, indicate which type(s) it is (separable and / or linear) and which methods
apply (separation of variables, integrating factor, and/or undetermined coefﬁcients)
by placing check marks in the appropriate boxes. Types Methods
DE Separable Linear Sep. Vars 4 Int. Factor Undet. Coeffs. J
@ = y + 1 J M d1: V
Q + 7} = 353/2 V
dzz: t l \/
d + cos y = 3:33;
d
93% ~ coszv 2: y V V’ 92f mass F02. gem sax CoataTmmszmmmw d ,
[6] b) Find the general solution to (T: + 3y 2 2 cos :1: — 5€_3$ using the method of undetermined coefﬁcients. *3“ . ..
{El “3k: jit3uj=0 ==> wkﬁ‘ WWW} .  a; r 35'? «:— {Lag
Q (3% ' )Wﬁfi Ki}? Aw“ f a , max ~3x
j?!ﬁg ‘4'" C "3X3 #3: f3??? =3 (3+3 ﬁr) am i» (33*19} ring: «rce‘ax basal
%§»#r«s0 :5? A3383 34:33 3'2, a) a gig Q’é‘é AMATH 250  Midterm Test 01 Fall 2010 Page 3 0f 5 d
[8] 2. Consider the DE a: = —2y + 6’”, whose solution is given by
1 a: ~2m
y(a:) = 36 + Ce 7 Where C is a constant. Provide a qualitative sketch of the solutions. Determine Where
solutions of the DE are increasiirig/decreasing7 and ﬁnd any equilibrium and exceptional
solutions. Also discuss the behaviour of the solutions as it tends to positive inﬁnity and negative inﬁnity. . a > Q 1 {(2. ex WW ‘*s\fieic: ,_ ‘wii‘x
\3 A,§*",,,T%‘FT‘5 a3x>+w,g>3a AMATH 250 — Midterm Test 01 Fall 2010 Page 4 of 5 3. The temperature of a hot cup of coffee can be modeled with Newton’s law of cooling,
described by the initial value problem Egg 2 “ Ta)7 : T07 which has the solution
T(t) : Ta + (T0 — my“. The parameter k is an unknown positive constant and T is the temperature of the coffee at time t.
[1] a) State what Ta represents.
{3mm 6 NT T? W9
[2] b) Suppose that the coffee, initially at 100°C, is placed in a room whose temperature is
10°C. Find an expression in terms of k for the time at which the temperature of the
coffee reaches 40°C. k 'E [1 WMWCéOMWJ e
nick ‘5' (@251) E} 'E I‘lié a}?! ‘1' J1 1 [7] 0) Suppose that when t = 0, the coffee is 100°C, and a small object with initial temper—
ature 130°C is placed inside[the coffee and starts to cool down. Use Newton’s law of cooling to find an expression for the temperature U (t) of the small object. Assume
that 0 temperature of the room is 10°C 0 the small object has the same thermal properties of the coffee so that they have
the same k 0 the object is small enough so that it does not affect the temperature of the coffee
0 the object is completely surrounded by coffee Hint: Newton’s law of cooling is also valid when Ta is not constant. 1; We «(M40 2 k (u(to+€zoe‘l‘*)) ...
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