Differential Equations: An Introduction for Engineers

Introduction

Preface and About the Author(s)

A brief outline of the history of this text, and the authors involved. Also includes license information for those that may be interested in using it.

Introduction to differential equations

We Introduction some basics of differential equations

Classification of differential equations

We introduce how to classify various properties of differential equations.

Linear First Order ODEs

Integrals as solutions

Stuff about Integrals as solutions

Slope fields

Stuff about Separable equations

Separable equations

Stuff about Separable equations

Linear equations and the integrating factor

Stuff about Linear equations and the integrating factor

Existence and Uniqueness of Solutions

Stuff about Existence and Uniqueness of Solutions

Numerical methods: Euler’s method

Stuff about Numerical methods: Euler’s method

Autonomous equations

We discuss autonomous equations

Bifurcation diagrams

We discuss bifurcation diagrams for autonomous equations with parameters.

Exact equations

We introduce how to classify various properties of differential equations.

Modeling with First Order Equations

Stuff about Modeling with First Order Equations

Modeling and Parameter Estimation

We discuss Modeling and Parameter Estimation

Substitution

Stuff about Substitution

First order linear PDE

We discuss First order linear PDE

Chapter Review

Review for Chapter on First Order ODEs.

Higher Order ODEs and Apps

Second order linear ODEs

We discuss Second order linear ODEs

Complex Roots and Euler’s Formula

We discuss Complex Roots and Euler’s Formula

Repeated Roots and Reduction of Order

We discuss Repeated Roots and Reduction of Order

Mechanical vibrations

We discuss Mechanical vibrations

Nonhomogeneous equations

We discuss Nonhomogeneous systems

Forced oscillations and resonance

We discuss Forced oscillations and resonance

Higher order linear ODEs

We discuss Higher order linear ODEs

Chapter Review

A Set of Problems for a Chapter Review on Higher Order ODEs

Linear Algebra

Vectors, mappings, and matrices

We discuss Vectors, mappings, and matrices

Matrix algebra

Elimination

We discuss Elimination

Linear independence, rank, and dimension

Determinant

We discuss the Determinant

Eigenvalues and Eigenvectors

We discuss Eigenvalues and Eigenvectors

Linear Systems of ODEs

Introduction to systems of ODEs

We discuss Introduction to systems of ODEs

Matrices and linear systems

We discuss Matrices and linear systems

Linear systems of ODEs

We discuss Linear systems of ODEs

Eigenvalue method

We discuss Eigenvalue method

Eigenvalue method with complex eigenvalues

We discuss Eigenvalue method with complex eigenvalues

Eigenvalue method with repeated eigenvalues

We discuss Eigenvalue method with repeated eigenvalues

Two-dimensional systems and their vector fields

We discuss Two-dimensional systems and their vector fields

Nonhomogeneous systems

We discuss Nonhomogeneous systems

Second order systems and applications

We discuss Second order systems and applications

Matrix exponentials

We discuss Matrix exponentials

Chapter Review

A Review of the chapter’s contents.

Non-Linear Systems of ODEs

Linearization, critical points, and stability

We discuss Linearization, critical points, and stability

Behavior of non-linear systems

We discuss Behavior of non-linear systems

Applications of nonlinear systems

We discuss Applications of nonlinear systems

Limit cycles

We discuss Limit cycles

Chaos

We discuss Chaos

Chapter Review

A review of the chapter’s content.

Laplace Transform

The Laplace transform

We discuss The Laplace transform

Transforms of derivatives and ODEs

We discuss Transforms of derivatives and ODEs

Convolution

We discuss Convolution

Dirac delta and impulse response

We discuss Dirac delta and impulse response

Power Series

Power series

We discuss Power series

Series solutions of linear second order ODEs

We discuss Series solutions of linear second order ODEs

Singular points and the method of Frobenius

We discuss Singular points and the method of Frobenius

Orthogonality

Inner product and projections

We discuss Inner product and projections

Special Matrices

We discuss Special Matrices

Vector Spaces of Functions

We discuss Vector Spaces of Functions

Fourier Series

Boundary value problems

We discuss Boundary value problems

The trigonometric series

We discuss The trigonometric series

More on the Fourier series

We discuss More on the Fourier series

Sine and cosine series

We discuss Sine and cosine series

Applications of Fourier series

We discuss Applications of Fourier series

Separability

First order linear PDE

We discuss First order linear PDE

Second order linear PDEs

We discuss Series solutions of linear second order ODEs

The heat equation

We discuss The heat equation

One-dimensional wave equation

We discuss One-dimensional wave equation

D’Alembert solution of the wave equation

We discuss D’Alembert solution of the wave equation

Steady state temperature and the Laplacian

We discuss Steady state temperature and the Laplacian

Dirichlet problem in the circle and the Poisson kernel

We discuss Dirichlet problem in the circle and the Poisson kernel

Fourier Transform

Fourier Integrals

We discuss Fourier Integrals

Fourier Transform

We discuss Fourier Transform

Applications of the Fourier Transform

We discuss Applications of the Fourier Transform

Eigenvalue Problems

Sturm–Liouville problems

We discuss Sturm–Liouville problems

Higher order eigenvalue problems

We discuss Higher order eigenvalue problems

Steady periodic solutions

We discuss Steady periodic solutions

Supplement and Review

Introduction to MATLAB

We present Introduction to MATLAB

Polynomials and Factoring

We review Polynomials and Factoring

Complex Numbers

We discuss review Complex Numbers

Differentiation and Integration Techniques

We review Differentiation and Integration Techniques

Table of Laplace Transforms

A reference table of Laplace Transforms

Introduction

Preface and About the Author(s)

Introduction to differential equations

Classification of differential equations

Linear First Order ODEs

Integrals as solutions

Slope fields

Separable equations

Linear equations and the integrating factor

Existence and Uniqueness of Solutions

Numerical methods: Euler’s method

Autonomous equations

Bifurcation diagrams

Exact equations

Modeling with First Order Equations

Modeling and Parameter Estimation

Substitution

First order linear PDE

Chapter Review

Higher Order ODEs and Apps

Second order linear ODEs

Complex Roots and Euler’s Formula

Repeated Roots and Reduction of Order

Mechanical vibrations

Nonhomogeneous equations

Forced oscillations and resonance

Higher order linear ODEs

Chapter Review

Linear Algebra

Vectors, mappings, and matrices

Matrix algebra

Elimination

Linear independence, rank, and dimension

Determinant

Eigenvalues and Eigenvectors

Related Topics in Linear Algebra

Linear Systems of ODEs

Introduction to systems of ODEs

Matrices and linear systems

Linear systems of ODEs

Eigenvalue method

Eigenvalue method with complex eigenvalues

Eigenvalue method with repeated eigenvalues

Two-dimensional systems and their vector fields

Nonhomogeneous systems

Second order systems and applications

Matrix exponentials

Chapter Review

Non-Linear Systems of ODEs

Linearization, critical points, and stability

Behavior of non-linear systems

Applications of nonlinear systems

Limit cycles

Chaos

Chapter Review

Laplace Transform

The Laplace transform

Transforms of derivatives and ODEs

Convolution

Dirac delta and impulse response

Power Series

Power series

Series solutions of linear second order ODEs

Singular points and the method of Frobenius

Orthogonality

Inner product and projections

Special Matrices

Vector Spaces of Functions

Fourier Series

Boundary value problems

The trigonometric series

More on the Fourier series

Sine and cosine series

Applications of Fourier series

Separability

First order linear PDE

Second order linear PDEs

The heat equation

One-dimensional wave equation

D’Alembert solution of the wave equation