Intro
Preface
Contents
Basic Notations
Sets
Hyperbolic Functions
Euler Integrals
1 First-Order Differential Equations
1.1 Introduction to Ordinary Differential Equations
1.2 Separable Equations
1.3 Homogeneous Equations
1.4 First-Order Linear Differential Equations
1.5 Bernoulli Equations
1.6 Riccati Equations
1.7 Exact Differential Equations
1.8 Lagrange Equations and Clairaut Equations
1.9 Existence and Uniqueness of Solution of the Cauchy Problem
1.10 Exercises
2 Higher-Order Differential Equations
2.1 Introduction

2.2 Homogeneous Linear Differential Equations of Order n
2.3 Non-Homogeneous Linear Differential Equations of Order n
2.4 Homogeneous Linear Equations with Constant Coefficients
2.5 Nonhomogeneous Linear Equations with Constant Coefficients
2.6 Euler Equations
2.7 Exercises
3 Systems of Differential Equations
3.1 Introduction
3.2 First-Order Systems and Differential Equations of Order n
3.3 Linear Systems of Differential Equations
3.4 Linear Systems with Constant Coefficients
3.4.1 The Homogeneous Case (the Algebraic Method)

3.4.2 The Non-Homogeneous Case (the Method of Undetermined Coefficients)
3.4.2.1 The Diagonalizable Case
3.4.2.2 The Non-Diagonalizable Case
3.4.3 Matrix Exponential and Linear Systems with Constant Coefficients
3.4.3.1 Fundamental Matrix
3.4.3.2 Matrix Exponential
3.4.3.3 The Exponential of a Diagonalizable Matrix
3.4.3.4 The Exponential of a Nondiagonalizable Matrix
3.4.4 Elimination Method for Linear Systems with Constant Coefficients
3.5 Autonomous Systems of Differential Equations
3.6 First-Order Partial Differential Equations

3.6.1 Linear Homogeneous First-Order PDE
3.6.2 Quasilinear First-Order Partial Differential Equations
3.7 Exercises
4 Fourier Series
4.1 Introduction: Periodic, Piecewise Smooth Functions
4.1.1 Periodic Functions
4.1.2 Piecewise Continuous and Piecewise Smooth Functions
4.2 Fourier Series Expansions
4.2.1 Series of Functions
4.2.2 A Basic Trigonometric System
4.2.3 Fourier Coefficients
4.3 Orthogonal Systems of Functions
4.3.1 Inner Product
4.3.2 Best Approximation in the Mean: Bessel's Inequality
4.4 The Convergence of Fourier Series

4.5 Differentiation and Integration of the Fourier Series
4.6 The Convergence in the Mean: Complete Systems
4.7 Examples of Fourier Expansions
4.8 The Complex form of the Fourier Series
4.9 Exercises
5 Fourier Transform
5.1 Improper Integrals
5.2 The Fourier Integral Formula
5.3 The Fourier Transform
5.4 Solving Linear Differential Equations
5.5 Moments Theorems
5.6 Sampling Theorem
5.7 Discrete Fourier Transform
5.8 Exercises
6 Laplace Transform
6.1 Introduction
6.2 Properties of the Laplace Transform
6.3 Inverse Laplace Transform