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Introduction to Differential Equations
Adapted for Math 204 at the University of Victoria
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Front Matter
Colophon
Abstract
0
Introduction
0.1
About this book
0.1.1
Math 204: Vector Calculus and Differential Equations
0.1.2
Original Textbook by Jiří Lebl
0.1.3
Differences between the versions
0.1.4
The video playlist
0.1.5
Computer resources
0.1.6
Acknowledgments
0.2
What are Differential Equations?
0.2.1
Differential equations
0.2.2
Solutions of differential equations
0.2.3
Differential equations in practice
0.2.4
Four fundamental equations
0.2.5
Exercises
0.3
Classification of differential equations
0.3.1
Exercises
1
First order equations
1.1
Integrals as solutions
1.1.1
Exercises
1.2
Slope fields
1.2.1
Slope fields
1.2.2
Existence and uniqueness
1.2.3
Exercises
1.3
Separable equations
1.3.1
Separable equations
1.3.2
Implicit solutions
1.3.3
Examples of separable equations
1.3.4
Exercises
1.4
Linear equations and the integrating factor
1.4.1
Exercises
1.5
Substitution
1.5.1
Substitution
1.5.2
Bernoulli equations
1.5.3
Homogeneous equations
1.5.4
Exercises
1.6
Autonomous equations
1.6.1
Exercises
1.7
Numerical methods: Euler’s method
1.7.1
Exercises
1.8
Exact equations
1.8.1
Solving exact equations
1.8.2
Integrating factors
1.8.3
Exercises
1.9
First order linear PDE
1.9.1
Exercises
2
Higher order linear ODEs
2.1
Second order linear ODEs
2.1.1
Exercises
2.2
Constant coefficient second order linear ODEs
2.2.1
Solving constant coefficient equations
2.2.2
Complex numbers and Euler’s formula
2.2.3
Complex roots
2.2.4
Exercises
2.3
Higher order linear ODEs
2.3.1
Linear independence
2.3.2
Theory of Higher Order ODEs
2.3.3
Constant coefficient higher order ODEs
2.3.4
Exercises
2.4
Mechanical vibrations
2.4.1
Some examples
2.4.2
Free undamped motion
2.4.3
Free damped motion
2.4.3.1
Overdamping
2.4.3.2
Critical damping
2.4.3.3
Underdamping
2.4.4
Exercises
2.5
Nonhomogeneous equations
2.5.1
Solving nonhomogeneous equations
2.5.2
Undetermined coefficients
2.5.3
Variation of parameters
2.5.4
Exercises
2.6
Forced oscillations and resonance
2.6.1
Undamped forced motion and resonance
2.6.2
Damped forced motion and practical resonance
2.6.3
Exercises
3
The Laplace transform
3.1
The Laplace transform
3.1.1
The transform
3.1.2
Existence and uniqueness
3.1.3
The inverse transform
3.1.4
Exercises
3.2
Transforms of derivatives and ODEs
3.2.1
Transforms of derivatives
3.2.2
Solving ODEs with the Laplace transform
3.2.3
Using the Heaviside function
3.2.4
Transfer functions
3.2.5
Transforms of integrals
3.2.6
Exercises
3.3
Convolution
3.3.1
The convolution
3.3.2
Solving ODEs
3.3.3
Volterra integral equation
3.3.4
Exercises
3.4
Dirac delta and impulse response
3.4.1
Rectangular pulse
3.4.2
The delta function
3.4.3
Impulse response
3.4.4
Periodic functions
3.4.5
Three-point beam bending
3.4.6
Exercises
3.5
Solving PDEs with the Laplace transform
3.5.1
Exercises
4
Power series methods
4.1
Power series
4.1.1
Definition
4.1.2
Radius of convergence
4.1.3
Analytic functions
4.1.4
Manipulating power series
4.1.5
Power series for rational functions
4.1.6
Analytic functions
4.1.7
Exercises
4.2
Series solutions of linear second order ODEs
4.2.1
Exercises
4.3
Singular points and the method of Frobenius
4.3.1
Examples
4.3.2
The method of Frobenius
4.3.3
Bessel functions
4.3.4
Exercises
5
Fourier series and PDEs
5.1
Boundary value problems
5.1.1
Boundary value problems
5.1.2
Eigenvalue problems
5.1.3
Orthogonality of eigenfunctions
5.1.4
Fredholm alternative
5.1.5
Application
5.1.6
Exercises
5.2
The trigonometric series
5.2.1
Periodic functions and motivation
5.2.2
The trigonometric series
5.2.3
Exercises
5.3
More on the Fourier series
5.3.1
2
L
-periodic functions
5.3.2
Convergence
5.3.3
Differentiation and integration of Fourier series
5.3.4
Rates of convergence and smoothness
5.3.5
Exercises
5.4
Sine and cosine series
5.4.1
Odd and even periodic functions
