Howell - Ordinary Differential Equations

Ordinary Differential Equations

Medium: Buch

ISBN: 978-1-4987-3381-6

Verlag: CRC Press

Erscheinungstermin: 10.12.2015

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THE BASICS
The Starting Point: Basic Concepts and TerminologyDifferential Equations: Basic Definitions and ClassificationsWhy Care about Differential Equations? Some Illustrative ExamplesMore on SolutionsAdditional Exercises
Integration and Differential EquationsDirectly-Integrable EquationsOn Using Indefinite IntegralsOn Using Definite IntegralsIntegrals of Piecewise-Defined FunctionsAdditional Exercises
FIRST-ORDER EQUATIONS
Some 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 Exercises
Separable 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 Exercises
Linear First-Order EquationsBasic NotionsSolving First-Order Linear EquationsOn Using Definite Integrals with Linear EquationsIntegrability, Existence and UniquenessAdditional Exercises
Simplifying Through SubstitutionBasic NotionsLinear SubstitutionsHomogeneous EquationsBernoulli EquationsAdditional Exercises
The 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 Exercises
Slope 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 Exercises
Euler’s Numerical MethodDeriving the Steps of the MethodComputing via Euler’s Method (Illustrated)What Can Go WrongReducing the ErrorError Analysis for Euler’s MethodAdditional Exercises
The 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 Exercises
SECOND- AND HIGHER-ORDER EQUATIONS
Higher-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 Exercises
Higher-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 Exercises
General Solutions to Homogeneous Linear Differential EquationsSecond-Order Equations (Mainly)Homogeneous Linear Equations of Arbitrary OrderLinear Independence and WronskiansAdditional Exercises
Verifying 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 Exercises
Second-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 Exercises
Springs: Part IModeling the ActionThe Mass/Spring Equation and Its SolutionsAdditional Exercises
Arbitrary Homogeneous Linear Equations with Constant CoefficientsSome AlgebraSolving the Differential EquationMore ExamplesOn Verifying Theorem 17.2On Verifying Theorem 17.3Additional Exercises
Euler EquationsSecond-Order Euler EquationsThe Special CasesEuler Equations of Any OrderThe Relation between Euler and Constant Coefficient EquationsAdditional Exercises
Nonhomogeneous Equations in GeneralGeneral Solutions to Nonhomogeneous EquationsSuperposition for Nonhomogeneous EquationsReduction of OrderAdditional Exercises
Method 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 Exercises
Springs: Part IIThe Mass/Spring SystemConstant ForceResonance and Sinusoidal ForcesMore on Undamped Motion under Nonresonant Sinusoidal ForcesAdditional Exercises
Variation of Parameters (A Better Reduction of Order Method)Second-Order Variation of ParametersVariation of Parameters for Even Higher Order EquationsThe Variation of Parameters FormulaAdditional Exercises
THE LAPLACE TRANSFORM
The 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 Exercises
Differentiation and the Laplace TransformTransforms of DerivativesDerivatives of TransformsTransforms of Integrals and Integrals of TransformsAppendix: Differentiating the TransformAdditional Exercises
The Inverse Laplace TransformBasic NotionsLinearity and Using Partial FractionsInverse Transforms of Shifted FunctionsAdditional Exercises
ConvolutionConvolution, the BasicsConvolution and Products of TransformsConvolution and Differential Equations (Duhamel’s Principle)Additional Exercises
Piecewise-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 Exercises
Delta FunctionsVisualizing Delta FunctionsDelta Functions in ModelingThe Mathematics of Delta FunctionsDelta Functions and Duhamel’s PrincipleSome "Issues" with Delta FunctionsAdditional Exercises
POWER SERIES AND MODIFIED POWER SERIES SOLUTIONS
Series Solutions: PreliminariesInfinite SeriesPower Series and Analytic FunctionsElementary Complex AnalysisAdditional Basic Material That May Be UsefulAdditional Exercises
Power 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 Exercises
Power 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 Exercises
Modified 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 Exercises
The 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 Exercises
Validating 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 Series
SYSTEMS OF DIFFERENTIAL EQUATIONS (A BRIEF INTRODUCTION)
Systems 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 Exercises
Critical 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 Exercises
Appendix: Author’s Guide to Using This TextOverviewChapter-by-Chapter Guide
Answers to Selected Exercises

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