Differential Equations By Paul Dawkins Free download

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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

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

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.