The lecture notes borrow very heavily from lecture notes written by
Dr. Giuseppe Tenti from the
Department of Applied Mathematics here at the University of Waterloo.
In some cases, the text is taken verbatim (with permission) and in others,
it has been expanded upon by Douglas Wilhelm Harder.
These pages use images with alt tags in order to display
mathematical formulas using the LaTeX document language. The web site LaTeX for the Visually
Impaired.
Day 
Primary Topic 
Secondary Topics 
Text Book 
1 
Ordinary Differential Equations and
Initial and Boundaryvalue Problems 
Introduction 
MEM 10.2.4, 10.3, 10.4.2,3,4 
2 
1storder ODEs 
MEM 10.5.3,9 
3 
4 
5 
2ndorder ODEs 
MEM 10.8, MEM 10.9 
6 
7 
8 
9 
IVPs and BVPs 
Notes 
10 
Oscillations 
MEM 10.10, MEM 10.12 
11 
12 
13 
14 
The Laplace Transform and its Applications 
Definitions and properties 
AMEM 2.1 AMEM 2.2.16 
15 
16 
17 
The inverse Laplace transform 
AMEM 2.2.79 
18 
Solutions to ODEs with the Laplace transform 
AMEM 2.3, AMEM 2.4 
19 
20 
The unit step and impulse functions,
convolutions, and the transfer function 
AMEM 2.5, AMEM 2.6

21 
22 
23 
Applications and consequences 
AMEM 2.7 Notes 
24 
25 
Fourier Series 
Definitions and properties 
AMEM 4.1, AMEM 4.2.18, AMEM 4.3.2 
26 
27 
Convergence 
AMEM 4.2.910, AMEM 4.3.1, Notes 
28 
Gibbs's phenomena 
AMEM 4.2.910, AMEM 4.3.1, Notes 
29 
Integration and differentiation of Fourier series 
AMEM 4.4, 4.5 
30 
The Response of a Fourierseries Representation of a Signal 
31 
Introduction to Partial DifferentialEquations 
Introduction, definitions, and classification 
AMEM 9.1,2.2,3 
32 
Separation of variables 
AMEM 9.3.2 
33 
The heat and Laplace equation 
AMEM 9.4.1,5.1 
34 
35 
36 
Suggestions to the Instructor
It is common in mathematics to refer to the impulse and step functions as the
Dirac delta and Heaviside functions.
Please use the more common engineering terms.
Use j instead of i for the imaginary unit.
When writing polynomials or differential equations with coefficients, label
the coefficients to correspond with the degree, for example,
a_{3}t^{3} +
a_{2}t^{2} +
a_{1}t + a_{0} = b
a_{2}y^{(2)}(t) +
a_{1}y^{(1)}(t) + a_{0}y(t) = b
are better than
a_{0}t^{3} +
a_{1}t^{2} +
a_{2}t + a_{3} = a_{4}
a_{0}y^{(2)}(t) +
a_{1}y^{(1)}(t) + a_{2}y(t) = a_{3}
In the first case, there is no question as to which coefficient corresponds with
which degree while with the second, there is a significant disjoint between the
coefficients and the term.
There is a straightforward derivative formula for calculating the partial fraction
decomposition which does not require coverup tricks to calculate.
There is no need to teach or discuss the bilateral (twosided) Laplace transform.
One suggestion from a previous instructor: when teaching the convolution is to begin by defining the
convolution as ,
but once the definition is given, always write the convolution as
convolution as . This
reinforces the integral definition which is hidden by the notation .
The components of the final examination could be weighted approximately
within the given ranges:
Major Topic  Minimum  Average  Maximum 
Differential Equations  0.25  0.33  0.33 
The Laplace Transform  0.25  0.33  0.33 
Fourier Series  0.17  0.17  0.25 
Partial Differential Equations  0.17  0.17  0.25 