DIFFERENTIAL EQUATIONS - Mathematics

1 DIFFERENTIAL EQUATIONS Paul Dawkins DIFFERENTIAL EQUATIONS 2007 Paul Dawkins i 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.

2 138 Variation of 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 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 Solutions to Systems ..272 Phase Plane ..274 Real, Distinct Eigenvalues ..279 Complex Eigenvalues ..289 Repeated Eigenvalues ..295 DIFFERENTIAL EQUATIONS 2007 Paul Dawkins ii Nonhomogeneous Systems ..302 Laplace Transforms ..306 Modeling ..308 Series Solutions to DIFFERENTIAL EQUATIONS .. 317 Introduction ..317 Review : Power Series.

3 318 Review : Taylor Series ..326 Series Solutions to DIFFERENTIAL EQUATIONS ..329 Euler EQUATIONS ..339 Higher Order DIFFERENTIAL EQUATIONS .. 345 Introduction ..345 Basic Concepts for nth Order Linear EQUATIONS ..346 Linear Homogeneous DIFFERENTIAL EQUATIONS ..349 Undetermined Coefficients ..354 Variation of Laplace Transforms ..362 Systems of DIFFERENTIAL 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 ..450 Separation of Variables ..453 Solving the Heat Equation.

4 464 Heat Equation with Non-Zero Temperature Boundaries ..477 Laplace s Equation ..480 Vibrating Summary of Separation of Variables ..494 DIFFERENTIAL EQUATIONS 2007 Paul Dawkins iii Pref ace 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.

5 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.

6 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 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.

7 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.

8 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.

9 Complex Roots Solving DIFFERENTIAL EQUATIONS whose characteristic equation complex real roots. DIFFERENTIAL EQUATIONS 2007 Paul Dawkins v 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.

10 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.