Ordinary Differential Equations: An Introduction to the Fundamentals(Textbooks in Mathematics)

Ordinary Differential Equations: An Introduction to the Fundamentals is a rigorous 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 covered Allows for a flexible presentation based on instructor preference and student ability Supports all claims with clear and solid proofs Includes material rarely found in introductory texts Ordinary 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™ or Mathematica®, is highly recommended, but not required.

Table of 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 Basics The Algebraic Method with First-Order Equations Validity of the Algebraic Method for First-Order Equations The Algebraic Method with Second-Order Equations Validity of the Algebraic Method for Second-Order Equations The Taylor Series Method Appendix: Using Induction Additional Exercises Power Series Solutions II: Generalizations and Theory Equations with Analytic Coefficients Ordinary and Singular Points, the Radius of Analyticity, and the Reduced Form The Reduced Forms Existence of Power Series Solutions Radius of Convergence for the Solution Series Singular Points and the Radius of Convergence Appendix: A Brief Overview of Complex Calculus Appendix: The "Closest Singular Point" Appendix: Singular Points and the Radius of Convergence for Solutions Additional Exercises Modified Power Series Solutions and the Basic Method of Frobenius Euler Equations and Their Solutions Regular and Irregular Singular Points (and the Frobenius Radius of Convergence) The (Basic) Method of Frobenius Basic Notes on Using the Frobenius Method About the Indicial and Recursion Formulas Dealing with Complex Exponents Appendix: On Tests for Regular Singular Points Additional Exercises The Big Theorem on the Frobenius Method, with Applications The Big Theorems Local Behavior of Solutions: Issues Local Behavior of Solutions: Limits at Regular Singular Points Local Behavior: Analyticity and Singularities in Solutions Case Study: The Legendre Equations Finding Second Solutions Using Theorem 33.2 Additional Exercises Validating the Method of Frobenius Basic Assumptions and Symbology The Indicial Equation and Basic Recursion Formula The Easily Obtained Series Solutions Second Solutions When r1 = r2 Second Solutions When r1 – r2 = K Convergence of the Solution Series SYSTEMS OF DIFFERENTIAL EQUATIONS (A BRIEF INTRODUCTION) Systems of Differential Equations: A Starting Point Basic Terminology and Notions A Few Illustrative Applications Converting Differential Equations to First-Order Systems Using Laplace Transforms to Solve Systems Existence, Uniqueness and General Solutions for Systems Single Nth-order Differential Equations Additional Exercises Critical Points, Direction Fields and Trajectories The Systems of Interest and Some Basic Notation Constant/Equilibrium Solutions "Graphing" Standard Systems Sketching Trajectories for Autonomous Systems Critical Points, Stability and Long-Term Behavior Applications Existence and Uniqueness of Trajectories Proving Theorem 36.2 Additional Exercises Appendix: Author’s Guide to Using This Text Overview Chapter-by-Chapter Guide Answers to Selected Exercises

About the Author :
Kenneth B. Howell earned bachelor degrees in both mathematics and physics from Rose-Hulman Institute of Technology, and master’s and doctoral degrees in mathematics from Indiana University. For more than thirty years, he was a professor in the Department of Mathematical Sciences of the University of Alabama in Huntsville (retiring in 2014). During his academic career, Dr. Howell published numerous research articles in applied and theoretical mathematics in prestigious journals, served as a consulting research scientist for various companies and federal agencies in the space and defense industries, and received awards from the College and University for outstanding teaching. He is also the author of Principles of Fourier Analysis (Chapman & Hall/CRC, 2001).