# Differential Equations

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- 10945

- Book: Differential Equations for Engineers (Lebl)
- 1: First order ODEs
- 2: Higher order linear ODEs
- 3: Systems of ODEs
- 3.1: Introduction to Systems of ODEs
- 3.2: Matrices and linear systems
- 3.3: Linear systems of ODEs
- 3.4: Eigenvalue Method
- 3.5: Two dimensional systems and their vector fields
- 3.6: Second order systems and applications
- 3.7: Multiple Eigenvalues
- 3.8: Matrix exponentials
- 3.9: Nonhomogeneous systems
- 3.E: Systems of ODEs (Exercises)

- 4: Fourier series and PDEs
- 4.1: Boundary value problems
- 4.2: The Trigonometric Series
- 4.3: More on the Fourier Series
- 4.4: Sine and cosine series
- 4.5: Applications of Fourier series
- 4.6: PDEs, separation of variables, and the heat equation
- 4.7: One dimensional wave equation
- 4.8: D’Alembert solution of the wave equation
- 4.9: Steady state temperature and the Laplacian
- 4.10: Dirichlet problem in the circle and the Poisson kernel
- 4.E: Fourier series and PDEs (Exercises)

- 5: Eigenvalue problems
- 6: The Laplace Transform
- 7: Power series methods
- 8: Nonlinear Equations

- Book: Elementary Differential Equations with Boundary Values Problems (Trench)
- Preface
- 1: Introduction
- 2: First Order Equations
- 3: Numerical Methods
- 4: Applications of First Order Equations
- 5: Linear Second Order Equations
- 6: Applications of Linear Second Order Equations
- 7: Series Solutions of Linear Second Order Equations
- Prelude to Series Solutions of Linear Second Order Equations
- 7.1: Review of Power Series
- 7.2: Series Solutions Near an Ordinary Point I
- 7.3: Series Solutions Near an Ordinary Point II
- 7.4: Regular Singular Points Euler Equations
- 7.5: The Method of Frobenius I
- 7.6: The Method of Frobenius II
- 7.7: The Method of Frobenius III

- 8: Laplace Transforms
- 8.1: Introduction to the Laplace Transform
- 8.2: The Inverse Laplace Transform
- 8.3: Solution of Initial Value Problems
- 8.4: The Unit Step Function
- 8.5: Constant Coefficient Equations with Piecewise Continuous Forcing Functions
- 8.6: Convolution
- 8.7: Constant Coefficient Equations with Impulses
- 8.8: A Brief Table of Laplace Transforms

- 9: Linear Higher Order Differential Equations
- 10: Linear Systems of Differential Equations
- 10.1: Introduction to Systems of Differential Equations
- 10.2: Linear Systems of Differential Equations
- 10.3: Basic Theory of Homogeneous Linear Systems
- 10.4: Constant Coefficient Homogeneous Systems I
- 10.5: Constant Coefficient Homogeneous Systems II
- 10.6: Constant Coefficient Homogeneous Systems III
- 10.7: Variation of Parameters for Nonhomogeneous Linear Systems

- 11: Boundary Value Problems and Fourier Expansions
- 12: Fourier Solutions of Partial Differential Equations
- 13: Boundary Value Problems for Second Order Linear Equations

- Book: Partial Differential Equations (Walet)
- 1: Introduction to Partial Differential Equations
- 2: Classiﬁcation of Partial Diﬀerential Equations
- 3: Boundary and Initial Conditions
- 4: Fourier Series
- 5: Separation of Variables on Rectangular Domains
- 6: D’Alembert’s Solution to the Wave Equation
- 7: Polar and Spherical Coordinate Systems
- 8: Separation of Variables in Polar Coordinates
- 9: Series Solutions of ODEs (Frobenius’ Method)
- 10: Bessel Functions and Two-Dimensional Problems
- 11: Separation of Variables in Three Dimensions

- Book: Partial Differential Equations (Miersemann)
- 1: Introduction
- 2: Equations of First Order
- 2.0: Prelude to First Order Equations
- 2.1: Linear Equations
- 2.2: Quasilinear Equations
- 2.2.1: A Linearization Method
- 2.2.2: Initial Value Problem of Cauchy
- 2.3: Nonlinear Equations in Two Variables
- 2.3.1: Initial Value Problem of Cauchy
- 2.4: Nonlinear Equations in \(\mathbb{R}^n\)
- 2.5: Hamilton-Jacobi Theory
- 2.E: Equations of First Order (Exercises)

- 3: Classification
- 3.1: Linear Equations of Second Order
- 3.1.1: Normal Form in Two Variables
- 3.2: Quasilinear Equations of Second Order
- 3.2.1: Quasilinear Elliptic Equations
- 3.3: Systems of First Order
- 3.3.1: Examples
- 3.4: Systems of Second Order
- 3.4.1: Examples
- 3.5: Theorem of Cauchy-Kovalevskaya
- 3.5.1 Appendix: Real Analytic Functions
- 3.E: Classification (Exercises)

- 4: Hyperbolic Equations
- 4.1: One-Dimensional Wave Equation
- 4.2: Higher Dimensions
- 4.2.1: Case n=3
- 4.2.2: Case n=2
- 4.3: Inhomogeneous Equations
- 4.4: A Method of Riemann
- 4.5: Initial-Boundary Value Problems
- 4.5.1: Oscillation of a String
- 4.5.2: Oscillation of a Membrane
- 4.5.3: Inhomogeneous Wave Equations
- 4.E: 4: Hyperbolic Equations (Exercises)

- 5: Fourier Transform
- 6: Parabolic Equations
- 7: Elliptic Equations of Second Order
- 7.1: Fundamental Solution
- 7.2: Representation Formula
- 7.2.1: Conclusions from the Representation Formula
- 7.3.1: Boundary Value Problems: Dirichlet Problem
- 7.3.2: Boundary Value Problems: Neumann Problem
- 7.3.3: Boundary Value Problems: Mixed Boundary Value Problem
- 7.4: Green's Function for \(\Delta\)
- 7.4.1: Green's Function for a Ball
- 7.4.2: Green's Function and Conformal Mapping
- 7.5: Inhomogeneous Equation
- 7.E: Elliptic Equations of Second Order (Exercises)

- Bibliography

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