Differential Equations: An Introduction for Engineers

Introduction
 
Preface and About the Author(s)
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Introduction to differential equations
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Practice for Solving Exact ODEs
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Classification of differential equations
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Practice for Classifying Equations
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Linear First Order ODEs
 
Integrals as solutions
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Practice for Solving by Direct Integration
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Slope fields
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Practice for Slope Fields
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Separable equations
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Practice for Separable ODEs
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Linear equations and the integrating factor
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Practice for Solving First Order Linear ODEs
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Existence and Uniqueness of Solutions
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Practice for Using the EU-Thm
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Numerical methods: Euler’s method
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Practice for Euler Method
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Autonomous equations
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Practice for Autonomous Equations
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Bifurcation diagrams
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Practice for bifurcation diagrams
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Exact equations
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Practice for Solving Exact ODEs
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Modeling with First Order Equations
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Practice for Modeling
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Modeling and Parameter Estimation
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Practice for Modeling Parameters
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Substitution
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Practice for Substitution
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First order linear PDE
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Practice for first PDE
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Chapter Review
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Higher Order ODEs and Apps
 
Second order linear ODEs
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Practice for Second Order Linear ODEs
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Complex Roots and Euler’s Formula
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Practice for Complex Roots
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Repeated Roots and Reduction of Order
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Practice for Second Order Repeated Roots
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Mechanical vibrations
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Practice for Mechanical Vibrations
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Nonhomogeneous equations
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Practice for Non-Homogeneous Equations
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Forced oscillations and resonance
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Practice for Forced Oscillations
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Higher order linear ODEs
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Practice for Higher Order ODEs
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Chapter Review
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Linear Algebra
 
Vectors, mappings, and matrices
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Practice for Forced Oscillations
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Matrix algebra
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Practice for Matrix Algebra
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Elimination
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Practice for Elimination Method
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Linear independence, rank, and dimension
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Practice for Forced Oscillations
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Determinant
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Practice for Determinants
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Eigenvalues and Eigenvectors
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Practice for Eigenvalues
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Related Topics in Linear Algebra
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Practice for Advanced Linear Algebra
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Linear Systems of ODEs
 
Introduction to systems of ODEs
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Practice for Intro to Linear Systems
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Matrices and linear systems
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Practice for Matrix Manipulations
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Linear systems of ODEs
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Practice for Linear Systems
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Eigenvalue method
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Practice for Eigenvalue Method
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Eigenvalue method with complex eigenvalues
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Practice for Eigenvalue Method with Complex Roots
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Eigenvalue method with repeated eigenvalues
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Practice for Eigenvalue Method with Repeated Roots
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Two-dimensional systems and their vector fields
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Practice for Advanced Linear Algebra
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Nonhomogeneous systems
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Practice for Non-Homogeneous Systems
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Second order systems and applications
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Practice for Second Order Systems
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Matrix exponentials
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Practice for Matrix Exponential
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Chapter Review
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Non-Linear Systems of ODEs
 
Linearization, critical points, and stability
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Practice for Linearization and Stability
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Behavior of non-linear systems
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Practice for Advanced Nonlinear Analysis
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Applications of nonlinear systems
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Practice for Nonlinear Applications
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Limit cycles
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Practice for Limit Cycles
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Chaos
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Practice for Chaos
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Chapter Review
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Laplace Transform
 
The Laplace transform
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Practice for the Laplace Transform
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Transforms of derivatives and ODEs
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Practice for Convolution
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Convolution
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Practice for Convolution
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Dirac delta and impulse response
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Practice for the Dirac Delta
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Power Series
 
Power series
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Practice for the Power Series
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Series solutions of linear second order ODEs
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Practice for Series Solutions
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Singular points and the method of Frobenius
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Practice for the Frobenius Method
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Orthogonality
 
Inner product and projections
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Practice for Inner Products
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Special Matrices
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Practice for Special Matrix
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Vector Spaces of Functions
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Practice for Function Vector Spaces
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Fourier Series
 
Boundary value problems
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Practice for Boundary Value Problems
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The trigonometric series
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Practice for Trig Series
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More on the Fourier series
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Practice for More Fourier
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Sine and cosine series
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Practice for Sine and Cosine Series
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Applications of Fourier series
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Practice for Applications of Fourier Series
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Separability
 
First order linear PDE
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Practice for PDEs
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Second order linear PDEs
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Practice for PDEs
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The heat equation
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Practice for Heat Equation
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One-dimensional wave equation
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Practice for One Dim Wave
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D’Alembert solution of the wave equation
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Practice for Dalemb Wave
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Steady state temperature and the Laplacian
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Practice for Laplacian
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Dirichlet problem in the circle and the Poisson kernel
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Practice for the Dirichlet Disc
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Fourier Transform
 
Fourier Integrals
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Fourier Transform
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Applications of the Fourier Transform
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Eigenvalue Problems
 
Sturm–Liouville problems
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Practice for Sturm Problems
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Higher order eigenvalue problems
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Practice for Higher Eval Problems
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Steady periodic solutions
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Practice for Steady Per Solutions
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Supplement and Review
 
Introduction to MATLAB
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Polynomials and Factoring
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Complex Numbers
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Differentiation and Integration Techniques
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Table of Laplace Transforms
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Overall
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