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Full Description
Ordinary Differential Equationsrigorous yet remarkably accessible textbook ideal for an introductory course in ordinary differential equations. Providing a useful resource both in and out of the classroom, the text:Employs a unique expository style that explains the how and why of each topic coveredAllows for a flexible presentation based on instructor preference and student abilitySupports all claims with clear and solid proofsIncludes material rarely found in introductory textsOrdinary Differential Equations: An Introduction to the Fundamentals also includes access to an author-maintained website featuring detailed solutions and a wealth of bonus material. Use of a math software package that can do symbolic calculations, graphing, and so forth, such as Maple (TM) or Mathematica (R), is highly recommended, but not required.
Contents
THE BASICSThe Starting Point: Basic Concepts and TerminologyDifferential Equations: Basic Definitions and ClassificationsWhy Care about Differential Equations? Some Illustrative ExamplesMore on SolutionsAdditional ExercisesIntegration and Differential EquationsDirectly-Integrable EquationsOn Using Indefinite IntegralsOn Using Definite IntegralsIntegrals of Piecewise-Defined FunctionsAdditional ExercisesFIRST-ORDER EQUATIONSSome Basics about First-Order EquationsAlgebraically Solving for the DerivativeConstant (or Equilibrium) SolutionsOn the Existence and Uniqueness of SolutionsConfirming the Existence of Solutions (Core Ideas)Details in the Proof of Theorem 3.1On Proving Theorem 3.2Appendix: A Little Multivariable CalculusAdditional ExercisesSeparable First-Order EquationsBasic NotionsConstant SolutionsExplicit Versus Implicit SolutionsFull Procedure for Solving Separable EquationsExistence, Uniqueness, and False SolutionsOn the Nature of Solutions to Differential EquationsUsing and Graphing Implicit SolutionsOn Using Definite Integrals with Separable EquationsAdditional ExercisesLinear First-Order EquationsBasic NotionsSolving First-Order Linear EquationsOn Using Definite Integrals with Linear EquationsIntegrability, Existence and UniquenessAdditional ExercisesSimplifying Through SubstitutionBasic NotionsLinear SubstitutionsHomogeneous EquationsBernoulli EquationsAdditional ExercisesThe Exact Form and General Integrating FactorsThe Chain RuleThe Exact Form, DefinedSolving Equations in Exact FormTesting for Exactness-Part I"Exact Equations": A SummaryConverting Equations to Exact FormTesting for Exactness-Part IIAdditional ExercisesSlope Fields: Graphing Solutions without the SolutionsMotivation and Basic ConceptsThe Basic ProcedureObserving Long-Term Behavior in Slope FieldsProblem Points in Slope Fields, and Issues of Existence and UniquenessTests for StabilityAdditional ExercisesEuler's Numerical MethodDeriving the Steps of the MethodComputing via Euler's Method (Illustrated)What Can Go WrongReducing the ErrorError Analysis for Euler's MethodAdditional ExercisesThe Art and Science of Modeling with First-Order EquationsPreliminariesA Rabbit RanchExponential Growth and DecayThe Rabbit Ranch, AgainNotes on the Art and Science of ModelingMixing ProblemsSimple ThermodynamicsAppendix: Approximations That Are Not ApproximationsAdditional ExercisesSECOND- AND HIGHER-ORDER EQUATIONSHigher-Order Equations: Extending First-Order ConceptsTreating Some Second-Order Equations as First-OrderThe Other Class of Second-Order Equations "Easily Reduced" to First-OrderInitial-Value ProblemsOn the Existence and Uniqueness of SolutionsAdditional ExercisesHigher-Order Linear Equations and the Reduction of Order MethodLinear Differential Equations of All OrdersIntroduction to the Reduction of Order MethodReduction of Order for Homogeneous Linear Second-Order EquationsReduction of Order for Nonhomogeneous Linear Second-Order EquationsReduction of Order in GeneralAdditional ExercisesGeneral Solutions to Homogeneous Linear Differential EquationsSecond-Order Equations (Mainly)Homogeneous Linear Equations of Arbitrary OrderLinear Independence and WronskiansAdditional ExercisesVerifying the Big Theorems and an Introduction to Differential OperatorsVerifying the Big Theorem on Second-Order, Homogeneous EquationsProving the More General Theorems on General Solutions and WronskiansLinear Differential OperatorsAdditional ExercisesSecond-Order Homogeneous Linear Equations with Constant CoefficientsDeriving the Basic ApproachThe Basic Approach, SummarizedCase 1: Two Distinct Real RootsCase 2: Only One RootCase 3: Complex RootsSummaryAdditional ExercisesSprings: Part IModeling the ActionThe Mass/Spring Equation and Its SolutionsAdditional ExercisesArbitrary Homogeneous Linear Equations with Constant CoefficientsSome AlgebraSolving the Differential EquationMore ExamplesOn Verifying Theorem 17.2On Verifying Theorem 17.3Additional ExercisesEuler EquationsSecond-Order Euler EquationsThe Special CasesEuler Equations of Any OrderThe Relation between Euler