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### Schaum's Outline of Differential Equations Fourth Edition by Richard Bronson and Gabriel B. Costa

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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.

• 1 Basic Concepts 1
Differential Equations 1
Notation 2
Solutions 2
Initial-Value and Boundary-Value Problems 2

• 2 An Introduction to Modeling and Qualitative Methods 9
Mathematical Models 9
The “Modeling Cycle” 9
Qualitative Methods 10

• 3 Classifications of First-Order Differential Equations 14
Standard Form and Differential Form 14
Linear Equations 14
Bernoulli Equations 14
Homogeneous Equations 15
Separable Equations 15
Exact Equations 15

• 4 Separable First-Order Differential Equations 21
General Solution 21
Solutions to the Initial-Value Problem 21
Reduction of Homogeneous Equations 22

• 5 Exact First-Order Differential Equations 31 Defining Properties 31 Method of Solution 31 Integrating Factors 32

• 6 Linear First-Order Differential Equations 42
Method of Solution 42
Reduction of Bernoulli Equations 42

• 7 Applications of First-Order Differential Equations 50 Growth and Decay Problems 50
Temperature Problems 50
Falling Body Problems 51
Dilution Problems 52
Electrical Circuits 52
Orthogonal Trajectories 53

• 8 Linear Differential Equations: Theory of Solutions 73
Linear Differential Equations 73
Linearly Independent Solutions 74
The Wronskian 74
Nonhomogeneous Equations 74

• 9 Second-Order Linear Homogeneous Differential Equations with Constant Coefficients 83
Introductory Remark 83
The Characteristic Equation 83
The General Solution 84

• 10 nth-Order Linear Homogeneous Differential Equations with Constant Coefficients 89
The Characteristic Equation 89
The General Solution 90

• 11 The Method of Undetermined Coefficients 94
Simple Form of the Method 94
Generalizations 95
Modifications 95
Limitations of the Method 95

• 12 Variation of Parameters 103
The Method 103
Scope of the Method 104

• 13 Initial-Value Problems for Linear Differential Equations 110

• 14 Applications of Second-Order Linear Differential Equations 114
Spring Problems 114
Electrical Circuit Problems 115
Buoyancy Problems 116
Classifying Solutions 117

• 15 Matrices 131
Matrices and Vectors 131
Scalar and Matrix Multiplication 132
Powers of a Square Matrix 132
Differentiation and Integration of Matrices 132
The Characteristic Equation 133

• 16 eAt 140
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

• 18 Graphical and Numerical Methods for Solving First-Order Differential Equations 157
Qualitative Methods 157
Direction Fields 157
Euler’s Method 158
Stability 158

• 19 Further Numerical Methods for Solving First-Order Differential Equations 176
General Remarks 176
Modified Euler’s Method 177
Runge–Kutta Method 177
Milne’s Method 177
Starting Values 178
Order of a Numerical Method 178

• 20 Numerical Methods for Solving Second-Order Differential Equations Via Systems 195
Second-Order Differential Equations 195
Euler’s Method 196
Runge–Kutta Method 196

• 21 The Laplace Transform 211
Definition 211
Properties of Laplace Transforms 211
Functions of Other Independent Variables 212

• 22 Inverse Laplace Transforms 224
Definition 224
Manipulating Denominators 224
Manipulating Numerators 225

• 23 Convolutions and the Unit Step Function 233
Convolutions 233
Unit Step Function 233
Translations 234

• 24 Solutions of Linear Differential Equations with Constant Coefficients by Laplace Transforms 242
Laplace Transforms of Derivatives 242
Solutions of Differential Equations 243

• 25 Solutions of Linear Systems by Laplace Transforms 249
The Method 249

• 26 Solutions of Linear Differential Equations with Constant Coefficients by Matrix Methods 254
Solution of the Initial-Value Problem 254
Solution with No Initial Conditions 255

• 27 Power Series Solutions of Linear Differential Equations with Variable Coefficients 262
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

• 28 Series Solutions Near a Regular Singular Point 275
Regular Singular Points 275
Method of Frobenius 275
General Solution 276

• 29 Some Classical Differential Equations 290
Classical Differential Equations 290
Polynomial Solutions and Associated Concepts 290

• 30 Gamma and Bessel Functions 295
Gamma Function 295
Bessel Functions 295
Algebraic Operations on Infinite Series 296

• 31 An Introduction to Partial Differential Equations 304
Introductory Concepts 304
Solutions and Solution Techniques 305

• 32 Second-Order Boundary-Value Problems 309
Standard Form 309
Solutions 310
Eigenvalue Problems 310
Sturm–Liouville Problems 310
Properties of Sturm–Liouville Problems 310

• 33 Eigenfunction Expansions 318
Piecewise Smooth Functions 318
Fourier Sine Series 319
Fourier Cosine Series 319

• 34 An Introduction to Difference Equations 325
Introduction 325
Classifications 325
Solutions 326

• A Laplace Transforms 330

• ANSWERS TO SUPPLEMENTARY PROBLEMS 338

• ##### Math Books of Differential Equations

Differential Equations I MATB44H3F
• Free
• English
• PDF 81
• Page 174

• Problem Solving by George Polya
• Free
• English
• PDF
• Page 433
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