# 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

of physics.

**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

coefficients .

**Lecture 17:** Solving second order linear non-homogeneous equations using
the method of variation

of parameters.

**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

coefficients .

**Lecture 21:** Solving higher order non-homogeneous equations using the
method of variation of

parameters.

**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.

**A discussion of the main properties of the Laplace transform which make it useful**

Lecture 26:

Lecture 26:

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

eigenvectors.

**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.

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