Notes on Diffy Qs: Differential Equations for Engineers E9HS8K3A52

URI	`m2://res/E9HS8K3A52`
Kind	textbook
Creators	Jiří Lebl
Scope	deferred (no test decides the record, and the classifier has no model)

Expressions

e1: edition

Contents

chapter 1: Table of Laplace Transforms
chapter 2: Linear algebra
section 2.1: Vectors, mappings, and matrices
subsection 2.1.1: Vectors and operations on vectors
subsection 2.1.2: Linear mappings and matrices
subsection 2.1.3: Exercises
section 2.2: Matrix algebra
subsection 2.2.1: One-by-one matrices
subsection 2.2.2: Matrix addition and scalar multiplication
subsection 2.2.3: Matrix multiplication
subsection 2.2.4: Some rules of matrix algebra
subsection 2.2.5: Inverse
subsection 2.2.6: Diagonal matrices
subsection 2.2.7: Transpose
subsection 2.2.8: Exercises
section 2.3: Elimination
subsection 2.3.1: Linear systems of equations
subsection 2.3.2: Row echelon form and elementary operations
subsection 2.3.3: Non-unique solutions and inconsistent systems
subsection 2.3.4: Linear independence and rank
subsection 2.3.5: Computing the inverse
subsection 2.3.6: Exercises
section 2.4: Subspaces, dimension, and the kernel
subsection 2.4.1: Subspaces, basis, and dimension
subsection 2.4.2: Kernel
subsection 2.4.3: Exercises
section 2.5: Inner product and projections
subsection 2.5.1: Inner product and orthogonality
subsection 2.5.2: Orthogonal projection
subsection 2.5.3: Orthogonal basis
subsection 2.5.4: The Gram--Schmidt process
subsection 2.5.5: Exercises
section 2.6: Determinant
subsection 2.6.1: Exercises
chapter 3: More on eigenvalue problems
section 3.1: Sturm--Liouville problems
subsection 3.1.1: Boundary value problems
subsection 3.1.2: Orthogonality
subsection 3.1.3: Fredholm alternative
subsection 3.1.4: Eigenfunction series
subsection 3.1.5: Exercises
section 3.2: Higher-order eigenvalue problems
subsection 3.2.1: Exercises
section 3.3: Steady periodic solutions
subsection 3.3.1: Forced vibrating string
subsection 3.3.2: Underground temperature oscillations
subsection 3.3.3: Exercises
chapter 4: First-order equations
section 4.1: Integrals as solutions
subsection 4.1.1: Exercises
section 4.2: Slope fields
subsection 4.2.1: Slope fields
subsection 4.2.2: Existence and uniqueness
subsection 4.2.3: Exercises
section 4.3: Separable equations
subsection 4.3.1: Separable equations
subsection 4.3.2: Implicit solutions
subsection 4.3.3: More examples of separable equations
subsection 4.3.4: Exercises
section 4.4: Linear equations and the integrating factor
subsection 4.4.1: Exercises
section 4.5: Substitution
subsection 4.5.1: Substitution
subsection 4.5.2: Bernoulli equations
subsection 4.5.3: Homogeneous equations
subsection 4.5.4: Exercises
section 4.6: Autonomous equations
subsection 4.6.1: Exercises
section 4.7: Numerical methods: Euler's method
subsection 4.7.1: Exercises
section 4.8: Exact equations
subsection 4.8.1: Solving exact equations
subsection 4.8.2: Integrating factors
subsection 4.8.3: Exercises
section 4.9: First-order linear PDEs
subsection 4.9.1: Exercises
chapter 5: Fourier series and PDEs
section 5.1: Boundary value problems
subsection 5.1.1: Boundary value problems
subsection 5.1.2: Eigenvalue problems
subsection 5.1.3: Orthogonality of eigenfunctions
subsection 5.1.4: Fredholm alternative
subsection 5.1.5: Application
subsection 5.1.6: Exercises
section 5.2: The trigonometric series
subsection 5.2.1: Periodic functions and motivation
subsection 5.2.2: Inner product and eigenvector decomposition
subsection 5.2.3: The trigonometric series
subsection 5.2.4: Exercises
section 5.3: More on the Fourier series
