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DIFFERENTIAL

EQUATIONS

Paul Dawkins

Differential Equations

© 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

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 ..................................................................................................................................71

Modeling with First Order Differential Equations ....................................................................................76

Equilibrium Solutions ...............................................................................................................................89

Euler's Method ..........................................................................................................................................93

Second Order Differential Equations ...................................................................................... 101

Introduction .............................................................................................................................................101

Basic Concepts ........................................................................................................................................103

Real, Distinct Roots ................................................................................................................................108

Complex Roots ........................................................................................................................................112

Repeated Roots .......................................................................................................................................117

Reduction of Order ..................................................................................................................................121

Fundamental Sets of Solutions ................................................................................................................125

More on the Wronskian ...........................................................................................................................130

Nonhomogeneous Differential Equations ...............................................................................................136

Undetermined Coefficients .....................................................................................................................138

Variation of Parameters...........................................................................................................................155

Mechanical Vibrations ............................................................................................................................161

Laplace Transforms .................................................................................................................. 180

Introduction .............................................................................................................................................180

The Definition .........................................................................................................................................182

Laplace Transforms .................................................................................................................................186

Inverse Laplace Transforms ....................................................................................................................190

Step Functions .........................................................................................................................................201

Solving IVP's with Laplace Transforms .................................................................................................214

Nonconstant Coefficient IVP's ...............................................................................................................221

IVP's With Step Functions......................................................................................................................225

Dirac Delta Function ...............................................................................................................................232

Convolution Integrals ..............................................................................................................................235

Systems of Differential Equations ............................................................................................ 240

Introduction .............................................................................................................................................240

Review : Systems of Equations ...............................................................................................................242

Review : Matrices and Vectors ...............................................................................................................248

Review : Eigenvalues and Eigenvectors .................................................................................................258

Systems of Differential Equations...........................................................................................................268

Solutions to Systems ...............................................................................................................................272

Phase Plane

Real, Distinct Eigenvalues ......................................................................................................................279

Complex Eigenvalues .............................................................................................................................289

Repeated Eigenvalues .............................................................................................................................295

Differential Equations

© 2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx

Nonhomogeneous Systems .....................................................................................................................302

Laplace Transforms .................................................................................................................................306

Modeling .................................................................................................................................................308

Series Solutions to Differential Equations ............................................................................... 317

Introduction .............................................................................................................................................317

Review : Power Series ............................................................................................................................318

Review : Taylor Series ............................................................................................................................326

Series Solutions to Differential Equations ..............................................................................................329

Euler Equations .......................................................................................................................................339

Higher Order Differential Equations ...................................................................................... 345

Introduction .............................................................................................................................................345

Basic Concepts for

n th

Order Linear Equations .......................................................................................346

Linear Homogeneous Differential Equations ..........................................................................................349

Undetermined Coefficients .....................................................................................................................354

Variation of Parameters...........................................................................................................................356

Laplace Transforms .................................................................................................................................362

Systems of Differential Equations...........................................................................................................364

Series Solutions .......................................................................................................................................369

Boundary Value Problems & Fourier Series .......................................................................... 373

Introduction .............................................................................................................................................373

Boundary Value Problems .....................................................................................................................374

Eigenvalues and Eigenfunctions .............................................................................................................380

Periodic Functions, Even/Odd Functions and Orthogonal Functions .....................................................397

Fourier Sine Series ..................................................................................................................................405

Fourier Cosine Series ..............................................................................................................................416

Fourier Series ..........................................................................................................................................425

Convergence of Fourier Series ................................................................................................................433

Partial Differential Equations .................................................................................................. 439

Introduction .............................................................................................................................................439

The Heat Equation ..................................................................................................................................441

The Wave Equation .................................................................................................................................448

Terminology

Separation of Variables ...........................................................................................................................453

Solving the Heat Equation ......................................................................................................................464

Heat Equation with Non-Zero Temperature Boundaries .........................................................................477

Laplace's Equation ..................................................................................................................................480

Vibrating String.......................................................................................................................................491

Summary of Separation of Variables ......................................................................................................494

Differential Equations

© 2007 Paul Dawkins iii http://tutorial.math.lamar.edu/terms.aspx

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 u sually 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.

Differential Equations

© 2007 Paul Dawkins iv http://tutorial.math.lamar.edu/terms.aspx

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 b e 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 diffe rential 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.

Differential Equations

© 2007 Paul Dawkins v http://tutorial.math.lamar.edu/terms.aspx

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 fo r 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,

Differential Equations

© 2007 Paul Dawkins vi http://tutorial.math.lamar.edu/terms.aspx 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 usingquotesdbs_dbs12.pdfusesText_18