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Calculus Differential Equations: Theory, Technique, & Practice, Simmons, Krantz

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  1. Feb 19, 2013 #1


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    Publisher's site: http://highered.mcgraw-hill.com/sites/0072863153/

    Table of Contents:

    Code (Text):
    1 What is a Differential Equation?
    1.1 Introductory Remarks
    1.2 The Nature of Solutions
    1.3 Separable Equations
    1.4 First-Order Linear Equations
    1.5 Exact Equations
    1.6 Orthogonal Trajectories and Families of Curves
    1.7 Homogeneous Equations
    1.8 Integrating Factors
    1.9 Reduction of Order
    1.9.1 Dependent Variable Missing
    1.9.2 Independent Variable Missing
    1.10 The Hanging Chain and Pursuit Curves
    1.10.1 The Hanging Chain
    1.10.2 Pursuit Curves
    1.11 Electrical Circuits
    Anatomy of an Application: The Design of a Dialysis Machine
    Problems for Review and Discovery

    2 Second-Order Linear Equations
    2.1 Second-Order Linear Equations with Constant Coefficients
    2.2 The Method of Undetermined Coefficients
    2.3 The Method of Variation of Parameters
    2.4 The Use of a Known Solution to Find Another
    2.5 Vibrations and Oscillations
    2.5.1 Undamped Simple Harmonic Motion
    2.5.2 Damped Vibrations
    2.5.3 Forced Vibrations
    2.5.4 A Few Remarks About Electricity
    2.6 Newton’s Law of Gravitation and Kepler’s Laws
    2.6.1 Kepler’s Second Law
    2.6.2 Kepler’s First Law
    2.6.3 Kepler’s Third Law
    2.7 Higher Order Linear Equations, Coupled Harmonic Oscillators
    Historical Note: Euler
    Anatomy of an Application: Bessel Functions and the Vibrating Membrane
    Problems for Review and Discovery

    3 Qualitative Properties and Theoretical Aspects
    3.1 Review of Linear Algebra
    3.1.1 Vector Spaces
    3.1.2 The Concept Linear Independence
    3.1.3 Bases
    3.1.4 Inner Product Spaces
    3.1.5 Linear Transformations and Matrices
    3.1.6 Eigenvalues and Eigenvectors
    3.2 A Bit of Theory
    3.3 Picard’s Existence and Uniqueness Theorem
    3.3.1 The Form of a Differential Equation
    3.3.2 Picard’s Iteration Technique
    3.3.3 Some Illustrative Examples
    3.3.4 Estimation of the Picard Iterates
    3.4 Oscillations and the Sturm Separation Theorem
    3.5 The Sturm Comparison Theorem
    Anatomy of an Application: The Green’s Function
    Problems for Review and Discovery

    4 Power Series Solutions and Special Functions
    4.1 Introduction and Review of Power Series
    4.1.1 Review of Power Series
    4.2 Series Solutions of First-Order Differential Equations
    4.3 Second-Order Linear Equations: Ordinary Points
    4.4 Regular Singular Points
    4.5 More on Regular Singular Points
    4.6 Gauss’s Hypergeometric Equation
    Historical Note: Gauss
    Historical Note: Abel
    Anatomy of an Application: Steady-State Temperature in a Ball
    Problems for Review and Discovery

    5 Fourier Series: Basic Concepts
    5.1 Fourier Coefficients
    5.2 Some Remarks about Convergence
    5.3 Even and Odd Functions: Cosine and Sine Series
    5.4 Fourier Series on Arbitrary Intervals
    5.5 Orthogonal Functions
    Historical Note: Riemann
    Anatomy of an Application: Introduction to the Fourier Transform
    Problems for Review and Discovery

    6 Partial Differential Equations and Boundary Value Problems
    6.1 Introduction and Historical Remarks
    6.2 Eigenvalues, Eigenfunctions, and the Vibrating String
    6.2.1 Boundary Value Problems
    6.2.2 Derivation of the Wave Equation
    6.2.3 Solution of the Wave Equation
    6.3 The Heat Equation
    6.4 The Dirichlet Problem for a Disc
    6.4.1 The Poisson Integral
    6.5 Sturm-Liouville Problems
    Historical Note: Fourier
    Historical Note: Dirichlet
    Anatomy of an Application: Some Ideas from Quantum Mechanics
    Problems for Review and Discovery

    7 Laplace Transforms
    7.1 Introduction
    7.2 Applications to Differential Equations
    7.3 Derivatives and Integrals of Laplace Transforms
    7.4 Convolutions
    7.3.1 Abel's Mechanical Problem
    7.5 The Unit Step and Impulse Functions
    Historical Note: Laplace
    Anatomy of an Application: Flow Initiated by an Impulsively-Started Flat Plate
    Problems for Review and Discovery

    8 The Calculus of Variations
    8.1 Introductory Remarks
    8.2 Euler’s Equation
    8.3 Isoperimetric Problems and the Like
    8.3.1 Lagrange Multipliers
    8.3.2 Integral Side Conditions
    8.3.3 Finite Side Conditions
    Historical Note: Newton
    Anatomy of an Application: Hamilton’s Principle and its Implications
    Problems for Review and Discovery

    9 Numerical Methods
    9.1 Introductory Remarks
    9.2 The Method of Euler
    9.3 The Error Term
    9.4 An Improved Euler Method
    9.5 The Runge-Kutta Method
    Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equations
    Problems for Review and Discovery

    10 Systems of First-Order Equations
    10.1 Introductory Remarks
    10.2 Linear Systems
    10.3 Homogeneous Linear Systems with Constant Coefficients
    10.4 Nonlinear Systems: Volterra’s Predator-Prey Equations
    Anatomy of an Application: Solution of Systems with Matrices and Exponentials
    Problems for Review and Discovery

    11 The Nonlinear Theory
    11.1 Some Motivating Examples
    11.2 Specializing Down
    11.3 Types of Critical Points: Stability
    11.4 Critical Points and Stability for Linear Systems
    11.5 Stability by Liapunov’s Direct Method
    11.6 Simple Critical Points of Nonlinear Systems
    11.7 Nonlinear Mechanics: Conservative Systems
    11.8 Periodic Solutions: The Poincaré-Bendixson Theorem
    Historical Note: Poincaré
    Anatomy of an Application: Mechanical Analysis of a Block on a Spring
    Problems for Review and Discovery

    12 Dynamical Systems
    12.1 Flows
    12.1.1 Dynamical Systems
    12.1.2 Stable and Unstable Fixed Points
    12.1.3 Linear Dynamics in the Plane
    12.2 Some Ideas from Topology
    12.2.1 Open and Closed Sets
    12.2.2 The Idea of Connectedness
    12.2.3 Closed Curves in the Plane
    12.3 Planar Autonomous Systems
    12.3.1 Ingredients of the Proof of Poincaré-Bendixson
    Anatomy of an Application: Lagrange’s Equations
    Problems for Review and Discovery

    I used, and still have, the 1972 edition of George Simmons's, Differential Equations with Applications and Historical Notes, which was one of the books in McGraw-Hill's International Series in Pure and Applied Mathematics. The text was revised in 1991.

    1991 Ed - https://www.amazon.com/Differential-Equations-Applications-Historical-Notes/dp/0070575401/

    Nathaniel Grossman, Professor of Mathematics, UCLA, writes:
    Last edited by a moderator: May 6, 2017
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