Fundamentals of Differential Equations

Chapter 1: Introduction

• • 1.1 Background
• • 1.2 Solutions and Initial Value Problems
• • 1.3 Direction Fields
• • 1.4 The Approximation Method of Euler

Chapter 2: First-Order Differential Equations

• • 2.1 Introduction: Motion of a Falling Body
• • 2.2 Separable Equations
• • 2.3 Linear Equations
• • 2.4 Exact Equations
• • 2.5 Special Integrating Factors
• • 2.6 Substitutions and Transformations

Chapter 3: Mathematical Models and Numerical Methods Involving First Order Equations

• • 3.1 Mathematical Modeling
• • 3.2 Compartmental Analysis
• • 3.3 Heating and Cooling of Buildings
• • 3.4 Newtonian Mechanics
• • 3.5 Electrical Circuits
• • 3.6 Improved Euler's Method
• • 3.7 Higher-Order Numerical Methods: Taylor and Runge-Kutta

Chapter 4: Linear Second-Order Equations

• • 4.1 Introduction: The Mass-Spring Oscillator
• • 4.2 Homogeneous Linear Equations: The General Solution
• • 4.3 Auxiliary Equations with Complex Roots
• • 4.4 Nonhomogeneous Equations: The Method of Undetermined Coefficients
• • 4.5 The Superposition Principle and Undetermined Coefficients Revisited
• • 4.6 Variation of Parameters
• • 4.7 Variable-Coefficient Equations
• • 4.8 Qualitative Considerations for Variable-Coefficient and Nonlinear Equations
• • 4.9 A Closer Look at Free Mechanical Vibrations
• • 4.10 A Closer Look at Forced Mechanical Vibrations

Chapter 5: Introduction to Systems and Phase Plane Analysis

• • 5.1 Interconnected Fluid Tanks
• • 5.2 Elimination Method for Systems with Constant Coefficients
• • 5.3 Solving Systems and Higher-Order Equations Numerically
• • 5.4 Introduction to the Phase Plane
• • 5.5 Applications to Biomathematics: Epidemic and Tumor Growth Models
• • 5.6 Coupled Mass-Spring Systems
• • 5.7 Electrical Systems
• • 5.8 Dynamical Systems, Poincaré Maps, and Chaos

Chapter 6: Theory of Higher-Order Linear Differential Equations

• • 6.1 Basic Theory of Linear Differential Equations
• • 6.2 Homogeneous Linear Equations with Constant Coefficients
• • 6.3 Undetermined Coefficients and the Annihilator Method
• • 6.4 Method of Variation of Parameters

Chapter 7: Laplace Transforms

• • 7.1 Introduction: A Mixing Problem
• • 7.2 Definition of the Laplace Transform
• • 7.3 Properties of the Laplace Transform
• • 7.4 Inverse Laplace Transform
• • 7.5 Solving Initial Value Problems
• • 7.6 Transforms of Discontinuous Functions
• • 7.7 Transforms of Periodic and Power Functions
• • 7.8 Convolution
• • 7.9 Impulses and the Dirac Delta Function
• • 7.10 Solving Linear Systems with Laplace Transforms

Chapter 8: Series Solutions of Differential Equations

• • 8.1 Introduction: The Taylor Polynomial Approximation
• • 8.2 Power Series and Analytic Functions
• • 8.3 Power Series Solutions to Linear Differential Equations
• • 8.4 Equations with Analytic Coefficients
• • 8.5 Cauchy-Euler (Equidimensional) Equations
• • 8.6 Method of Frobenius
• • 8.7 Finding a Second Linearly Independent Solution
• • 8.8 Special Functions

Chapter 9: Matrix Methods for Linear Systems

• • 9.1 Introduction
• • 9.2 Review 1: Linear Algebraic Equations
• • 9.3 Review 2: Matrices and Vectors
• • 9.4 Linear Systems in Normal Form
• • 9.5 Homogeneous Linear Systems with Constant Coefficients
• • 9.6 Complex Eigenvalues
• • 9.7 Nonhomogeneous Linear Systems
• • 9.8 The Matrix Exponential Function

Chapter 10: Partial Differential Equations

• • 10.1 Introduction: A Model for Heat Flow
• • 10.2 Method of Separation of Variables
• • 10.3 Fourier Series
• • 10.4 Fourier Cosine and Sine Series
• • 10.5 The Heat Equation
• • 10.6 The Wave Equation
• • 10.7 Laplace's Equation

Chapter Appendices: Appendices

• • 1. Newton’s Method
• • 2. Simpson’s Rule
• • 3. Cramer’s Rule
• • 4. Method of Least Squares
• • 5. Runge-Kutta Procedure for n Equations