DIFFERENTIAL EQUATIONS - Mathematics

DIFFERENTIALEQUATIONSPaul Dawkins

Differential EquationsTable of ContentsPreface . 3Outline . ivBasic Concepts . 1Introduction . 1Definitions. 2Direction Fields . 8Final Thoughts .19First Order Differential Equations . 20Introduction .20Linear Differential Equations .21Separable Differential Equations .34Exact Differential Equations .45Bernoulli Differential Equations .56Substitutions .63Intervals of Validity .71Modeling with First Order Differential Equations .76Equilibrium Solutions .89Euler’s Method .93Second Order Differential Equations . 101Introduction .101Basic Concepts .103Real, Distinct Roots .108Complex Roots .112Repeated Roots .117Reduction of Order.121Fundamental Sets of Solutions .125More on the Wronskian.130Nonhomogeneous Differential Equations .136Undetermined Coefficients .138Variation of Parameters.155Mechanical Vibrations .161Laplace Transforms . 180Introduction .180The Definition .182Laplace Transforms.186Inverse Laplace Transforms .190Step Functions .201Solving IVP’s with Laplace Transforms .214Nonconstant Coefficient IVP’s .221IVP’s With Step Functions.225Dirac Delta Function .232Convolution Integrals .235Systems of Differential Equations . 240Introduction .240Review : Systems of Equations .242Review : Matrices and Vectors .248Review : Eigenvalues and Eigenvectors .258Systems of Differential Equations.268Solutions to Systems .272Phase Plane .274Real, Distinct Eigenvalues .279Complex Eigenvalues .289Repeated Eigenvalues .295 2007 Paul Dawkinsihttp://tutorial.math.lamar.edu/terms.aspx

Differential EquationsNonhomogeneous Systems .302Laplace Transforms.306Modeling .308Series Solutions to Differential Equations . 317Introduction .317Review : Power Series .318Review : Taylor Series .326Series Solutions to Differential Equations .329Euler Equations .339Higher Order Differential Equations . 345Introduction .345Basic Concepts for nth Order Linear Equations .346Linear Homogeneous Differential Equations .349Undetermined Coefficients .354Variation of Parameters.356Laplace Transforms.362Systems of Differential Equations.364Series Solutions .369Boundary Value Problems & Fourier Series . 373Introduction .373Boundary Value Problems .374Eigenvalues and Eigenfunctions .380Periodic Functions, Even/Odd Functions and Orthogonal Functions .397Fourier Sine Series .405Fourier Cosine Series .416Fourier Series .425Convergence of Fourier Series .433Partial Differential Equations . 439Introduction .439The Heat Equation .441The Wave Equation .448Terminology .450Separation of Variables .453Solving the Heat Equation .464Heat Equation with Non-Zero Temperature Boundaries .477Laplace’s Equation .480Vibrating String.491Summary of Separation of Variables .494 2007 Paul Dawkinsiihttp://tutorial.math.lamar.edu/terms.aspx

Differential EquationsPrefaceHere are my online notes for my differential equations course that I teach here at LamarUniversity. Despite the fact that these are my “class notes”, they should be accessible to anyonewanting to learn how to solve differential equations or needing a refresher on differentialequations.I’ve tried to make these notes as self contained as possible and so all the information needed toread through them is either from a Calculus or Algebra class or contained in other sections of thenotes.A couple of warnings to my students who may be here to get a copy of what happened on a daythat you missed.1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learndifferential equations I have included some material that I do not usually have time tocover 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 thesenotes 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 ofthe moment and so in this class my class work will follow these notes fairly close as faras worked problems go. With that being said I will, on occasion, work problems off thetop of my head when I can to provide more examples than just those in my notes. Also, Ioften don’t have time in class to work all of the problems in the notes and so you willfind that some sections contain problems that weren’t worked in class due to timerestrictions.3. Sometimes questions in class will lead down paths that are not covered here. I try toanticipate as many of the questions as possible in writing these up, but the reality is that Ican’t anticipate all the questions. Sometimes a very good question gets asked in classthat leads to insights that I’ve not included here. You should always talk to someone whowas in class on the day you missed and compare these notes to their notes and see whatthe differences are.4. This is somewhat related to the previous three items, but is important enough to merit itsown 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 notednot everything in these notes is covered in class and often material or insights not in thesenotes is covered in class. 2007 Paul x

Differential EquationsOutlineHere is a listing and brief description of the material in this set of notes.Basic ConceptsDefinitions – Some of the common definitions and concepts in a differentialequations courseDirection Fields – An introduction to direction fields and what they can tell usabout the solution to a differential equation.Final Thoughts – A couple of final thoughts on what we will be looking atthroughout this course.First Order Differential EquationsLinear Equations – Identifying and solving linear first order differentialequations.Separable Equations – Identifying and solving separable first order differentialequations. We’ll also start looking at finding the interval of validity from thesolution to a differential equation.Exact Equations – Identifying and solving exact differential equations. We’lldo a few more interval of validity problems here as well.Bernoulli Differential Equations – In this section we’ll see how to solve theBernoulli Differential Equation. This section will also introduce the idea ofusing a substitution to help us solve differential equations.Substitutions – We’ll pick up where the last section left off and take a look at acouple of other substitutions that can be used to solve some differential equationsthat we couldn’t otherwise solve.Intervals of Validity – Here we will give an in-depth look at intervals of validityas well as an answer to the existence and uniqueness question for first orderdifferential equations.Modeling with First Order Differential Equations – Using first orderdifferential equations to model physical situations. The section will show somevery real applications of first order differential equations.Equilibrium Solutions – We will look at the behavior of equilibrium solutionsand autonomous differential equations.Euler’s Method – In this section we’ll take a brief look at a method forapproximating solutions to differential equations.Second Order Differential EquationsBasic Concepts – Some of the basic concepts and ideas that are involved insolving second order differential equations.Real Roots – Solving differential equations whose characteristic equation hasreal roots.Complex Roots – Solving differential equations whose characteristic equationcomplex real roots. 2007 Paul Dawkinsivhttp://tutorial.math.lamar.edu/terms.aspx

Differential EquationsRepeated Roots – Solving differential equations whose characteristic equationhas repeated roots.Reduction of Order – A brief look at the topic of reduction of order. This willbe one of the few times in this chapter that non-constant coefficient differentialequation will be looked at.Fundamental Sets of Solutions – A look at some of the theory behind thesolution to second order differential equations, including looks at the Wronskianand fundamental sets of solutions.More on the Wronskian – An application of the Wronskian and an alternatemethod for finding it.Nonhomogeneous Differential Equations – A quick look into how to solvenonhomogeneous differential equations in general.Undetermined Coefficients – The first method for solving nonhomogeneousdifferential equations that we’ll be looking at in this section.Variation of Parameters – Another method for solving nonhomogeneousdifferential equations.Mechanical Vibrations – An application of second order differential equations.This section focuses on mechanical vibrations, yet a simple change of notationcan move this into almost any other engineering field.Laplace TransformsThe Definition – The definition of the Laplace transform. We will also computea couple Laplace transforms using the definition.Laplace Transforms – As the previous section will demonstrate, computingLaplace transforms directly from the definition can be a fairly painful process. Inthis section we introduce the way we usually compute Laplace transforms.Inverse La

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