Ordinary differential equations : an introduction to the fundamentals /: an introduction to the fundamentals. (2016)

• Record Type: Book
• Further Information: Note: Kenneth B. Howell.
• Authors: Howell, Kenneth B
• Contents: THE BASICS The Starting Point: Basic Concepts and Terminology; Differential Equations: Basic Definitions and Classifications; Why Care about Differential Equations? Some Illustrative Examples; More on Solutions; Additional Exercises Integration and Differential Equations; Directly-Integrable Equations; On Using Indefinite Integrals; On Using Definite Integrals; Integrals of Piecewise-Defined Functions; Additional Exercises FIRST-ORDER EQUATIONS Some Basics about First-Order Equations; Algebraically Solving for the Derivative; Constant (or Equilibrium) Solutions; On the Existence and Uniqueness of Solutions; Confirming the Existence of Solutions (Core Ideas); Details in the Proof of Theorem 3.1; On Proving Theorem 3.2; Appendix: A Little Multivariable Calculus; Additional Exercises Separable First-Order Equations; Basic Notions; Constant Solutions; Explicit Versus Implicit Solutions; Full Procedure for Solving Separable Equations; Existence, Uniqueness, and False Solutions; On the Nature of Solutions to Differential Equations; Using and Graphing Implicit Solutions; On Using Definite Integrals with Separable Equations; Additional Exercises Linear First-Order Equations; Basic Notions; Solving First-Order Linear Equations; On Using Definite Integrals with Linear Equations; Integrability, Existence and Uniqueness; Additional Exercises Simplifying Through Substitution; Basic Notions; Linear Substitutions; Homogeneous Equations; Bernoulli Equations; Additional Exercises The Exact Form and General Integrating Factors; The Chain Rule; The Exact Form, Defined; Solving Equations in Exact Form; Testing for Exactness—Part I; "Exact Equations": A Summary; Converting Equations to Exact Form; Testing for Exactness—Part II; Additional Exercises Slope Fields: Graphing Solutions without the Solutions; Motivation and Basic Concepts; The Basic Procedure; Observing Long-Term Behavior in Slope Fields; Problem Points in Slope Fields, and Issues of Existence and Uniqueness; Tests for Stability; Additional Exercises Euler’s Numerical Method; Deriving the Steps of the Method; Computing via Euler’s Method (Illustrated); What Can Go Wrong; Reducing the Error; Error Analysis for Euler’s Method; Additional Exercises The Art and Science of Modeling with First-Order Equations; Preliminaries; A Rabbit Ranch; Exponential Growth and Decay; The Rabbit Ranch, Again; Notes on the Art and Science of Modeling; Mixing Problems; Simple Thermodynamics; Appendix: Approximations That Are Not Approximations; Additional Exercises SECOND- AND HIGHER-ORDER EQUATIONS Higher-Order Equations: Extending First-Order Concepts; Treating Some Second-Order Equations as First-Order; The Other Class of Second-Order Equations "Easily Reduced" to First-Order; Initial-Value Problems; On the Existence and Uniqueness of Solutions; Additional Exercises Higher-Order Linear Equations and the Reduction of Order Method; Linear Differential Equations of All Orders; Introduction to the Reduction of Order Method; Reduction of Order for Homogeneous Linear Second-Order Equations; Reduction of Order for Nonhomogeneous Linear Second-Order Equations; Reduction of Order in General; Additional Exercises General Solutions to Homogeneous Linear Differential Equations; Second-Order Equations (Mainly); Homogeneous Linear Equations of Arbitrary Order; Linear Independence and Wronskians; Additional Exercises Verifying the Big Theorems and an Introduction to Differential Operators; Verifying the Big Theorem on Second-Order, Homogeneous Equations; Proving the More General Theorems on General Solutions and Wronskians; Linear Differential Operators; Additional Exercises Second-Order Homogeneous Linear Equations with Constant Coefficients; Deriving the Basic Approach; The Basic Approach, Summarized; Case 1: Two Distinct Real Roots; Case 2: Only One Root; Case 3: Complex Roots; Summary; Additional Exercises Springs: Part I; Modeling the Action; The Mass/Spring Equation and Its Solutions; Additional Exercises Arbitrary Homogeneous Linear Equations with Constant Coefficients; Some Algebra; Solving the Differential Equation; More Examples; On Verifying Theorem 17.2; On Verifying Theorem 17.3; Additional Exercises Euler Equations; Second-Order Euler Equations; The Special Cases; Euler Equations of Any Order; The Relation between Euler and Constant Coefficient Equations; Additional Exercises Nonhomogeneous Equations in General; General Solutions to Nonhomogeneous Equations; Superposition for Nonhomogeneous Equations; Reduction of Order; Additional Exercises Method of Undetermined Coefficients (aka: Method of Educated Guess); Basic Ideas; Good First Guesses for Various Choices of g ; When the First Guess Fails; Method of Guess in General; Common Mistakes; Using the Principle of Superposition; On Verifying Theorem 20.1; Additional Exercises Springs: Part II; The Mass/Spring System; Constant Force; Resonance and Sinusoidal Forces; More on Undamped Motion under Nonresonant Sinusoidal Forces; Additional Exercises Variation of Parameters (A Better Reduction of Order Method); Second-Order Variation of Parameters; Variation of Parameters for Even Higher Order Equations; The Variation of Parameters Formula; Additional Exercises THE LAPLACE TRANSFORM The Laplace Transform (Intro); Basic Definition and Examples; Linearity and Some More Basic Transforms; Tables and a Few More Transforms; The First Translation Identity (And More Transforms); What Is "Laplace Transformable"? (and Some Standard Terminology); Further Notes on Piecewise Continuity and Exponential Order; Proving Theorem 23.5; Additional Exercises Differentiation and the Laplace Transform; Transforms of Derivatives; Derivatives of Transforms; Transforms of Integrals and Integrals of Transforms; Appendix: Differentiating the Transform; Additional Exercises The Inverse Laplace Transform; Basic Notions; Linearity and Using Partial Fractions; Inverse Transforms of Shifted Functions; Additional Exercises Convolution; Convolution, the Basics; Convolution and Products of Transforms; Convolution and Differential Equations (Duhamel’s Principle); Additional Exercises Piecewise-Defined Functions and Periodic Functions; Piecewise-Defined Functions; The "Translation along the -T -Axis" Identity; Rectangle Functions and Transforms of More Piecewise-Defined Functions; Convolution with Piecewise-Defined Functions; Periodic Functions; An Expanded Table of Identities; Duhamel’s Principle and Resonance; Additional Exercises Delta Functions; Visualizing Delta Functions; Delta Functions in Modeling; The Mathematics of Delta Functions; Delta Functions and Duhamel’s Principle; Some "Issues" with Delta Functions; Additional Exercises POWER SERIES AND MODIFIED POWER SERIES SOLUTIONS Series Solutions: Preliminaries; Infinite Series; Power Series and Analytic Functions; Elementary Complex Analysis; Additional Basic Material That May Be Useful; Additional Exercises Power Series Solutions I: Basic Computational Methods; </STRONG&gt
• Edition: 1st
• Publisher Details: Boca Raton : CRC Press
• Publication Date: 2016
• Extent: 1 online resource, illustrations (black and white)
• Subjects: 515.35; Differential equations
• Languages: English
• ISBNs: 9781498734202; 9781498733847; 9781498734219
• Related ISBNs: 9781498733816
• Notes: Note: Description based on CIP data; item not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.