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Lectures

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 Boundary-value Problems Introduction MEM 10.2.4, 10.3, 10.4.2,3,4
2 1st-order ODEs MEM 10.5.3,9
3
4
5 2nd-order 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 2.2.1-6
15
16
17 The inverse Laplace transform AMEM 2.2.7-9
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 4.2.1-8, AMEM 4.3.2
26
27 Convergence AMEM 4.2.9-10, AMEM 4.3.1, Notes
28 Gibbs's phenomena AMEM 4.2.9-10, AMEM 4.3.1, Notes
29 Integration and differentiation of Fourier series AMEM 4.4, 4.5
30 The Response of a Fourier-series Representation of a Signal
31 Introduction to Partial Differential-Equations 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,

a3t3 + a2t2 + a1t + a0 = b

a2y(2)(t) + a1y(1)(t) + a0y(t) = b

are better than

a0t3 + a1t2 + a2t + a3 = a4

a0y(2)(t) + a1y(1)(t) + a2y(t) = a3

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 straight-forward derivative formula for calculating the partial fraction decomposition which does not require cover-up tricks to calculate.

There is no need to teach or discuss the bilateral (two-sided) Laplace transform.

One suggestion from a previous instructor: when teaching the convolution is to begin by defining the convolution as tex:$$(f * g)(t) = \int_{-\infty}^\infty f(\tau)g(t - \tau) d\tau$$, but once the definition is given, always write the convolution as convolution as tex:$$\int_{-\infty}^\infty f(\tau)g(t - \tau) d\tau$$. This reinforces the integral definition which is hidden by the notation tex:$$f * g$$.

The components of the final examination could be weighted approximately within the given ranges:

Major TopicMinimumAverageMaximum
Differential Equations0.250.330.33
The Laplace Transform0.250.330.33
Fourier Series0.170.170.25
Partial Differential Equations0.170.170.25