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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
    Matrix Addition 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
    Adams–Bashford–Moulton 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
    Adams–Bashford–Moulton 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

  • B Some Comments about Technology 336 Introductory Remarks 336

  • ANSWERS TO SUPPLEMENTARY PROBLEMS 338

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