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DIFFERENTIAL
EQUATIONS

Paul Dawkins

Differential Equations

Preface ............................................................................................................................................ 3
Outline ........................................................................................................................................... iv
Basic Concepts ............................................................................................................................... 1
Introduction ................................................................................................................................................ 1
Definitions .................................................................................................................................................. 2
Direction Fields .......................................................................................................................................... 8
Final Thoughts ......................................................................................................................................... 19
First Order Differential Equations ............................................................................................ 20
Introduction .............................................................................................................................................. 20
Linear Differential Equations ................................................................................................................... 21
Separable Differential Equations ............................................................................................................. 34
Exact Differential Equations .................................................................................................................... 45
Bernoulli Differential Equations .............................................................................................................. 56
Substitutions ............................................................................................................................................. 63
Intervals of Validity ................................................................................................................................. 72
Modeling with First Order Differential Equations ................................................................................... 77
Equilibrium Solutions .............................................................................................................................. 90
Euler’s Method ......................................................................................................................................... 94
Second Order Differential Equations ...................................................................................... 102
Introduction .............................................................................................................................................102
Basic Concepts ........................................................................................................................................104
Real, Distinct Roots ................................................................................................................................109
Complex Roots ........................................................................................................................................113
Repeated Roots .......................................................................................................................................118
Reduction of Order ..................................................................................................................................122
Fundamental Sets of Solutions ................................................................................................................126
More on the Wronskian ...........................................................................................................................131
Nonhomogeneous Differential Equations ...............................................................................................137
Undetermined Coefficients .....................................................................................................................139
Variation of Parameters ...........................................................................................................................156
Mechanical Vibrations ............................................................................................................................162
Laplace Transforms .................................................................................................................. 181
Introduction .............................................................................................................................................181
The Definition .........................................................................................................................................183
Laplace Transforms .................................................................................................................................187
Inverse Laplace Transforms ....................................................................................................................191
Step Functions .........................................................................................................................................202
Solving IVP’s with Laplace Transforms .................................................................................................215
Nonconstant Coefficient IVP’s ...............................................................................................................222
IVP’s With Step Functions ......................................................................................................................226
Dirac Delta Function ...............................................................................................................................233
Convolution Integrals ..............................................................................................................................236
Systems of Differential Equations ............................................................................................ 241
Introduction .............................................................................................................................................241
Review : Systems of Equations ...............................................................................................................243
Review : Matrices and Vectors ...............................................................................................................249
Review : Eigenvalues and Eigenvectors .................................................................................................259
Systems of Differential Equations ...........................................................................................................269
Solutions to Systems ...............................................................................................................................273
Phase Plane .............................................................................................................................................275
Real, Distinct Eigenvalues ......................................................................................................................280
Complex Eigenvalues..............................................................................................................................290
Repeated Eigenvalues .............................................................................................................................296
i
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Differential Equations
Nonhomogeneous Systems .....................................................................................................................303
Laplace Transforms .................................................................................................................................307
Modeling .................................................................................................................................................309
Series Solutions to Differential Equations ............................................................................... 318
Introduction .............................................................................................................................................318
Review : Power Series ............................................................................................................................319
Review : Taylor Series ............................................................................................................................327
Series Solutions to Differential Equations ..............................................................................................330
Euler Equations .......................................................................................................................................340
Higher Order Differential Equations ...................................................................................... 346
Introduction .............................................................................................................................................346
Basic Concepts for nth Order Linear Equations .......................................................................................347
Linear Homogeneous Differential Equations ..........................................................................................350
Undetermined Coefficients .....................................................................................................................355
Variation of Parameters ...........................................................................................................................357
Laplace Transforms .................................................................................................................................363
Systems of Differential Equations ...........................................................................................................365
Series Solutions .......................................................................................................................................370
Boundary Value Problems & Fourier Series .......................................................................... 374
Introduction .............................................................................................................................................374
Boundary Value Problems .....................................................................................................................375
Eigenvalues and Eigenfunctions .............................................................................................................381
Periodic Functions, Even/Odd Functions and Orthogonal Functions .....................................................398
Fourier Sine Series ..................................................................................................................................406
Fourier Cosine Series ..............................................................................................................................417
Fourier Series ..........................................................................................................................................426
Convergence of Fourier Series ................................................................................................................434
Partial Differential Equations .................................................................................................. 440
Introduction .............................................................................................................................................440
The Heat Equation ..................................................................................................................................442
The Wave Equation .................................................................................................................................449
Terminology ............................................................................................................................................451
Separation of Variables ...........................................................................................................................454
Solving the Heat Equation ......................................................................................................................465
Heat Equation with Non-Zero Temperature Boundaries .........................................................................478
Laplace’s Equation ..................................................................................................................................481
Vibrating String .......................................................................................................................................492
Summary of Separation of Variables ......................................................................................................495

ii
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Differential Equations

Preface

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.

1. 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.

2. 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.

3. 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.

4. 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.

iii
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Differential Equations

Outline

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.
iv
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Differential Equations
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,
v
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Differential Equations
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 nth 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 2nd 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 3rd order
differential equation just so say that we looked at one with order higher than 2nd.
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.
vi
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Differential Equations
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 2nd.

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 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.

vii
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Differential Equations

Basic Concepts

Introduction
There isn’t really a whole lot to this chapter it is mainly here so we can get some basic definitions
and concepts out of the way. Most of the definitions and concepts introduced here can be
introduced without any real knowledge of how to solve differential equations. Most of them are
terms that we’ll use throughout a class so getting them out of the way right at the beginning is a
good idea.

During an actual class I tend to hold off on a couple of the definitions and introduce them at a
later point when we actually start solving differential equations. The reason for this is mostly a
time issue. In this class time is usually at a premium and some of the definitions/concepts require
a differential equation and/or its solution so I use the first couple differential equations that we
will solve to introduce the definition or concept.

Here is a quick list of the topics in this Chapter.

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.

1
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Differential Equations
Definitions

Differential Equation
The first definition that we should cover should be that of differential equation. A differential
equation is any equation which contains derivatives, either ordinary derivatives or partial
derivatives.

There is one differential equation that everybody probably knows, that is Newton’s Second Law
of Motion. If an object of mass m is moving with acceleration a and being acted on with force F
then Newton’s Second Law tells us.

F = ma (1)

To see that this is in fact a differential equation we need to rewrite it a little. First, remember that
we can rewrite the acceleration, a, in one of two ways.
2
dv
d u

a =
OR
a =
(2)
2
dt
dt

Where v is the velocity of the object and u is the position function of the object at any time t. We
should also remember at this point that the force, F may also be a function of time, velocity,
and/or position.

So, with all these things in mind Newton’s Second Law can now be written as a differential
equation in terms of either the velocity, v, or the position, u, of the object as follows.
dv

m
= F (t,v) (3)
dt
2
d u

du

m
= F t,u,

(4)
2
dt

dt

So, here is our first differential equation. We will see both forms of this in later chapters.

Here are a few more examples of differential equations.

ay′′ + by′ + cy = g (t ) (5)
2
d y
dy

sin ( y)
= (1− y)
2
5
y
+ y e (6)
2
dx
dx
(4)

y
+10y′′′ − 4y′ + 2y = cos(t) (7)
2u
u

2
α ∂

=
(8)
2
x

t

2
a u = u (9)
xx
tt
3
u
u

= 1+
(10)
2
x t

y

Order
The order of a differential equation is the largest derivative present in the differential equation. In
the differential equations listed above (3) is a first order differential equation, (4), (5), (6), (8),
2
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