This book is designed for students in science, engineering and mathematics who have completed calculus through partial differentiation.
The Table of Contents 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 62 2.5 Exact Equations 73 2.6 Integrating Factors 82
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
5.5 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 290 6.4 Motion Under a Central Force 296
Chapter 7 Series Solutions of Linear Second Order Equations 7.1 Review of Power Series 306 7.2 Series Solutions Near an Ordinary Point I 319 7.3 Series Solutions Near an Ordinary Point II 334 7.4 Regular Singular Points Euler Equations 342 7.5 The Method of Frobenius I 347 7.6 The Method of Frobenius II 364 7.7 The Method of Frobenius III 378
Chapter 8 Laplace Transforms 8.1 Introduction to the Laplace Transform 393 8.2 The Inverse Laplace Transform 405 8.3 Solution of Initial Value Problems 413 8.4 The Unit Step Function 419 8.5 Constant Coefficient Equations with Piecewise Continuous Forcing Functions 430 8.6 Convolution 440 8.7 Constant Cofficient Equations with Impulses 452 8.8 A Brief Table of Laplace Transforms
Chapter 9 Linear Higher Order Equations 9.1 Introduction to Linear Higher Order Equations 465 9.2 Higher Order Constant Coefficient Homogeneous Equations 475 9.3 Undetermined Coefficients for Higher Order Equations 487 9.4 Variation of Parameters for Higher Order Equations 497
Chapter 10 Linear Systems of Differential Equations 10.1 Introduction to Systems of Differential Equations 507 10.2 Linear Systems of Differential Equations 515 10.3 Basic Theory of Homogeneous Linear Systems 521 10.4 Constant Coefficient Homogeneous Systems I 529vi Contents 10.5 Constant Coefficient Homogeneous Systems II 542 10.6 Constant Coefficient Homogeneous Systems II 556 10.7 Variation of Parameters for Nonhomogeneous Linear Systems 568
Chapter 11 Boundary Value Problems and Fourier Expansions 580 11.1 Eigenvalue Problems for y 00 + λy = 0 580 11.2 Fourier Series I 586 11.3 Fourier Series II 603
Chapter 12 Fourier Solutions of Partial Differential Equations
Chapter 13 Boundary Value Problems for Second Order Linear Equations
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/
