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Crash Course- Fundamentals of Differential Equations and Boundary Value Problems R. Nagle, Edward Saff, Arthur Snider

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Textbook used as reference: Nagle, R., Edward Saff, and Arthur Snider. Fundamentals of Differential Equations and Boundary Value Problems. 7th ed., Pearson, 2017.

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 Numerical Methods: A Closer Look At Euler’s Algorithm
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 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

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

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