Contents of Differential Equations Book
Chapter 1. Basic Concepts
Differential Equations Notation
Solutions
Initial-Value and Boundary-Value Problems
Chapter 2. An Introduction to Modeling and Qualitative Methods Mathematical Models
The “Modeling Cycle”
Qualitative Methods
Chapter 3. Classifications of First-Order Differential Equations
Standard Form and Differential Form
Linear Equations
Bernoulli Equations
Homogeneous Equations
Separable Equations
Exact Equations
Chapter 4. Separable First-Order Differential Equations General Solution
Solutions to the Initial-Value Problem
Reduction of Homogeneous Equations
Chapter 5. Exact First-Order Differential Equations
Defining Properties Method of Solution
Integrating Factors
Chapter 6. Linear First-Order Differential Equations Method of Solution
Reduction of Bernoulli Equations
Chapter 7. Applications of First-Order Differential Equations
Growth and Decay Problems
Temperature Problems
Falling Body Problems Dilution Problems
Electrical Circuits Orthogonal Trajectories
Chapter 8. Linear Differential Equations: Theory of Solutions
Linear Differential Equations
Linearly Independent Solutions
The Wronskian Nonhomogeneous Equations
Chapter 9. Second-Order Linear Homogeneous Differential Equations with Constant Coefficients
Introductory Remark
The Characteristic Equation
The General Solution
Chapter 10. nth-Order Linear Homogeneous Differential Equations with Constant Coefficients
The Characteristic Equation
The General Solution
Chapter 11. The Method of Undetermined Coefficients
Simple Form of the Method Generalizations Modifications
Limitations of the Method
Chapter 12. Variation of Parameters
The Method
Scope of the Method
Chapter 13. Initial-Value Problems for Linear Differential Equations
Chapter 14. Applications of Second-Order Linear Differential Equations
Spring Problems
Electrical Circuit Problems
Buoyancy Problems
Classifying Solutions
Chapter 15. Matrices Matrices and Vectors Matrix Addition
Scalar and Matrix Multiplication
Powers of a Square Matrix
Differentiation and Integration of Matrices
The Characteristic Equation
Chapter 16. eAt 140 Definition
Computation of eAt
Chapter 17. Reduction of Linear Differential Equations to a System of First-Order Equations
An Example
Reduction of an nth Order Equation
Reduction of a System
Chapter 18. Graphical and Numerical Methods for Solving First-Order Differential Equations
Qualitative Methods Direction Fields
Euler’s Method
Stability
Chapter 19. Further Numerical Methods for Solving First-Order Differential Equations General Remarks Modified Euler’s Method
Runge–Kutta Method
Adams–Bashford–Moulton Method Milne’s Method
Starting Values Order of a Numerical Method
Chapter 20. Numerical Methods for Solving Second-Order Differential Equations Via Systems
Second-Order Differential Equations
Euler’s Method
Runge–Kutta Method
Adams–Bashford–Moulton Method
Chapter 21. The Laplace Transform
Definition
Properties of Laplace Transforms
Functions of Other Independent Variables
Chapter 22. Inverse Laplace Transforms Definition Manipulating Denominators
Manipulating Numerators
Chapter 23. Convolutions and the Unit Step Function
Convolutions Unit Step Function
Translations
Chapter 24. Solutions of Linear Differential Equations with Constant Coefficients by Laplace Transforms
Laplace Transforms of Derivatives
Solutions of Differential Equations
Chapter 25. Solutions of Linear Systems by Laplace
Transforms
The Method
Chapter 26. Solutions of Linear Differential Equations with Constant Coefficients by Matrix Methods
Solution of the Initial-Value Problem
Solution with No Initial Conditions
Chapter 27. Power Series Solutions of Linear Differential
Equations with Variable Coefficients
Second-Order Equations
Analytic Functions and Ordinary Points
Solutions Around the Origin of Homogeneous Equations
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