MAT2125 Elementary real analysis (Winter 2016)
Homework assignment 4
Solutions
1. Prove (by using the definition) uniform continuity of the function
f (x) = 2x 3 on the real line R.
Solution. Since |f (x) f (x )| = 2|x x | by the definition of the
functio
MAT2125 Elementary real analysis (Winter 2016)
Homework assignment 1
Solutions
1. Prove by induction that the sum of the first n odd integers is n2 .
Solution. Put
Sn =
n
X
k=1
(2k 1) ,
so that one has to prove that Sn = n2 for any integer n. X An inducti
MAT 2125 Mid-Term Examination 2015 Solutions
1. a) If cfw_an n
1
is a real sequence, and a 2 R, give the denition of
lim an = a.
n!1
Solution:
8" > 0 9N 2 N such that k
or
8" > 0 9N 2 N such that 8k
[2 point(s)]
Now dene a sequence cfw_an n
1
N ) |a
an |
University of Ottawa
MAT 2324 Midterm Exam
Feb 26, 2015. Duration: 80 Minutes. Instructor: Robert Smith?
Family Name:
First Name:
Do not write your student ID number on this front page. Please write your student ID
number in the space provided on the seco
MAT 2324
2384-Practice Problems on independence of solutions of ODEs and the
Wronskian
Question 1 For each of the following higher orer ODEs, use the Wronskian to show that the given
functions form a basis of solutions.
1. y (4) = 0,
2. x2 y
3. y
000
000
2324
MAT 2384-Practice Problems on Linear First Order ODEs and Bernoulli
Equation
For each of the following ODEs, Find the General Solution. If an initial condition is
given, nd also the corresponding particular solution.
1
1. x2 y 0 + 3xy = x , y(1) =
1
MAT 2324
2384-Practice Problems on higher order homogeneous ODEs
Question 1 Find an ODE for which the given functions form a basis of solutions.
1. ex , e2x , e3x
2. ex , e
x
, cos x, sin x
3. 1, x, cos 2x, sin 2x
Question 2 Solve the given ODE.
1. y 000
A brief introduction to the
course
Dr. Robert Smith?
Department of Mathematics and Faculty of Medicine
The University of Ottawa
Who is Robert Smith?
Most profs never tell you how to address
them
You can call me either Robert or Dr. Smith?
Either is fin
20. te
MAT 2324
2384-Practice Problems on
3t cos t
21. f (t) =
(
0
t>2
t2
0<t<2
22. f (t) =
(
0
t>2
23. f (t) =
(
0
24. f (t) =
Laplace Transforms-
(
1
Find the Laplace transform of each of the
following functions.
1. e2
2.
3t
cos2 (2t)
3. et cosh(2t)
4.
MAT 2324
2384-Practice Problems on Nonhomogeneous higher order ODEs-Methods of
Undermined Coe cients and the Variation of Parameters
For each of the following ODEs, Find the General Solution. If an initial condition is given, nd also
the corresponding par
MAT 2324
2384-Practice Problems on Nonhomogeneous second order ODEs-Methods of
Undermined Coe cients and the Variation of Parameters
For each of the following ODEs, Find the General Solution. If an initial condition is given, nd also
the corresponding par
MAT 2324
2384-Practice Problems on Exact ODEs and Integrating Factors
For each of the following ODEs, test for exactness. If exact solve. If not, use an
integrating factor to solve. If the ODE is equipped with an initial condition, use it
to nd the partic
MAT 2324
2384-Practice Problems on Linear Second Order ODEs with
constant coe cients
For each of the following ODEs, Find the General Solution. If an initial condition is
given, nd also the corresponding particular solution.
1. 4y 00
20y 0 + 25y = 0, y(0)
ADM 2350A
October 8, 2014
Quiz #4 Examination
Version #1 Solutions
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluation
ADM 2350A
October 22, 2014
Quiz #5 Examination
Version #1 Solutions
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluation
ADM 2350A
September 17, 2014
Quiz #1 Examination
Version #1 Solutions
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that
may result in a false academic evaluati
ADM 2350A
September 24, 2014
Quiz #2 Examination
Version #1 Solutions
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluati
ADM 2350A
October 1, 2014
Quiz #3 Examination
Version #1 Solutions
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluation
ADM 2350A
October 31, 2014
Quiz #6 Examination
Version #1 Solutions
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluation
ADM 2350A
November 7, 2014
Quiz #7 Examination
Version #1 Solutions
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluation
ADM 2350A
November 21, 2014
Quiz #8 Examination
Version #1
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluation of that
MAT 2324
2384-Practice Problems on Homogeneous Euler-Cauchy Equations
For each of the following ODEs, Find the General Solution. If an initial condition is
given, nd also the corresponding particular solution.
