Main subject categories: • Ordinary differential equations • Problems & solutions of ODEs • Mathematical models • Linear DEs of higher order • Systems of Linear DEs • Laplace transformsThis unique book on ordinary differential equations addresses practical issues of composing and solving such equations by large number of examples and homework problems with solutions. These problems originate in engineering, finance, as well as science at appropriate levels that readers with the basic knowledge of calculus, physics or economics are assumed able to follow. Contents 10 PREFACE 6 CHAPTER 1 FIRST-ORDER DIFFERENTIAL EQUATIONS 14 1.1 DEFINITION OF DIFFERENTIAL EQUATIONS 14 1.2 MATHEMATICAL MODELS 22 1.2.1 Newton’s Law of Cooling 22 1.2.2 Newton’s Law of Motion 24 1.2.3 Torricelli’s Law for Draining 26 1.2.4 Population Models 30 1.2.5 A Swimmer’s Problem 31 1.2.6 Slope Fields & Solution Curves 34 1.3 SEPARATION OF VARIABLES 39 1.4 LINEAR FIRST-ORDER DES 45 1.5 SUBSTITUTION METHODS 53 1.5.1 Polynomial Substitution 53 1.5.2 Homogeneous DEs 55 1.5.3 Bernoulli DEs 57 1.6 THE EXACT DES 64 1.7 RICCATI DES 85 CHAPTER 2 MATHEMATICAL MODELS 89 2.1 POPULATION MODEL 89 2.1.1 General Population Equation 89 2.1.2 The Logistic Equation 91 2.1.3 Doomsday vs. Extinction 94 2.2 ACCELERATION-VELOCITY MODEL 100 2.2.1 Velocity and Acceleration Models 100 2.2.2 Air Resistance Model 101 2.2.3 Gravitational Acceleration 106 2.3 AN EXAMPLE IN FINANCE 114 CHAPTER 3 LINEAR DES OF HIGHER ORDER 120 3.1 CLASSIFICATION OF DES 120 3.2 LINEAR INDEPENDENCE 125 3.3 CONSTANT COEFFICIENT HOMOGENEOUS DES 134 3.4 CAUCHY-EULER DES 148 3.5 INHOMOGENEOUS HIGHER ORDER DES 152 3.6 VARIATION OF PARAMETERS 164 CHAPTER 4 SYSTEMS OF LINEAR DES 175 4.1 BASICS OF DE SYSTEMS 175 4.2 FIRST-ORDER SYSTEMS AND APPLICATIONS 178 4.3 SUBSTITUTION METHOD 186 4.4 OPERATOR METHOD 192 4.5 EIGEN-ANALYSIS METHOD 197 CHAPTER 5 LAPLACE TRANSFORMS 206 5.1 LAPLACE TRANSFORMS 206 5.2 PROPERTIES OF LAPLACE TRANSFORMS 208 5.2.1 Laplace Transforms for Polynomials 209 5.2.2 The Translator Property 212 5.2.3 Shifting Property 216 5.2.4 The t-multiplication property 219 5.2.5 Periodic Functions 222 5.2.6 Differentiation and Integration Property 223 5.3 INVERSE LAPLACE TRANSFORMS 228 5.4 THE CONVOLUTION OF TWO FUNCTIONS 233 5.5 APPLICATION OF LAPLACE TRANSFORMS 237 APPENDIX A SOLUTIONS TO SELECTED PROBLEMS 251 CHAPTER 1 FIRST-ORDER DES 251 1.1 Definition of DEs 251 1.2 Mathematical Models 257 1.3 Separation of Variables 267 1.4 Linear First-Order DEs 275 1.5 Substitution Methods 286 1.6 The Exact DEs 309 1.7 Riccati DEs 321 CHAPTER 2 MATHEMATICAL MODELS 329 2.1 Population Model 329 2.2 Acceleration-Velocity Model 339 2.3 An example in Finance 371 CHAPTER 3 LINEAR DES OF HIGHER ORDER 381 3.1 Classification of DEs 381 3.2 Linear Independence 383 3.3 Constant Coefficient Homogeneous DEs 391 3.4 Cauchy-Euler DEs 404 3.5 Inhomogeneous Higher Order DEs 409 3.6 Variation of Parameters 428 CHAPTER 4 SYSTEMS OF LINEAR DES 439 4.2 First-Order Systems and Applications 439 4.3 Substitution Method 444 4.4 Operator Method 452 4.5 Eigen-Analysis Method 460 CHAPTER 5 LAPLACE TRANSFORMS 468 5.2 Properties of Laplace Transforms 468 5.3 Inverse Laplace Transforms 476 5.4 The Convolution of Two Functions 480 5.5 Application of Laplace Transforms 483 APPENDIX B LAPLACE TRANSFORMS 518 SELECTED LAPLACE TRANSFORMS 518 SELECTED PROPERTIES OF LAPLACE TRANSFORMS 519 APPENDIX C DERIVATIVES & INTEGRALS 522 APPENDIX D ABBREVIATIONS 524 APPENDIX E TEACHING PLANS 526 REFERENCES 528 INDEX 530
