Lecture Summaries for Differential Equations
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.
Lecture 04: The Existence and Uniqueness Theorem for solving general first order linear equations.
Lecture 05: Applications of first order ODEs involving continuous compounding, and population
dynamics using the logistic equation .
Lecture 06: Solving the logistic equation, and an application of first order ODEs to a problem
Lecture 07: Solving exact equations.
Lecture 08: Sketching a proof of the Existence and Uniqueness Theorem for first order ODEs.
Lecture 09: An introduction to difference equations and their solutions, focusing on first order
linear difference equations.
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.
Lecture 11: An introduction to second order ODEs and initial value problems, and a discussion
of solutions to second order homogeneous constant coefficient equations.
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.
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.
Lecture 14: Solving ODEs with characteristic equation having non-real complex roots.
Lecture 15: Solving ODEs with characteristic equation having repeated roots.
Lecture 16: Solving second order linear non-homogeneous equations using the method of undetermined
Lecture 17: Solving second order linear non-homogeneous equations using the method of variation
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.
Lecture 19: Solving higher order constant
coefficient homogeneous equations.
Lecture 20: Solving higher order non-homogeneous equations using the method of undetermined
Lecture 21: Solving higher order non-homogeneous equations using the method of variation of
Lecture 22: A review of the most fundamental properties of power series.
Lecture 23: Solving differential equations and initial value problems using power series.
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.
Lecture 25: A review of improper integration and an introduction to the Laplace transform.
Lecture 26: A discussion of the main properties of the Laplace transform which make it useful
for solving initial value problems .
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.
Lecture 28: An introduction to the convolution of two functions, and an examination of how
the Laplace transform acts on such a convolution.
Lecture 29: An introduction to systems of equations and the basic existence and uniqueness
result for the corresponding initial value problems .
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.
Lecture 31: Solving constant coefficient linear homogeneous systems using eigenvalues and
Lecture 32: Solving constant coefficient linear homogeneous systems in the case where an
eigenvalue is complex .
Lecture 33: Solving constant coefficient linear homogeneous systems in the case where there is
a repeated eigenvalue.
Lecture 34: Viewing solutions to linear homogeneous systems in terms of fundamental matrices
and the exponential of a matrix .
Lecture 35: Solving non-homogeneous systems using diagonalization and variation of parameters.