subsection 5.3.1: $2L$-periodic functions
subsection 5.3.2: Convergence
subsection 5.3.3: Differentiation and integration of Fourier series
subsection 5.3.4: Rates of convergence and smoothness
subsection 5.3.5: Exercises
section 5.4: Sine and cosine series
subsection 5.4.1: Odd and even periodic functions
subsection 5.4.2: Sine and cosine series
subsection 5.4.3: Application
subsection 5.4.4: Exercises
section 5.5: Applications of Fourier series
subsection 5.5.1: Periodically forced oscillation
subsection 5.5.2: Resonance
subsection 5.5.3: Exercises
section 5.6: PDEs, separation of variables, and the heat equation
subsection 5.6.1: Heat on an insulated wire
subsection 5.6.2: Separation of variables
subsection 5.6.3: Insulated ends
subsection 5.6.4: Exercises
section 5.7: One-dimensional wave equation
subsection 5.7.1: Exercises
section 5.8: D'Alembert solution of the wave equation
subsection 5.8.1: Change of variables
subsection 5.8.2: D'Alembert's formula
subsection 5.8.3: Another way to solve for the side conditions
subsection 5.8.4: Some remarks
subsection 5.8.5: Exercises
section 5.9: Steady state temperature and the Laplacian
subsection 5.9.1: Exercises
section 5.10: Dirichlet problem in the circle and the Poisson kernel
subsection 5.10.1: Laplace in polar coordinates
subsection 5.10.2: Series solution
subsection 5.10.3: Poisson kernel
subsection 5.10.4: Exercises
chapter 6: Higher-order linear ODEs
section 6.1: Second-order linear ODEs
subsection 6.1.1: Exercises
section 6.2: Constant-coefficient second-order linear ODEs
subsection 6.2.1: Solving constant-coefficient equations
subsection 6.2.2: Complex numbers and Euler's formula
subsection 6.2.3: Complex roots
subsection 6.2.4: Exercises
section 6.3: Higher-order linear ODEs
subsection 6.3.1: Linear independence
subsection 6.3.2: Constant-coefficient higher-order ODEs
subsection 6.3.3: Exercises
section 6.4: Mechanical vibrations
subsection 6.4.1: Some examples
subsection 6.4.2: Free undamped motion
subsection 6.4.3: Free damped motion
subsection 6.4.4: Exercises
section 6.5: Nonhomogeneous equations
subsection 6.5.1: Solving nonhomogeneous equations
subsection 6.5.2: Undetermined coefficients
subsection 6.5.3: Variation of parameters
subsection 6.5.4: Exercises
section 6.6: Forced oscillations and resonance
subsection 6.6.1: Undamped forced motion and resonance
subsection 6.6.2: Damped forced motion and practical resonance
subsection 6.6.3: Exercises
chapter: Introduction
section 6.7: Notes about these notes
subsection 6.7.1: Organization
subsection 6.7.2: Typical courses
subsection 6.7.3: Computer resources
subsection 6.7.4: Acknowledgments
section 6.8: Introduction to differential equations
subsection 6.8.1: Differential equations
subsection 6.8.2: Solutions of differential equations
subsection 6.8.3: Differential equations in practice
subsection 6.8.4: Four fundamental equations
subsection 6.8.5: Exercises
section 6.9: Classification of differential equations
subsection 6.9.1: Exercises
chapter 7: The Laplace transform
section 7.1: The Laplace transform
subsection 7.1.1: The transform
subsection 7.1.2: Existence and uniqueness
subsection 7.1.3: The inverse transform
subsection 7.1.4: Exercises
section 7.2: Transforms of derivatives and ODEs
subsection 7.2.1: Transforms of derivatives
subsection 7.2.2: Solving ODEs with the Laplace transform
subsection 7.2.3: Using the Heaviside function
subsection 7.2.4: Transfer functions
subsection 7.2.5: Transforms of integrals
subsection 7.2.6: Periodic functions
subsection 7.2.7: Exercises
section 7.3: Convolution
subsection 7.3.1: The convolution
subsection 7.3.2: Solving ODEs
