Course Outline

Summary:

1. Ordinary Differential Equations
2. Second Order Linear Ordinary Differential Equations
3. Laplace Transforms
4. Partial Differentiation
5. Series

A list of learning outcomes is available.

Detailed Contents:

1.	Ordinary Differential Equations
1.1	First Order Separable ODEs
1.2	Examples for First Order Separable ODEs
1.3	First Order Linear ODEs
1.4	Reduction of Order
1.5	Numerical Methods for First Order First Degree ODEs
1.6	Exact First Order ODEs [not examinable]
1.7	Integrating Factor [not examinable]
1.8	Derivation of the General Solution of First Order Linear ODEs [not examinable]
1.9	Integration by Parts   [Lecture in Week 2]
2.	Second Order Linear Ordinary Differential Equations
2.1	Complementary Function
2.2	Particular Solution (Variation of Parameters)
2.3	Particular Solution (Undetermined Coefficients)
2.4	Higher Order Linear Ordinary Differential Equations
2.5	Numerical Methods [not examinable]

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3.	Laplace Transforms
3.01	Definition, Linearity, Laplace Transforms of Polynomial Functions
3.02	Laplace Transforms of Derivatives
3.03	Review of Complex Numbers
3.04	First Shift Theorem, Transform of Exponential, Cosine and Sine Functions
3.05	Applications to Initial Value Problems
3.06	Laplace Transform of an Integral
3.07	Heaviside Function, Second Shift Theorem; Example for RC Circuit
3.08	Dirac Delta Function, Example for Mass-Spring System
3.09	Laplace Transform of Periodic Functions; Square and Sawtooth Waves [not examinable]
3.10	Derivative of a Laplace Transform
3.11	Convolution; Integro-Differential Equations; Circuit Example
4.	Partial Differentiation
4.1	Partial Derivatives - introduction, chain rule, practice
4.2	Higher Partial Derivatives, Clairaut’s theorem, Laplace’s PDE
4.3	Differentials; error estimation; chain rule [again]; implicit functions; partial derivatives on curves of intersection
4.4	The Jacobian - implicit and explicit forms; plane polar; spherical polar
4.5	Gradient Vector, directional derivative, potential function, central force law
4.6	Extrema; Second Derivative Test for   z = f (x, y)
4.7	Lagrange Multipliers; nearest point on curve of intersection to given point
4.8	Miscellaneous Additional Examples
5.	Series
5.01	Sequences; general term, limits, convergence
5.02	Series; summation notation, convergence, divergence test
5.03	Series; telescoping series, geometric series, p-series
5.04	Tests for Convergence: comparison and limit comparison tests
5.05	Tests for Convergence: alternating series; absolute and conditional convergence
5.06	Tests for Convergence: ratio and root tests
5.07	Power Series, radius and interval of convergence
5.08	Taylor and Maclaurin Series, remainder term
5.09	Binomial Series
5.10	Introduction to Fourier Series
5.11	Introduction to Series Solutions of ODEs
5.A	Integral Test [not examinable; for reference only]

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Appendix:
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