5.4.2
Sine and cosine series
5.4.3
Application
5.4.4
Exercises
5.5
Applications of Fourier series
5.5.1
Periodically forced oscillation
5.5.2
Resonance
5.5.3
Exercises
5.6
PDEs, separation of variables, and the heat equation
5.6.1
Heat on an insulated wire
5.6.2
Separation of variables
5.6.3
Insulated ends
5.6.4
Exercises
5.7
One-dimensional wave equation
5.7.1
Exercises
5.8
D’Alembert solution of the wave equation
5.8.1
Change of variables
5.8.2
D’Alembert’s formula
5.8.3
Another way to solve for the side conditions
5.8.4
Some remarks
5.8.5
Exercises
5.9
Steady state temperature and the Laplacian
5.9.1
Exercises
5.10
Dirichlet problem in the circle and the Poisson kernel
5.10.1
Laplace in polar coordinates
5.10.2
Series solution
5.10.3
Poisson kernel
5.10.4
Exercises
6
More on eigenvalue problems
6.1
Sturm–Liouville problems
6.1.1
Boundary value problems
6.1.2
Orthogonality
6.1.3
Fredholm alternative
6.1.4
Eigenfunction series
6.1.5
Exercises
6.2
Higher order eigenvalue problems
6.2.1
Exercises
6.3
Steady periodic solutions
6.3.1
Forced vibrating string
6.3.2
Underground temperature oscillations
6.3.3
Exercises
7
Systems of ODEs
7.1
Introduction to systems of ODEs
7.1.1
Systems
7.1.2
Applications
7.1.3
Changing to first order
7.1.4
Autonomous systems and vector fields
7.1.5
Picard’s theorem
7.1.6
Exercises
7.2
Matrices and linear systems
7.2.1
Matrices and vectors
7.2.2
Matrix multiplication
7.2.3
The determinant
7.2.4
Solving linear systems
7.2.5
Computing the inverse
7.2.6
Exercises
7.3
Linear systems of ODEs
7.3.1
Exercises
7.4
Eigenvalue method
7.4.1
Eigenvalues and eigenvectors of a matrix
7.4.2
The eigenvalue method with distinct real eigenvalues
7.4.3
Complex eigenvalues
7.4.4
Exercises
7.5
Two-dimensional systems and their vector fields
7.5.1
Exercises
7.6
Second order systems and applications
7.6.1
Undamped mass-spring systems
7.6.2
Examples
7.6.3
Forced oscillations
7.6.4
Exercises
7.7
Multiple eigenvalues
7.7.1
Geometric multiplicity
7.7.2
Defective eigenvalues
7.7.3
Exercises
7.8
Matrix exponentials
7.8.1
Definition
7.8.2
Simple cases
7.8.3
General matrices
7.8.4
Fundamental matrix solutions
7.8.5
Approximations
7.8.6
Exercises
7.9
Nonhomogeneous systems
7.9.1
First order constant coefficient
7.9.1.1
Integrating factor
7.9.1.2
Eigenvector decomposition
7.9.1.3
Undetermined coefficients
7.9.2
First order variable coefficient
7.9.2.1
Variation of parameters
7.9.3
Second order constant coefficients
7.9.3.1
Undetermined coefficients
7.9.3.2
Eigenvector decomposition
7.9.4
Exercises
Back Matter
A
Linear algebra
A.1
Vectors, mappings, and matrices
A.1.1
Vectors and operations on vectors
A.1.2
Linear mappings and matrices
A.1.3
Exercises
A.2
Matrix algebra
A.2.1
One-by-one matrices
A.2.2
Matrix addition and scalar multiplication
A.2.3
Matrix multiplication
A.2.4
Some rules of matrix algebra
A.2.5
Inverse
A.2.6
Diagonal matrices
A.2.7
Transpose
A.2.8
Exercises
A.3
Elimination
A.3.1
Linear systems of equations
A.3.2
Row echelon form and elementary operations
A.3.3
Non-unique solutions and inconsistent systems
A.3.4
Linear independence and rank
A.3.5
Computing the inverse
A.3.6
Exercises
A.4
Subspaces, dimension, and the kernel
A.4.1
Subspaces, basis, and dimension
A.4.2
Kernel
A.4.3
Exercises
A.5
Inner product and projections
A.5.1
Inner product and orthogonality
A.5.2
Orthogonal projection
A.5.3
Orthogonal basis
A.5.4
The Gram–Schmidt process
A.5.5
Exercises
A.6
Determinant
A.6.1
Exercises
B
Table of Laplace Transforms
Further Reading
Index
🔗
Chapter
5
Fourier series and PDEs
5.1
Boundary value problems
5.2
The trigonometric series
5.3
More on the Fourier series
5.4
Sine and cosine series
5.5
Applications of Fourier series
5.6
PDEs, separation of variables, and the heat equation
5.7
One-dimensional wave equation
5.8
D’Alembert solution of the wave equation
5.9
Steady state temperature and the Laplacian
5.10
Dirichlet problem in the circle and the Poisson kernel
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