This book is designed for students in science, engineering and mathematics who have completed calculus through partial differentiation.
The Table of Contents for this book is as follows:
Chapter 1 Introduction 1 1.1 Applications Leading to Differential Equations 1.2 First Order Equations 5 1.3 Direction Fields for First Order Equations 16
Chapter 2 First Order Equations 30 2.1 Linear First Order Equations 30 2.2 Separable Equations 45 2.3 Existence and Uniqueness of Solutions of Nonlinear Equations 55 2.4 Transformation of Nonlinear Equations into Separable Equations 63 2.5 Exact Equations 73 2.6 Integrating Factors 83
Chapter 3 Numerical Methods 3.1 Euler’s Method 96 3.2 The Improved Euler Method and Related Methods 109 3.3 The Runge-Kutta Method 119
Chapter 4 Applications of First Order Equations1em 130 4.1 Growth and Decay 130 4.2 Cooling and Mixing 140 4.3 Elementary Mechanics 151 4.4 Autonomous Second Order Equations 162 4.5 Applications to Curves 179
Chapter 5 Linear Second Order Equations 5.1 Homogeneous Linear Equations 194 5.2 Constant Coefficient Homogeneous Equations 210 5.3 Nonhomgeneous Linear Equations 221 5.4 The Method of Undetermined Coefficients I 229 iv5.5 The Method of Undetermined Coefficients II 238 5.6 Reduction of Order 248 5.7 Variation of Parameters 255
Chapter 6 Applcations of Linear Second Order Equations 268 6.1 Spring Problems I 268 6.2 Spring Problems II 279 6.3 The RLC Circuit 291 6.4 Motion Under a Central Force 297
Chapter 7 Series Solutions of Linear Second Order Equations 7.1 Review of Power Series 307 7.2 Series Solutions Near an Ordinary Point I 320 7.3 Series Solutions Near an Ordinary Point II 335 7.4 Regular Singular Points Euler Equations 343 7.5 The Method of Frobenius I 348 7.6 The Method of Frobenius II 365 7.7 The Method of Frobenius III 379
Chapter 8 Laplace Transforms 8.1 Introduction to the Laplace Transform 394 8.2 The Inverse Laplace Transform 406 8.3 Solution of Initial Value Problems 414 8.4 The Unit Step Function 421 8.5 Constant Coefficient Equations with Piecewise Continuous Forcing Functions 431 8.6 Convolution 441 8.7 Constant Cofficient Equations with Impulses 453 8.8 A Brief Table of Laplace Transforms
Chapter 9 Linear Higher Order Equations 9.1 Introduction to Linear Higher Order Equations 466 9.2 Higher Order Constant Coefficient Homogeneous Equations 476 9.3 Undetermined Coefficients for Higher Order Equations 488 9.4 Variation of Parameters for Higher Order Equations 498
Chapter 10 Linear Systems of Differential Equations 10.1 Introduction to Systems of Differential Equations 508 10.2 Linear Systems of Differential Equations 516 10.3 Basic Theory of Homogeneous Linear Systems 522 10.4 Constant Coefficient Homogeneous Systems I 530vi Contents 10.5 Constant Coefficient Homogeneous Systems II 543 10.6 Constant Coefficient Homogeneous Systems II 557 10.7 Variation of Parameters for Nonhomogeneous Linear Systems 570
NOTE: This book meets the evaluation criteria set by the Editorial Board of the American Institute of Mathematics in connection with the Institute's Open Textbook Initiative. http://www.aimath.org/textbooks/
