## Table of contents for Differential equations : theory, technique, and practice / George F. Simmons and Steven G. Krantz.

Bibliographic record and links to related information available from the Library of Congress catalog

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Preface
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 Poincare;-Bendixson Theorem
Historical Note: Poincare;
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 Poincare;-Bendixson
Anatomy of an Application: Lagrange’s Equations
Problems for Review and Discovery
Bibliography

Library of Congress subject headings for this publication:

Differential equations -- Textbooks.