Differential Equations (Math 3301)
Here are my online notes for my differential equations
course that I teach here at Lamar University.
Despite the fact that these are my “class notes”, they should be
accessible to anyone wanting to learn how to solve differential equations or
needing a refresher on differential equations.

I’ve tried to make these notes as self contained as possible
and so all the information needed to read through them is either from a
Calculus or Algebra class or contained in other sections of the notes.

A couple of warnings to my students who may be here to get a
copy of what happened on a day that you missed.

- Because
I wanted to make this a fairly complete set of notes for anyone wanting to
learn differential equations I have included some material that I do not
usually have time to cover in class and because this changes from semester
to semester it is not noted here.
You will need to find one of your fellow class mates to see if
there is something in these notes that wasn’t covered in class.

- In
general I try to work problems in class that are different from my
notes. However, with Differential
Equation many of the problems are difficult to make up on the spur of the
moment and so in this class my class work will follow these notes fairly
close as far as worked problems go.
With that being said I will, on occasion, work problems off the top
of my head when I can to provide more examples than just those in my
notes. Also, I often don’t have
time in class to work all of the problems in the notes and so you will find
that some sections contain problems that weren’t worked in class due to
time restrictions.

- Sometimes
questions in class will lead down paths that are not covered here. I try to anticipate as many of the
questions as possible in writing these up, but the reality is that I can’t
anticipate all the questions.
Sometimes a very good question gets asked in class that leads to
insights that I’ve not included here.
You should always talk to someone who was in class on the day you
missed and compare these notes to their notes and see what the differences
are.

- This
is somewhat related to the previous three items, but is important enough
to merit its own item. THESE NOTES
ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!
Using these notes as a substitute for class is liable to get you in
trouble. As already noted not everything in these notes is covered in
class and often material or insights not in these notes is covered in
class.

Here is a listing and brief description of the material in
this set of notes.

**Basic Concepts**

**Definitions** Some of the common definitions and concepts in
a differential equations course

**Direction Fields** An introduction to direction fields and what
they can tell us about the solution to a differential equation.

**Final Thoughts** A couple of final thoughts on what we will be
looking at throughout this course.

**First Order Differential Equations**

**Linear Equations** Identifying and solving linear first order
differential equations.

**Separable Equations** Identifying and solving separable first order
differential equations. We’ll also start
looking at finding the interval of validity from the solution to a differential
equation.

**Exact Equations** Identifying and solving exact differential
equations. We’ll do a few more interval
of validity problems here as well.

**Bernoulli Differential Equations** In this section we’ll see how to solve the
Bernoulli Differential Equation. This
section will also introduce the idea of using a substitution to help us solve
differential equations.

**Substitutions** We’ll pick up where the last section left off
and take a look at a couple of other substitutions that can be used to solve
some differential equations that we couldn’t otherwise solve.

**Intervals of Validity** Here we will give an in-depth look at
intervals of validity as well as an answer to the existence and uniqueness
question for first order differential equations.

**Modeling with First Order Differential
Equations** Using first order differential equations to
model physical situations. The section
will show some very real applications of first order differential equations.

**Equilibrium Solutions** We will look at the behavior of equilibrium
solutions and autonomous differential equations.

**Euler’s Method** In this section we’ll take a brief look at a
method for approximating solutions to differential equations.

**Second Order
Differential Equations**

**Basic Concepts** Some of the basic concepts and ideas that are
involved in solving second order differential equations.

**Real Roots** Solving differential equations whose
characteristic equation has real roots.

**Complex Roots** Solving differential equations whose
characteristic equation complex real roots.

**Repeated Roots** Solving differential equations whose
characteristic equation has repeated roots.

**Reduction of Order** A brief look at the topic of reduction of
order. This will be one of the few times
in this chapter that non-constant coefficient differential equation will be
looked at.

**Fundamental
Sets of Solutions** A look at some of the theory behind the
solution to second order differential equations, including looks at the
Wronskian and fundamental sets of solutions.

**More on the Wronskian** An application of the Wronskian and an
alternate method for finding it.

**Nonhomogeneous Differential Equations**
A quick look into how to solve nonhomogeneous
differential equations in general.

**Undetermined
Coefficients** The first method for solving nonhomogeneous
differential equations that we’ll be looking at in this section.

**Variation of Parameters** Another method for solving nonhomogeneous
differential equations.

**Mechanical Vibrations** An application of second order differential
equations. This section focuses on
mechanical vibrations, yet a simple change of notation can move this into
almost any other engineering field.

** **

** ****Laplace Transforms**

**The Definition** The definition of the Laplace
transform. We will also compute a couple
Laplace transforms using the definition.

**Laplace Transforms** As the previous section will demonstrate,
computing Laplace transforms directly from the
definition can be a fairly painful process.
In this section we introduce the way we usually compute Laplace transforms.

**Inverse Laplace Transforms** In this section we ask the opposite
question. Here’s a Laplace
transform, what function did we originally have?

**Step Functions** This is one of the more important functions in
the use of Laplace transforms. With the introduction of this function the
reason for doing Laplace transforms starts to
become apparent.

