# Introduction To Differential Equations, An: Deterministic Modeling, Methods And Analysis (Volume 1)

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## Product Details

ISBN-13: 9789814368902 World Scientific Publishing Company, Incorporated 07/23/2012 New Edition 544 6.50(w) x 9.60(h) x 1.20(d)

Preface vii

1 Elements of Matrices, Determinants, and Calculus 1

1.1 Introduction 1

1.2 Problem-Solving Process 1

1.2.1 Problem analysis 2

1.2.2 Solutions (proof) strategies 3

1.3 Algebra of Matrices 11

1.4 Determinants 31

1.5 Matrix Calculus 52

2 First-Order Differential Equations 91

2.1 Introduction 91

2.2 Mathematical Modeling 92

2.3 Integrable Equations 100

2.3.1 General problem 101

2.3.2 Procedure for finding a general solution 101

2.3.3 Initial value problem 102

2.3.4 Procedure for solving the IVP 103

2.4 Linear Homogeneous Equations 109

2.4.1 General problem 109

2.4.2 Procedure for finding a general solution 110

2.4.3 Initial value problem 114

2.4.4 Procedure for solving the IVP 115

2.5 Linear Nonhomogeneous Equations 121

2.5.1 General problem 122

2.5.2 Procedure for finding a general solution 122

2.5.3 Initial value problem 128

2.5.4 Procedure for solving the IVP 128

2.6 Fundamental Conceptual Algorithms and Analysis 142

3 First-Order Nonlinear Differential Equations 165

3.1 Introduction 165

3.2 Mathematical Modeling 166

3.3 Energy Function Method 186

3.3.1 General problem 186

3.3.2 Procedure for finding a general solution representation 187

3.4 Integrable Reduced Equations 189

3.4.1 General problem 189

3.4.2 Procedure for finding a general solution representation 189

3.5 Linear Nonhomogeneous Reduced Equations 200

3.5.1 General problem 201

3.5.2 Procedures for finding a general solution representation 201

3.6 Variable Separable Equations 208

3.6.1 General problem 209

3.6.2 Procedure for finding a general solution 209

3.7 Homogeneous Equations 221

3.7.1 General problem 222

3.7.2 Procedure for finding a general solution 222

3.8 Bernoulli Equations 233

3.8.1 General problem 233

3.8.2 Procedure for finding a general solution 233

3.9 Essentially Time-Invariant Equations 245

3.9.1 General problem 245

3.9.2 Procedure for finding a general solution 245

4 First-Order Systems of Linear Differential Equations 253

4.1 Introduction 253

4.2 Mathematical Modeling 254

4.3 Linear Homogeneous Systems 285

4.3.1 General problem 286

4.3.2 Procedure for finding a general solution 287

4.3.3 Initial value problem 294

4.3.4 Procedure for the solving IVP 295

4.4 Procedure for Finding the Fundamental Matrix Solution 307

4.5 General Linear Homogeneous Systems 340

4.5.1 General problem 341

4.5.2 Procedure for finding a general solution 341

4.5.3 Initial value problem 354

4.5.4 Procedure for solving the IVP 354

4.6 Linear Nonhomogeneous Systems 361

4.6.1 General problem 361

4.6.2 Procedure for finding a general solution 361

4.6.3 Initial value problem 378

4.6.4 Procedure for solving the IVP 378

4.7 Fundamental Conceptual Algorithm Analysis 381

5 Higher-Order Linear Differential Equations 405

5.1 Introduction 405

5.2 Mathematical Modeling 405

5.3 Linear Homogeneous Equations 415

5.3.1 General problem 415

5.3.2 Procedure for finding a general solution 416

5.3.3 Initial value problem 418

5.3.4 Procedure for solving the IVP 420

5.4 Companion System 428

5.5 Higher-Order Linear Nonhomogeneous Equations 436

5.6 The Laplace Transform 440

5.7 Applications of the Laplace Transform 448

6 Topics in Differential Equations 457

6.1 Introduction 457

6.2 Fundamental Conceptual Algorithms and Analysis 458

6.3 Method of Variation of Parameters 468

6.4 Generalized Method of Variation of Parameters 473

6.5 Differential Inequalities and Comparison Theorems 477

6.6 Energy/Lyapunov Function and Comparison Theorems 481

6.7 Variational Comparison Method 489

6.8 Linear Hybrid Systems 494

6.8.1 General problem 499

6.9 Linear Hereditary Systems 501

6.9.1 General problem 502

6.10 Qualitative Properties of the Solution Process 505

6.11 Linear Stochastic Systems 507