Schaum's Outline of Differential Equations

Cover
Title Page
Copyright Page
Dedication
Contents
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
	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
	Solutions Around the Origin of Nonhomogeneous Equations
	Initial-Value Problems
	Solutions Around Other Points
Chapter 28 Series Solutions Near a Regular Singular Point
	Regular Singular Points
	Method of Frobenius
	General Solution
Chapter 29 Some Classical Differential Equations
	Classical Differential Equations
	Polynomial Solutions and Associated Concepts
Chapter 30 Gamma and Bessel Functions
	Gamma Function
	Bessel Functions
	Algebraic Operations on Infinite Series
Chapter 31 An Introduction to Partial Differential Equations
	Introductory Concepts
	Solutions and Solution Techniques
Chapter 32 Second-Order Boundary-Value Problems
	Standard Form
	Solutions
	Eigenvalue Problems
	Sturm–Liouville Problems
	Properties of Sturm–Liouville Problems
Chapter 33 Eigenfunction Expansions
	Piecewise Smooth Functions
	Fourier Sine Series
	Fourier Cosine Series
Chapter 34 An Introduction to Difference Equations
	Introduction
	Classifications
	Solutions
Chapter 35 Solving Differential Equations Using Mathematica
	Introduction
Chapter 36 Solving Systems of Differential Equations via Eigenvalues Using Mathematica
	Introduction
	Terminology
Chapter 37 Qualitative Methods
	Introduction
	Terminology
Chapter 38 Euler’s Method Using Microsoft Excel®
	The Method
Chapter 39 Some Interesting Modeling Problems
	Examples
APPENDIX Laplace Transforms
ANSWERS TO SUPPLEMENTARY PROBLEMS
INDEX