**Solving IVP’s with Laplace Transforms**
Here’s how we used Laplace
transforms to solve IVP’s.

**Nonconstant
Coefficient IVP’s** We will see how Laplace
transforms can be used to solve some nonconstant coefficient IVP’s

**IVP’s with Step Functions** Solving IVP’s that contain step
functions. This is the section where the
reason for using Laplace transforms really
becomes apparent.

**Dirac Delta Function** One last function that often shows up in Laplace transform problems.

**Convolution Integral** A brief introduction to the convolution
integral and an application for Laplace
transforms.

**Table of Laplace Transforms** This is a small table of Laplace Transforms
that we’ll be using here.

**Systems of
Differential Equations**

**Review : Systems of Equations** The traditional starting point for a linear
algebra class. We will use linear algebra
techniques to solve a system of equations.

**Review : Matrices and Vectors** A brief introduction to matrices and
vectors. We will look at arithmetic
involving matrices and vectors, inverse of a matrix, determinant of a matrix,
linearly independent vectors and systems of equations revisited.

**Review : Eigenvalues and Eigenvectors**
Finding the eigenvalues and eigenvectors of a
matrix. This topic will be key to
solving systems of differential equations.

**Systems of Differential Equations** Here we will look at some of the basics of
systems of differential equations.

**Solutions to Systems** We will take a look at what is involved in
solving a system of differential equations.

**Phase Plane** A brief introduction to the phase plane and
phase portraits.

**Real Eigenvalues** Solving systems of differential equations with
real eigenvalues.

**Complex Eigenvalues** Solving systems of differential equations with
complex eigenvalues.

**Repeated Eigenvalues** Solving systems of differential equations with
repeated eigenvalues.

**Nonhomogeneous Systems** Solving nonhomogeneous systems of differential
equations using undetermined coefficients and variation of parameters.

**Laplace Transforms** A very brief look at how Laplace transforms
can be used to solve a system of differential equations.

**Modeling** In this section we’ll take a quick look at
some extensions of some of the modeling we did in previous sections that lead
to systems of equations.

**Series Solutions**

**Review : Power Series** A brief review of some of the basics of power
series.

**Review : Taylor Series** A reminder on how to construct the Taylor series for a
function.

**Series Solutions** In this section we will construct a series
solution for a differential equation about an ordinary point.

**Euler Equations** We will look at solutions to Euler’s
differential equation in this section.

** ****Higher Order Differential Equations**

**Basic Concepts for ***n*^{th} Order Linear Equations We’ll start the chapter off with a quick look
at some of the basic ideas behind solving higher order linear differential
equations.

**Linear Homogeneous Differential Equations**** ** In this section we’ll take a look at extending
the ideas behind solving 2^{nd} order differential equations to higher
order.

**Undetermined Coefficients** Here we’ll look at undetermined coefficients
for higher order differential equations.

**Variation of Parameters** We’ll look at variation of parameters for
higher order differential equations in this section.

**Laplace Transforms** In this section we’re just going to work an
example of using Laplace transforms to solve a differential equation on a 3^{rd}
order differential equation just so say that we looked at one with order higher
than 2^{nd}.

**Systems of Differential Equations** Here we’ll take a quick look at extending the
ideas we discussed when solving 2 x 2 systems of differential equations to
systems of size 3 x 3.

**Series Solutions** This section serves the same purpose as the
Laplace Transform section. It is just
here so we can say we’ve worked an example using series solutions for a
differential equations of order higher than 2^{nd}.

**Boundary Value Problems & Fourier
Series**

**Boundary Value Problems** In this section we’ll define the boundary
value problems as well as work some basic examples.

**Eigenvalues and
Eigenfunctions** Here we’ll take a look at the eigenvalues and
eigenfunctions for boundary value problems.

**Periodic Functions and Orthogonal Functions**
We’ll take a look at periodic functions and
orthogonal functions in section.

**Fourier Sine Series** In this section we’ll start looking at Fourier
Series by looking at a special case : Fourier Sine Series.

**Fourier Cosine Series** We’ll continue looking at Fourier Series by
taking a look at another special case : Fourier Cosine Series.

**Fourier Series** Here we will look at the full Fourier series.

**Convergence
of Fourier Series** Here we’ll take a look at some ideas involved
in the just what functions the Fourier series converge to as well as
differentiation and integration of a Fourier series.

**Partial Differential Equations**

**The Heat Equation** We do a partial derivation of the heat
equation in this section as well as a discussion of possible boundary values.

**The Wave Equation**
Here we do a partial derivation of the wave
equation.

**Terminology**
In this section we take a quick look at some
of the terminology used in the method of separation of variables.

**Separation
of Variables** We take a look at the first step in the method
of separation of variables in this section.
This first step is really the step that motivates the whole process.

**Solving the Heat Equation** In this section we go through the complete
separation of variables process and along the way solve the heat equation with
three different sets of boundary conditions.

**Heat Equation with Non-Zero Temperature
Boundaries** Here we take a quick look at solving the heat
equation in which the boundary conditions are fixed, non-zero temperature
conditions.

**Laplace’s
Equation** We discuss solving Laplace’s equation on both
a rectangle and a disk in this section.

**Vibrating
String** Here we solve the wave equation for a vibrating
string.

**Summary of
Separation of Variables** In this final section we give a quick summary
of the method of separation of variables.