Differential Equations: Theory, Technique, and Practice

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9780071254373: Differential Equations: Theory, Technique, and Practice

This traditional text is intended for mainstream one- or two-semester differential equations courses taken by undergraduates majoring in engineering, mathematics, and the sciences. Written by two of the world’s leading authorities on differential equations, Simmons/Krantz provides a cogent and accessible introduction to ordinary differential equations written in classical style. Its rich variety of modern applications in engineering, physics, and the applied sciences illuminate the concepts and techniques that students will use through practice to solve real-life problems in their careers.

This text is part of the Walter Rudin Student Series in Advanced Mathematics.

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Contenuti:

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 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

Bibliography

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