UNIVERSITY OF WATERLOO
FINAL EXAMINATION
FALL TERM 2012
Name (Please Print Legibly)
Signature
Student ID Number
COURSE NUMBER AMath 350
COURSE TITLE Differential Equations for Business and Economics
COURSE SECTION(s) 001 002
DATE OF EXAM Saturday, Decembe
y = 3x cos2 (y )
,
22
y
sec2 y dy =
tan y =
3x2
+C ,
2
y = tan1
y = 8x3 ey ,
dy
= 3xdx
cos2 y
3xdx
3x2
+C
2
y.
y (1) = 0
ey dy =
8x3 dx
ey = 2x4 + C
y = ln 2x4 + C .
1 = 2 + C
C = 1
y = ln 2x4
AMath 350
Assignment #7
Fall 2014
Solutions
1. ~ 0 = A~ + F (t);
x
x ~
A=
2
1
1
2
~
, and F (t) =
8te
6
t
.
Solution:
We rst need to nd the solution to the corresponding homogeneous equation:
A
I=
2
1
1
2
so the characteristic equation is ( + 2)2 1 = 0, o
AMath 350
Assignment #6
Fall 2011
Due Monday, October 31st
1.
a) Show that the boundary value problem
y + ky = 0,
y (0) = 0,
y (0) = 1
has a solution for every value of k .
b) Find the eigenvalues and eigenfunctions for the boundary value problem
y + 2ky
AMath 350
Assignment #2
Winter 2015
SOLUTIONS
1. (Are the given equations separable, linear, neither, or both?)
a) Not separable, but linear.
b) Separable, but not linear.
c) Both.
d) Neither.
e) Both.
f) Separable, but not linear.
2.
a) Find the solution
AMath 350
Assignment #1 (Review)
Winter 2015
Solutions
C
u = y, dv = sin (y) , du = 1, v =
Z
Z
y sin (y) dy = y cos (y) + cos (y) dy
=
u=t
cos (y)
y cos (y) + sin (y) + C.
2 du = dt
Z
4
Z
1
dt =
4t + t2
Z
t
2
du =
u = ln (2x)
ln (2x)
dx =
x
2
sin (3) d =
Dierential Equations
for Business and Economics
Course Notes for AMATH 350
Sue Ann Campbell
Department of Applied Mathematics
University of Waterloo
Winter 2013 Edition
Edited by David Harmsworth
c S.A. Campbell 2010
Contents
1 Introduction
1
2 First Orde
Theorems on Ordinary Dierential Equations
(Proofs as Presented in Class)
1. Existence and Uniqueness Theorem for IVPs Involving 1stOrder Equations
(see course notes - proof not discussed)
2. Existence and Uniqueness Theorem for IVPs Involving Linear Equat
Must-Know Formulas for Calculus
In Math 117 and Math 119 we have tried to encourage you to concentrate on understanding the material,
rather than memorizing formulas. However, there are a few things that you simply have to KNOW, by heart.
Trying to write
The Greek Alphabet
D.Harmsworth, 2011
Upper Case
Lower Case
Alpha
a (as in father)
Beta
b
g (as in good)
Delta
to Classical Greek Pronunciation
Gamma
A
Closest English Equivalent
Version of Name
B
Traditional English
d
E
or "*
Epsilon
e (as in set)
Z
Zet
AMath 350
Assignment #7
Winter 2014
Solutions
1. ~ 0 = A~ + F (t);
x
x ~
A=
2
1
1
2
~
, and F (t) =
8te
6
t
.
Solution:
We rst need to nd the solution to the corresponding homogeneous equation:
A
I=
2
1
1
2
so the characteristic equation is ( + 2)2 1 = 0,
AMath 350
Assignment #3
Winter 2015
Solutions
1. (Find all solutions to the following equations:)
a) x2
dy
= xy
dx
y2
dy
y y2
=
, and we
dx
x x2
y
can see that it is a so-called homogeneous equation. So, we let u = ,
x
and then, in the usual way, we can w
AMath 350
Assignment #8
Winter 2014
Solutions
1. Following the hint, we have
Z 1
Z
(x x2 ) dx =
e
0
1
0
=e
Now, letting u = x
1
2
2
e (x x) dx =
1
4
Z
1
e (x
Z
1
0
h
x
e (
1 2
2
)
1
4
i
dx
1 2
2
) dx.
