# differential equation

### Text-only Preview

DIFFERENTIAL

EQUATIONS

Paul Dawkins

Differential Equations

**Table of Contents****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

© 2007 Paul Dawkins

i

http://tutorial.math.lamar.edu/terms.aspx

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

© 2007 Paul Dawkins

ii

http://tutorial.math.lamar.edu/terms.aspx

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.

© 2007 Paul Dawkins

iii

http://tutorial.math.lamar.edu/terms.aspx

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.

© 2007 Paul Dawkins

iv

http://tutorial.math.lamar.edu/terms.aspx

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,

© 2007 Paul Dawkins

v

http://tutorial.math.lamar.edu/terms.aspx

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**– We’ll start the chapter off

*n*th Order Linear Equationswith 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.

© 2007 Paul Dawkins

vi

http://tutorial.math.lamar.edu/terms.aspx

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.

© 2007 Paul Dawkins

vii

http://tutorial.math.lamar.edu/terms.aspx

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.

© 2007 Paul Dawkins

1

http://tutorial.math.lamar.edu/terms.aspx

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*

+10

*y*′′′ − 4

*y*′ + 2

*y*= cos(

*t*)

**(7)**

2

*u*

*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),

© 2007 Paul Dawkins

2

http://tutorial.math.lamar.edu/terms.aspx

# Document Outline

- ÿ
- ÿ
- ÿ

- ÿ
- ÿ
- ÿ
- ÿ
- ÿ
- ÿ

- ÿ
- ÿ
- ÿ
- ÿ
- ÿ
- ÿ

- ÿ
- ÿ
- ÿ
- ÿ
- ÿ
- ÿ

- ÿ
- ÿ
- ÿ
- ÿ
- ÿ
- ÿ
- ÿ

- ÿ
- ÿ
- ÿ

- ÿ
- ÿ
- ÿ
- ÿ

- ÿ
- ÿ
- ÿ
- ÿ
- ÿ

- ÿ
- ÿ
- ÿ
- ÿ
- ÿ