and Constant Coefficient EquationsAdditional ExercisesNonhomogeneous Equations in GeneralGeneral Solutions to Nonhomogeneous EquationsSuperposition for Nonhomogeneous EquationsReduction of OrderAdditional ExercisesMethod of Undetermined Coefficients (aka: Method of Educated Guess)Basic IdeasGood First Guesses for Various Choices of gWhen the First Guess FailsMethod of Guess in GeneralCommon MistakesUsing the Principle of SuperpositionOn Verifying Theorem 20.1Additional ExercisesSprings: Part IIThe Mass/Spring SystemConstant ForceResonance and Sinusoidal ForcesMore on Undamped Motion under Nonresonant Sinusoidal ForcesAdditional ExercisesVariation of Parameters (A Better Reduction of Order Method)Second-Order Variation of ParametersVariation of Parameters for Even Higher Order EquationsThe Variation of Parameters FormulaAdditional ExercisesTHE LAPLACE TRANSFORMThe Laplace Transform (Intro)Basic Definition and ExamplesLinearity and Some More Basic TransformsTables and a Few More TransformsThe First Translation Identity (And More Transforms)What Is "Laplace Transformable"? (and Some Standard Terminology)Further Notes on Piecewise Continuity and Exponential OrderProving Theorem 23.5Additional ExercisesDifferentiation and the Laplace TransformTransforms of DerivativesDerivatives of TransformsTransforms of Integrals and Integrals of TransformsAppendix: Differentiating the TransformAdditional ExercisesThe Inverse Laplace TransformBasic NotionsLinearity and Using Partial FractionsInverse Transforms of Shifted FunctionsAdditional ExercisesConvolutionConvolution, the BasicsConvolution and Products of TransformsConvolution and Differential Equations (Duhamel's Principle)Additional ExercisesPiecewise-Defined Functions and Periodic FunctionsPiecewise-Defined FunctionsThe "Translation along the -T -Axis" IdentityRectangle Functions and Transforms of More Piecewise-Defined FunctionsConvolution with Piecewise-Defined FunctionsPeriodic FunctionsAn Expanded Table of IdentitiesDuhamel's Principle and ResonanceAdditional ExercisesDelta FunctionsVisualizing Delta FunctionsDelta Functions in ModelingThe Mathematics of Delta FunctionsDelta Functions and Duhamel's PrincipleSome "Issues" with Delta FunctionsAdditional ExercisesPOWER SERIES AND MODIFIED POWER SERIES SOLUTIONSSeries Solutions: PreliminariesInfinite SeriesPower Series and Analytic FunctionsElementary Complex AnalysisAdditional Basic Material That May Be UsefulAdditional ExercisesPower Series Solutions I: Basic Computational MethodsBasicsThe Algebraic Method with First-Order EquationsValidity of the Algebraic Method for First-Order EquationsThe Algebraic Method with Second-Order EquationsValidity of the Algebraic Method for Second-Order EquationsThe Taylor Series MethodAppendix: Using InductionAdditional ExercisesPower Series Solutions II: Generalizations and TheoryEquations with Analytic CoefficientsOrdinary and Singular Points, the Radius of Analyticity, and the Reduced FormThe Reduced FormsExistence of Power Series SolutionsRadius of Convergence for the Solution SeriesSingular Points and the Radius of ConvergenceAppendix: A Brief Overview of Complex CalculusAppendix: The "Closest Singular Point"Appendix: Singular Points and the Radius of Convergence for SolutionsAdditional ExercisesModified Power Series Solutions and the Basic Method of FrobeniusEuler Equations and Their SolutionsRegular and Irregular Singular Points (and the Frobenius Radius of Convergence)The (Basic) Method of FrobeniusBasic Notes on Using the Frobenius MethodAbout the Indicial and Recursion FormulasDealing with Complex ExponentsAppendix: On Tests for Regular Singular PointsAdditional ExercisesThe Big Theorem on the Frobenius Method, with ApplicationsThe Big TheoremsLocal Behavior of Solutions: IssuesLocal Behavior of Solutions: Limits at Regular Singular PointsLocal Behavior: Analyticity and Singularities in SolutionsCase Study: The Legendre EquationsFinding Second Solutions Using Theorem 33.2Additional ExercisesValidating the Method of FrobeniusBasic Assumptions and SymbologyThe Indicial Equation and Basic Recursion FormulaThe Easily Obtained Series SolutionsSecond Solutions When r1 = r2Second Solutions When r1 - r2 = KConvergence of the Solution SeriesSystems of Differential Equations: A Starting PointBasic Terminology and NotionsA Few Illustrative ApplicationsConverting Differential Equations to First-Order SystemsUsing Laplace Transforms to Solve SystemsExistence, Uniqueness and General Solutions for SystemsSingle Nth-order Differential EquationsAdditional ExercisesCritical Points, Direction Fields and TrajectoriesThe Systems of Interest and Some Basic NotationConstant/Equilibrium Solutions"Graphing" Standard SystemsSketching Trajectories for Autonomous SystemsCritical Points, Stability and Long-Term BehaviorApplicationsExistence and Uniqueness of TrajectoriesProving Theorem 36.2Additional ExercisesAppendix: Author's Guide to Using This TextOverviewChapter-by-Chapter GuideAnswers to Selected Exercises
