Fundamentals of Differential Equations and Boundary Value Problems – R. Nagle, E. Saff, D. Snider – 6th Edition

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 and Periodic Functions
7.7 Convolution
7.8 Impulses and the Dirac Delta Function
7.9 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 Nonuniform 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
A. Review of Integration Techniques
B. Newton's Method
C. Simpson's Rule
D. Cramer's Rule
E. Method of Least Squares
F. Runge-Kutta Procedure for n Equations