Solution to MAT B44 Midterm Exam, Saturday, March 12, 2011
1. Consider the system
dx
=y
dt
dy
= 4 sin(x).
dt
(a) Find the critical points.
At each critical point, answer the following questions:
(b) What is the type of the corresponding linear system (nod

MAT C46S
SUMMARY
I. Laplace Transforms
E := cfw_f | C such that |f (x) Cex for sufficiently large x
(exponential order )
Throughout this section, assume all functions considered lie in E for some .
R
1. L(f )(s) f(s) := 0 est f (t) dt
Properties:
a) lims

MAT C46 - Winter 2014
Instructor: Prof. Lisa Jeffrey
Course outline:
Textbook: Boyce and Di Prima, Elementary Differential Equations and
Boundary Value Problems, 9th edition (all references below are to this textbook)
Chapter 6 (Laplace Transforms) all t

MAT C46S
SUMMARY OF MATERIAL AFTER MIDTERM
I. Partial Dierential Equations
Partial dierential equations involve more than one independent variable.
1, Heat equation:
2 uxx = ut
Wave equation:
a2 uxx = utt
2. Laplace equation: In two dimensions
uxx + uyy =

MAT C46 Final Exam, Friday, April 15, 2011
7 pm 10 pm
There are seven questions No books or calculators are allowed.
1. (15 pts)
Consider the system
dx
= x(1 0.5x 0.5y)
dt
y = y(0.25 + 0.5x)
(5 pts (a) Find all the critical points of the system
For each c

University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
Final Examination
MATB46H Dierential Equations II
Examiner: L. Jerey
Date: April 29, 2010
Duration: 3 hours
DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO.
There are

MAT C46 Final Exam, Friday, April 15, 2011
7 pm 10 pm
There are seven questions No books or calculators are allowed.
1. (15 pts)
Consider the system
dx
= x(1 0.5x 0.5y)
dt
y = y(0.25 + 0.5x)
(5 pts (a) Find all the critical points of the system
Solution:

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
Term Test
MATC46H Dierential Equations II
Examiner: L. Jerey
Date: March 10, 2014
Duration: 2 hours
FAMILY NAME:
GIVEN NAMES:
STUDENT NUMBER:
SIGNATURE:
DO NOT OPEN THIS BOOK

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
Term Test
MATC46H Dierential Equations II
Examiner: L. Jerey
Date: March 7, 2014
Duration: 2 hours
FAMILY NAME:
GIVEN NAMES:
STUDENT NUMBER:
SIGNATURE:
DO NOT OPEN THIS BOOKL

TABLE 6.2.1 Elementary Laplace Transforms
F (S) = EU (1)}
_ f0) = 11-31%» _
1
1. 1 —, s > 0 ' r
s
1
2 e‘“ , s > a
s —- a
n . . . n! I
3. t , n = posmve mteger Sn“ , s > 0
F 1
4. W, p > ——1 (Sp—:1), s > O
. a
5. smat I 32+ai, s>0
s
6. cosat 52+

The Inverse Laplace Transform
1. If Lcfw_f (t) = F (s), then the inverse Laplace transform of F (s) is
L1 cfw_F (s) = f (t).
(1)
The inverse transform L1 is a linear operator:
L1 cfw_F (s) + G(s) = L1 cfw_F (s) + L1 cfw_G(s),
(2)
L1 cfw_cF (s) = cL1 cfw_F

MATC46 solutions to Assignment 5, 201314
1. Use the sturmliouville program in the MathLab to nd approximations to the eigenvalues of
y = (1 + 4x2 )y,
y(0) = 0,
y(1) = 0
which are less than 100. Print the output of the program showing the eigenfunctions fo

University of Toronto at Scarborough
Department of Computer and Mathematical Sciences, Mathematics
MAT C46S
2013/14
Problem Set #4
Due date: Thursday, March 20, 2014 at the beginning of class
Part A:
1. 1. Let g(x) be a solution of the BVP, (p(x)y ) + q(x

Mathematics MATC46, Assignment 1
Solutions to Selected Problems
6.2 #3. Find the inverse Laplace transform of the following
function:
2
+ 3s 4
2
Answer: s + 3s 4 = (s + 4)(s 1) so we search for a decomposition
F (s) =
s2
s2
A
B
1
=
+
+ 3s 4
s+4 s1
A(s 1)

Mathematics MATC46, Assignment 2
Solutions to Selected Problems
First question:
dx
dy
= x(y 1)
= (2y + (x 1)(x 4)
dt
dt
Critical points: x = 0 or y = 1, and 2y + (x 1)(x 4) = 0 implies if
x = 0 then y = 2). If y = 1 then
(x 1)(x 4) + 2 = 0
or
x2 5x + 6 =

University of Toronto at Scarborough
Department of Computer and Mathematical Sciences, Mathematics
MAT C46S
2013/14
Problem Set #3
Due date: Thursday, February 27, 2014 at the beginning of class
PART A
1. Show that
dy
dx
= y + x(1 x2 y 2 )3 ,
= x + y(1 x2

University of Toronto at Scarborough
Division of Physical Sciences, Mathematics
MAT C46S
2013/2014
Problem Set #5
Due date: Thursday, April 3, 2014 at the beginning of class
1. Use the sturmliouville program to nd approximations to the eigenvalues of
y =

MAT C46
Partial Differential Equations II
Course Outline.
LW1, LW2. Systems of Nonlinear Differential Equations & Stability.
Autonomous equations, Phase Portrait of nonlinear and almost linear systems.
Qualitative solutions and stability.
LW3, LW4. The La

University of Toronto at Scarborough
Department of Computer and Mathematical Sciences, Mathematics
MAT C46S
2013/14
Problem Set #4
Due date: Thursday, March 20, 2014 at the beginning of class
Part A:
1. Let g(x) be a solution of the BVP, (p(x)y ) + q(x)y

University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MAT C46S
2013/14
Problem Set #1
Due date: Tuesday, January 21, 2014 at the beginning of class
Do the following problems from Boyce-Di Prima.
S. 6.2:
#3, 23
S. 6.3 #13, 20
S

Dierential Equations II
MATC46H3S
Lisa Jerey
Paul Selick
E-mail address, Lisa Jerey: jeffrey@math.toronto.edu
E-mail address, Paul Selick: selick@math.toronto.edu
(Lisa Jerey) Bahen Centre, room BA6211, 40 St. George Street, Toronto, Ontario, M5S
2E4
(Pau

University of Toronto at Scarborough
Department of Computer and Mathematical Sciences, Mathematics
MAT C46S
2013/14
Problem Set #2
Due date: Thursday, January 30, 2014 at the beginning of class
For the system
dx
dy
= x(y 1)
= (2y + (x 1)(x 4)
dt
dt
nd the