Lesson Contents
- Lecture 01: An introduction to the very basic definitions and
terminology of differential equations, as well as a discussion
of central issues and objectives for the course.
- Lecture 02: Solving first order linear differential equations and
initial value problems using "integrating factors".
- Lecture 03: Solving "separable" equations.
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Lecture 04: The Existence and Uniqueness Theorem for solving general
first order linear equations.
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Lecture 05: Applications of first order ODEs involving continuous
compounding, and population dynamics using the "logistic" equation.
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Lecture 06: Solving the logistic equation, and an application of
first order ODEs to a problem of physics.
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Lecture 07: Solving "exact" equations.
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Lecture 08: Sketching a proof of the Existence and Uniqueness Theorem
for first order ODEs.
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Lecture 09: An introduction to "difference" equations and their
solutions,
focusing on first order linear difference equations.
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Lecture 10: An application of first order linear difference equations,
as well as a brief discussion of non-linear difference equations, their
solutions, and "stairstep diagrams".
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Lecture 11: An introduction to second order ODEs and initial value
problems, and a discussion of solutions to second order homogeneous
constant coefficient equations.
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Lecture 12: A discussion of existence and uniqueness results for second
order linear ODEs, and of fundamental sets of solutions and the importance
of
the "Wronskian" of solutions.
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Lecture 13: A discussion of the structure of the set of solutions to a
linear homogeneous ODE from a linear algebra perspective; concepts such
as "linear independence", "span", and "basis" are used to better
understand
fundamental sets of solutions.
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Lecture 14: Solving ODEs with characteristic equation having non-real
complex roots.
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Lecture 15: Solving ODEs with characteristic equation having repeated
roots.
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Lecture 16: Solving second order linear non-homogeneous equations
using the method of "undetermined coefficients."
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Lecture 17: Solving second order linear non-homogeneous equations
using the method of "variation of parameters."
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Lecture 18: A discussion of the structure of solution sets to higher
order linear equations, the basic Existence and Uniqueness Theorem, and
a generalization of the Wronskian.
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Lecture 19: Solving higher order constant coefficient homogeneous
equations.
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Lecture 20: Solving higher order non-homogeneous equations using the
method of undetermined coefficients.
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Lecture 21: Solving higher order non-homogeneous equations using the
method of variation of parameters.
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Lecture 22: A review of the most fundamental properties of power
series.
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Lecture 23: Solving differential equations and initial value problems
using power series.
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Lecture 24: An example of how to use power series to solve non-constant
coefficient ODEs, and a discussion of the basic theorem underlying the
use of power series to solve ODEs.
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Lecture 25: A review of improper integration and an introduction to
the Laplace transform.
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Lecture 26: A discussion of the main properties of the Laplace
transform which make it useful for solving initial value problems.
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Lecture 27: A discussion of how the Laplace transform and its inverse
act on unit step functions, exponentials, and products of these functions
with others.
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Lecture 28: An introduction to the "convolution" of two functions, and
an examination of how the Laplace transform acts on such a convolution.
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Lecture 29: An introduction to "systems" of equations and the basic
existence and uniqueness result for the corresponding initial value
problems.
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Lecture 30: An introduction to vector function notation, and a discussion
of the structure of solution sets to homogeneous systems and the
importance of the Wronskian.
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Lecture 31: Solving constant coefficient linear homogeneous systems
using eigenvalues and eigenvectors.
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Lecture 32: Solving constant coefficient linear homogeneous systems in
the case where an eigenvalue is complex.
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Lecture 33: Solving constant coefficient linear homogeneous systems in
the case where there is a repeated eigenvalue.
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Lecture 34: Viewing solutions to linear homogeneous systems in terms of
"fundamental matrices" and the exponential of a matrix.
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Lecture 35: Solving non-homogeneous systems using diagonalization and
variation of parameters.
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