subsection 7.3.3: Volterra integral equation
subsection 7.3.4: Exercises
section 7.4: Dirac delta and impulse response
subsection 7.4.1: Rectangular pulse
subsection 7.4.2: The delta function
subsection 7.4.3: Impulse response
subsection 7.4.4: Three-point beam bending
subsection 7.4.5: Exercises
section 7.5: Solving PDEs with the Laplace transform
subsection 7.5.1: Exercises
chapter 8: Nonlinear systems
section 8.1: Linearization, critical points, and equilibria
subsection 8.1.1: Autonomous systems and phase plane analysis
subsection 8.1.2: Linearization
subsection 8.1.3: Exercises
section 8.2: Stability and classification of isolated critical points
subsection 8.2.1: Isolated critical points and almost linear systems
subsection 8.2.2: Stability and classification of isolated critical points
subsection 8.2.3: The trouble with centers
subsection 8.2.4: Conservative equations
subsection 8.2.5: Exercises
section 8.3: Applications of nonlinear systems
subsection 8.3.1: Pendulum
subsection 8.3.2: Predator-prey or Lotka--Volterra systems
subsection 8.3.3: Exercises
section 8.4: Limit cycles
subsection 8.4.1: Exercises
section 8.5: Chaos
subsection 8.5.1: Duffing equation and strange attractors
subsection 8.5.2: The Lorenz system
subsection 8.5.3: Exercises
chapter 9: Power-series methods
section 9.1: Power series
subsection 9.1.1: Definition
subsection 9.1.2: Radius of convergence
subsection 9.1.3: Analytic functions
subsection 9.1.4: Manipulating power series
subsection 9.1.5: Power series for rational functions
subsection 9.1.6: Exercises
section 9.2: Series solutions of linear second-order ODEs
subsection 9.2.1: Exercises
section 9.3: Singular points and the method of Frobenius
subsection 9.3.1: Examples
subsection 9.3.2: The method of Frobenius
subsection 9.3.3: Bessel functions
subsection 9.3.4: Exercises
chapter 10: Systems of ODEs
section 10.1: Introduction to systems of ODEs
subsection 10.1.1: Systems
subsection 10.1.2: Applications
subsection 10.1.3: Changing to first-order systems
subsection 10.1.4: Autonomous systems and vector fields
subsection 10.1.5: Picard's theorem
subsection 10.1.6: Exercises
section 10.2: Matrices and linear systems
subsection 10.2.1: Matrices and vectors
subsection 10.2.2: Matrix multiplication
subsection 10.2.3: The determinant
subsection 10.2.4: Solving linear systems
subsection 10.2.5: Computing the inverse
subsection 10.2.6: Exercises
section 10.3: Linear systems of ODEs
subsection 10.3.1: Exercises
section 10.4: Eigenvalue method
subsection 10.4.1: Eigenvalues and eigenvectors of a matrix
subsection 10.4.2: The eigenvalue method with distinct real eigenvalues
subsection 10.4.3: Complex eigenvalues
subsection 10.4.4: Exercises
section 10.5: Two-dimensional systems and their vector fields
subsection 10.5.1: Exercises
section 10.6: Second-order systems and applications
subsection 10.6.1: Undamped mass-spring systems
subsection 10.6.2: Examples
subsection 10.6.3: Forced oscillations
subsection 10.6.4: Exercises
section 10.7: Repeated eigenvalues
subsection 10.7.1: Geometric multiplicity
subsection 10.7.2: Defective eigenvalues
subsection 10.7.3: Exercises
section 10.8: Matrix exponentials
subsection 10.8.1: Definition
subsection 10.8.2: Simple cases
subsection 10.8.3: General matrices
subsection 10.8.4: Fundamental matrix solutions
subsection 10.8.5: Approximations
subsection 10.8.6: Exercises
section 10.9: Nonhomogeneous systems
subsection 10.9.1: First-order constant-coefficient
subsection 10.9.2: First-order variable-coefficient
subsection 10.9.3: Second-order constant-coefficients
subsection 10.9.4: Exercises

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