1. x2 y 00
6y = 0
2. x2 y 00
7xy 0 + 16y = 0
MAT 2324
2384-Practice Problems on First-order Separable- Homogeneous
ODEs
1. Find the general solution of each of the following ODEs.
(a) y 0 = 2 sec(2y)
(b) yy 0 + 25x = 0
(c) y 0 sin(x) = y cos(x)
(d) y 0 e
2x
= y2 + 1
(e) (x3 + y 3 )dx
3xy 2 dy = 0
(x
MAT 2324
2384-Practice Problems on Systems of Dierential Equations
~
~
~
In each case, solve the Systems of Dierential Equations Y 0 = AY + F (x) for the given matrix
~
A and the vector F (x). If an initial condition is given, solve the corresponding IVP.
MAT 2324
2384-Practice Problems on Solving ODEs using Laplace TransformsUse Laplace transforms to solve each of the following IVPs.
1. y 00 + 10y 0 + 24y = 144t2 , y(0) = 19 , y 0 (0) = 5
12
(
0 if 0 < t < 1
2. y 00 + 3y 0 + 2y = r(t), r(t) =
, y(0) = 0,
20.
Find the Inverse Laplace transform of each
of the following functions.
1.
s2
(s2 +4)2
23. ln s+2
s 3
2s+3
s2 +9
24.
2s+1
s2 7s+6
3.
s
e 3s
s2 +6s 7
4.
3
2s
s (e
5.
2s+1
s2 +2s+1
6.
2s+3
e 3s
s2
7.
s2 +4
8.
1
e 2s
(s 1)2
9.
1
s2 +8s+25
10.
100(s+25)
s(
MAT2125 Elementary real analysis (Winter 2016)
Homework assignment 4
1. Prove (by using the definition) uniform continuity of the function f (x) = 2x3
on the real line R.
2. Do the same for the function
x1
f (x) =
2x + 3
on [0, ).
3. Let f (x) = x2 cos( x
MAT2125 Elementary real analysis (Winter 2016)
Homework assignment 1
1. Prove by induction that the sum of the first n odd integers is n2 .
2. Prove by induction that 13n 8n is divisible by 5 for any natural n.
3. Let a1 = a2 = 5, and
an+1 = an + 6an1
for
2 \20 E0 i Ema/Qt 206% ,
1. a)~ If A C R give the denition of the supremum of A, usually denoted sup A. C?)
b) Let cfw_anhzl be a bounded realsequence and let
a := sup cfw_an I n 2 1. MM 0)
<~ . E , ,/
. . . )klfncfamlccb cfw_:37
Suppose, in addition, tha
1. a) Dene what is meant by K C R is compact H OFT/22011 5319 WA
b) State the BolzanoWeierstrass theorem.
(1) State the Maximum theorem for a continuous function on a closed and
bounded interval [03,19].
Now let 9 : R > R be a continuous function and supp
MBA 3170.9
1. a) If A C R give the denition of the supremum of A, usu ly denoted sup A
b) State necessary and sufcient conditions for the existence of sup A.
c) Prove that if A C B, then sup A S sup B.
d) Let f : [a, b] > R be continuous on [a, b] and den
W
1. Let 0,1;th e R. Prove that: (/1 ,r 0.33% cfw_EMA (if u
1 ,1 I
Janl + 7 52mm 0
(3/13, 3)ma.Xcfw_SB y " y + 73 / ,1: E
Ob)|m|>b>0=>|ll_<_lil- (4:13-70) (2)
1: x:r MM WI 5E1 )
+49 :2 f NEW-cfw_M3
[Lava WW Xi? ._ 3:?"4 1? 3f 1 K 1
if! f 17 L
6. (Bonus) Suppose f : R ! R is continuous on R and that f ([0, 1]) [0, 1], i.e.
8x 2 [0, 1], f (x) 2 [0, 1]. Prove that there exists a 2 [0, 1] such that f (a) = a.
Hint: Consider the function g : [0, 1] ! R defined by g(x) = f (x)
x.
Solution:
Since f (