0
, this becomes
e
1
4
Z
1
2
e
u2
du
e
u2
du.
1
2
and
AMath 350
Assignment #9
Winter 2014
Due Wednesday, April 2nd
Hand in #4, #5, and #6ab.
1. Use the denition or Theorem 5.31 of the Course Notes to determine if the
Fourier Transform of each of the following functions exists.
a) f (x) = x
2
b) f (x) = xe x
AMath 350
Assignment #6
Winter 2014
Solutions
1.
a)
ax2
d2 y
dy
+ bx + cy = 0
2
dx
dx
Let x = ez (which, of course, means z = ln x, if x > 0). Then
dy
dy dz
1 dy
=
=
.
dx
dz dx
x dz
We also need the second derivative:
d2 y
d dy
d 1 dy
=
=
dx2
dx dx
dx x
AMath 350
Assignment #8
Winter 2015
Solutions
1. Following the hint, we have
Z 1
Z
(x x2 ) dx =
e
0
1
0
=e
Now, letting u = x
1
2
2
e (x x) dx =
1
4
Z
1
e (x
Z
1
0
h
x
e (
1 2
2
)
1
4
i
dx
1 2
2
) dx.
0
, this becomes
e
1
4
Z
1
2
e
u2
du
e
u2
du.
1
2
and
AMath 350
Assignment #7
Winter 2015
Solutions
1. ~ 0 = A~ + F (t);
x
x ~
A=
2
1
1
2
~
, and F (t) =
8te
6
t
.
Solution:
We rst need to nd the solution to the corresponding homogeneous equation:
A
I=
2
1
1
2
so the characteristic equation is ( + 2)2 1 = 0,
AMath 350
Assignment #6
Winter 2015
Solutions
1.
a)
ax2
d2 y
dy
+ bx + cy = 0
2
dx
dx
Let x = ez (which, of course, means z = ln x, if x > 0). Then
dy
dy dz
1 dy
=
=
.
dx
dz dx
x dz
We also need the second derivative:
d2 y
d dy
d 1 dy
=
=
dx2
dx dx
dx x
AMath 350
Assignment #5
Winter 2015
Solutions
1. Solve y 0 + 3y = 5e
3x
+ 6xe3x :
By inspection, the complementary function is yh = Ce
3x
.
For a particular solution, the 5e 3x term would normally require a term of the
form Ae 3x , but since this matches
AMath 350
Possible Final Exam Content
Methods and Concepts
Youre responsible for everything weve discussed, except for the method for solving
PDEs known as separation of variables:
ODEs:
1st-order Equations
separable equations
linear equations (integr
AMATH 350
Page 11 of 13
Final Examination - Fall 2014
Formulas (I)
An invertible transformation = (x, y), = (x, y) will convert a second-order
linear PDE
a (x, y) uxx + b (x, y) uxy + c (x, y) uyy + d (x, y) ux + e (x, y) uy + f (x, y) = g (x, y)
into the
University of Waterloo
AMath 350
Midterm Examination
Winter 2014
Friday February 28th , 4:00-5:30pm or 5:00-6:30pm (90 minutes)
Name (print):
I.D. Number:
Signature:
Closed book.
MARKS
Question
Marks Available
1
7
2
8
3
16
4
9
5
10
6
10
Total
60
Marks Awa
AMath 350
Assignment #9
Fall 2014
Due Friday, November 27th
Hand in #4, #5, and #6ab.
1. Use the denition or Theorem 5.31 of the Course Notes to determine if the
Fourier Transform of each of the following functions exists.
a) f (x) = x
2
b) f (x) = xe x
(
AMath 350
Assignment #4
Fall 2014
due Friday, October 9th
Hand in #1, #2c, #5, #6, and #7.
1. Find the general solutions to the following ODEs:
a) y 00
7y 0 + 12y = 0
b) y 00
8y 0 + 16y = 0
2
0
2y + 1 + 4 y = 0
c) y 00
d) y 000
3y 00 + 3y 0
y=0
e) y 000 +
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
is
sh
Th
https:/www.coursehero.com/file/11992708/Midterm-Solutions/
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
is
sh
Th
https:/www.coursehero.com/file/11992708/Midterm-Solutions/
ar