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Here's the perfect self-teaching guide to help anyone master differential equations--a common stumbling block for students looking to progress to advanced topics in both science and math. Covers First Order Equations, Second Order Equations and Higher, Properties, Solutions, Series Solutions, Fourier Series and Orthogonal Systems, Partial Differential Equations and Boundary Value Problems, Numerical Techniques, and more.
I. What is a differential equation?
A. Examples
B. Growth and Decay
C. Curves, Trajectories, and Vector Fields
II. First Order Equations
A. Linear theory
B. Separation of variables
C. Homogeneous equations
D. Reduction of Order
E. he Hanging Cable
F. Electrical Circuits
III. Second Order Equations and Higher
A. Key Elements
B. The Homogeneous Equation
C. Constant Coefficients
D. Using One Solution to Find Another
E. Undetermined Coefficients
F. Variation of Parameters
G. Vibrations and Oscillations
IV. Qualitative Properties
A. Vector Fields
B. Flows and Trajectories
C. Poincare-Bendixon Theory
V. Power Series Solutions
A. Review of Key Properties of Power Series
B. Series Solutions for First Order Equations
C. Second Order Linear: Ordinary Points
D. Regular Singular Points
VI. Fourier Series and Orthogonal Systems
A. Calculating Fourier Coefficients
B. Convergence of Fourier Series
C. Even and Odd Functions: Cosine Series and Sine Series
D. Arbitrary Intervals
E. Orthogonality
F. Mean Convergence: Cesaro Summability
VII. Partial Differential Equations and Boundary Value Problems
A. The Laplacian
B. The Dirichlet Problem
VIII. The Laplace Transform
A. The Idea of the Laplace Transform
B. Inversion of the Laplace Transform
C. Application to Differential Equations
D. Derivatives and Integrals
E. Dirac Point Masses and Heaviside Functions
F. Applications
IX. Numerical Techniques
A. Introduction
B. Euler's Method
C. The Runge-Kutta Method
D. Multi-Step Methods
E. Rates of Convergence
F. Estimation of Error Terms
G. Higher Order Methods
H.The Use of a Computer Algebra System
Here's the perfect self-teaching guide to help anyone master differential equations--a common stumbling block for students looking to progress to advanced topics in both science and math. Covers First Order Equations, Second Order Equations and Higher, Properties, Solutions, Series Solutions, Fourier Series and Orthogonal Systems, Partial Differential Equations and Boundary Value Problems, Numerical Techniques, and more.
Table of Contents
I. What is a differential equation?
A. Examples
B. Growth and Decay
C. Curves, Trajectories, and Vector Fields
II. First Order Equations
A. Linear theory
B. Separation of variables
C. Homogeneous equations
D. Reduction of Order
E. he Hanging Cable
F. Electrical Circuits
III. Second Order Equations and Higher
A. Key Elements
B. The Homogeneous Equation
C. Constant Coefficients
D. Using One Solution to Find Another
E. Undetermined Coefficients
F. Variation of Parameters
G. Vibrations and Oscillations
IV. Qualitative Properties
A. Vector Fields
B. Flows and Trajectories
C. Poincare-Bendixon Theory
V. Power Series Solutions
A. Review of Key Properties of Power Series
B. Series Solutions for First Order Equations
C. Second Order Linear: Ordinary Points
D. Regular Singular Points
VI. Fourier Series and Orthogonal Systems
A. Calculating Fourier Coefficients
B. Convergence of Fourier Series
C. Even and Odd Functions: Cosine Series and Sine Series
D. Arbitrary Intervals
E. Orthogonality
F. Mean Convergence: Cesaro Summability
VII. Partial Differential Equations and Boundary Value Problems
A. The Laplacian
B. The Dirichlet Problem
VIII. The Laplace Transform
A. The Idea of the Laplace Transform
B. Inversion of the Laplace Transform
C. Application to Differential Equations
D. Derivatives and Integrals
E. Dirac Point Masses and Heaviside Functions
F. Applications
IX. Numerical Techniques
A. Introduction
B. Euler's Method
C. The Runge-Kutta Method
D. Multi-Step Methods
E. Rates of Convergence
F. Estimation of Error Terms
G. Higher Order Methods
H.The Use of a Computer Algebra System