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**Schaum's Outline of Differential Equations Fourth Edition ** written by
**Richard Bronson ** Professor of Mathematics and Computer Science, Fairleigh Dickinson University and
**Gabriel B. Costa** Associate Professor of Mathematical Sciences, United States Military Academy, Associate Professor of Mathematics and Computer Science, Seton Hall University.

**Schaum's Outline of Differential Equations Fourth Edition ** written by
**Richard Bronson ** and
**Gabriel B. Costa**
cover the following topics.

Differential Equations 1

Notation 2

Solutions 2

Initial-Value and Boundary-Value Problems 2

Mathematical Models 9

The “Modeling Cycle” 9

Qualitative Methods 10

Standard Form and Differential Form 14

Linear Equations 14

Bernoulli Equations 14

Homogeneous Equations 15

Separable Equations 15

Exact Equations 15

General Solution 21

Solutions to the Initial-Value Problem 21

Reduction of Homogeneous Equations 22

Method of Solution 42

Reduction of Bernoulli Equations 42

Temperature Problems 50

Falling Body Problems 51

Dilution Problems 52

Electrical Circuits 52

Orthogonal Trajectories 53

Linear Differential Equations 73

Linearly Independent Solutions 74

The Wronskian 74

Nonhomogeneous Equations 74

Introductory Remark 83

The Characteristic Equation 83

The General Solution 84

The Characteristic Equation 89

The General Solution 90

Simple Form of the Method 94

Generalizations 95

Modifications 95

Limitations of the Method 95

The Method 103

Scope of the Method 104

Spring Problems 114

Electrical Circuit Problems 115

Buoyancy Problems 116

Classifying Solutions 117

Matrices and Vectors 131

Matrix Addition 131

Scalar and Matrix Multiplication 132

Powers of a Square Matrix 132

Differentiation and Integration of Matrices 132

The Characteristic Equation 133

Definition 140

Computation of eAt 140

Chapter 17 Reduction of Linear Differential Equations to a System of First-Order Equations 148

An Example 148

Reduction of an nth Order Equation 149

Reduction of a System 150

Qualitative Methods 157

Direction Fields 157

Euler’s Method 158

Stability 158

General Remarks 176

Modified Euler’s Method 177

Runge–Kutta Method 177

Adams–Bashford–Moulton Method 177

Milne’s Method 177

Starting Values 178

Order of a Numerical Method 178

Second-Order Differential Equations 195

Euler’s Method 196

Runge–Kutta Method 196

Adams–Bashford–Moulton Method 196

Definition 211

Properties of Laplace Transforms 211

Functions of Other Independent Variables 212

Definition 224

Manipulating Denominators 224

Manipulating Numerators 225

Convolutions 233

Unit Step Function 233

Translations 234

Laplace Transforms of Derivatives 242

Solutions of Differential Equations 243

The Method 249

Solution of the Initial-Value Problem 254

Solution with No Initial Conditions 255

Second-Order Equations 262

Analytic Functions and Ordinary Points 262

Solutions Around the Origin of Homogeneous Equations 263

Solutions Around the Origin of Nonhomogeneous Equations 263

Initial-Value Problems 264

Solutions Around Other Points 264

Regular Singular Points 275

Method of Frobenius 275

General Solution 276

Classical Differential Equations 290

Polynomial Solutions and Associated Concepts 290

Gamma Function 295

Bessel Functions 295

Algebraic Operations on Infinite Series 296

Introductory Concepts 304

Solutions and Solution Techniques 305

Standard Form 309

Solutions 310

Eigenvalue Problems 310

Sturm–Liouville Problems 310

Properties of Sturm–Liouville Problems 310

Piecewise Smooth Functions 318

Fourier Sine Series 319

Fourier Cosine Series 319

Introduction 325

Classifications 325

Solutions 326

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