## Description

### Table of contents:

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 Numerical Methods: A Closer Look At Euler’s Algorithm

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 Differential Operators and the Elimination Method for Systems

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

Appendix A Review of Integration Techniques

Appendix B Newton’s Method

Appendix C Simpson’s Rule

Appendix D Cramer’s Rule

Appendix E Method of Least Squares

Appendix F Runge-Kutta Procedure for n Equations

Appendix G Software for Analyzing Differential Equations