Fundamentals of Differential Equations and Boundary Value Problems

1. Introduction

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

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

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

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

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

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

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

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

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

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

11. Eigenvalue Problems and Sturm-Liouville Equations

• 11.1 Introduction: Heat Flow in a Non-uniform Wire
• 11.2 Eigenvalues and Eigenfunctions
• 11.3 Regular Sturm-Liouville Boundary Value Problems
• 11.4 Nonhomogeneous Boundary Value Problems and the Fredholm Alternative
• 11.5 Solution by Eigenfunction Expansion
• 11.6 Green's Functions
• 11.7 Singular Sturm-Liouville Boundary Value Problems.
• 11.8 Oscillation and Comparison Theory

12. Stability of Autonomous Systems

• 12.1 Introduction: Competing Species
• 12.2 Linear Systems in the Plane
• 12.3 Almost Linear Systems
• 12.4 Energy Methods
• 12.5 Lyapunov's Direct Method
• 12.6 Limit Cycles and Periodic Solutions
• 12.7 Stability of Higher-Dimensional Systems

13. Existence and Uniqueness Theory

• 13.1 Introduction: Successive Approximations
• 13.2 Picard's Existence and Uniqueness Theorem
• 13.3 Existence of Solutions of Linear Equations
• 13.4 Continuous Dependence of Solutions

Appendices

1. Review of Integration Techniques
2. Newton’s Method
3. Simpson’s Rule
4. Cramer’s Rule
5. Method of Least Squares
6. Runge-Kutta Procedure for n Equations
7. Software for Analyzing Differential Equations