**Ordinary Differential Equations: Brief Revision**

First-Order Equations

Homogeneous Linear Equations with Constant Coefficients

Nonhomogeneous Linear Equations with Constant Coefficients

Cauchy–Euler Equations

Functions and Operators

**Fourier Series**

The Full Fourier Series

Fourier Sine Series

Fourier Cosine Series

Convergence and Differentiation

**Sturm–Liouville Problems**

Regular Sturm–Liouville Problems

Other Problems

Bessel Functions

Legendre Polynomials

Spherical Harmonics

** **

**Some Fundamental Equations of Mathematical Physics**

The Heat Equation

The Laplace Equation

The Wave Equation

Other Equations

**The Method of Separation of Variables**

The Heat Equation

The Wave Equation

The Laplace Equation

Other Equations

Equations with More than Two Variables

**Linear Nonhomogeneous Problems**

Equilibrium Solutions

Nonhomogeneous Problems

**The Method of Eigenfunction Expansion**

The Heat Equation

The Wave Equation

The Laplace Equation

Other Equations

**The Fourier Transformations**

The Full Fourier Transformation

The Fourier Sine and Cosine Transformations

Other Applications

**The Laplace Transformation**

Definition and Properties

Applications

**The Method of Green’s Functions**

The Heat Equation

The Laplace Equation

The Wave Equation

**General Second-Order Linear Partial Differential Equations with Two Independent Variables**

The Canonical Form

Hyperbolic Equations

Parabolic Equations

Elliptic Equations

**The Method of Characteristics**

First-Order Linear Equations

First-Order Quasilinear Equations

The One-Dimensional Wave Equation

Other Hyperbolic Equations

**Perturbation and Asymptotic Methods**

Asymptotic Series

Regular Perturbation Problems

Singular Perturbation Problems

**Complex Variable Methods **

Elliptic Equations

Systems of Equations

**Answers to Odd-Numbered Exercises **

**Appendix **

**Bibliography **

**Index**