Elementary Differential Equations and Boundary Value Problems, International Adaptation

ISBN-13: 9781119820512 / Angielski / Miękka / 2022

Douglas B. (University of South Carolina - Columbia) Meade

Spis treści

TABLE OF CONTENTS1 Introduction1.1 Some Basic Mathematical Models1.2 Solutions of Some Differential Equations1.3 Classification of Differential Equations2 First-Order Differential Equations2.1 Linear Differential Equations; Method of Integrating Factors2.2 Separable Differential Equations2.3 Modeling with First-Order Linear Differential Equations2.4 Differences Between Linear and Nonlinear Differential Equations2.5 Autonomous Differential Equations and Population Dynamics2.6 Exact Differential Equations and Integrating Factors2.7 Numerical Approximations: Euler's Method2.8 The Existence and Uniqueness Theorem2.9 First-Order Difference Equations3 Second-Order Linear Differential Equations3.1 Homogeneous Differential Equations with Constant Coefficients3.2 Solutions of Linear Homogeneous Equations; the Wronskian3.3 Complex Roots of the Characteristic Equation3.4 Repeated Roots; Reduction of Order3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients3.6 Variation of Parameters3.7 Mechanical and Electrical Vibrations3.8 Forced Periodic Vibrations3.9 Central Gravitational Forces and Kepler's Laws4 Higher-Order Linear Differential Equations4.1 General Theory of nth Order Linear Differential Equations4.2 Homogeneous Differential Equations with Constant Coefficients4.3 The Method of Undetermined Coefficients4.4 The Method of Variation of Parameters5 Series Solutions of First-Order and Second-Order Linear Equations5.1 Review of Power Series5.2 Series Solution of First Order Equations5.3 Series Solutions Near an Ordinary Point, Part I5.4 Series Solutions Near an Ordinary Point, Part II5.5 Euler Equations; Regular Singular Points5.6 Series Solutions Near a Regular Singular Point, Part I5.7 Series Solutions Near a Regular Singular Point, Part II5.8 Bessel's Equation6 Systems of First-Order Linear Equations6.1 Introduction6.2 Matrices6.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors6.4 Basic Theory of Systems of First-Order Linear Equations6.5 Homogeneous Linear Systems with Constant Coefficients6.6 Complex-Valued Eigenvalues6.7 Fundamental Matrices6.8 Repeated Eigenvalues6.9 Nonhomogeneous Linear Systems7 The Laplace Transform7.1 Definition of the Laplace Transform7.2 Solution of Initial Value Problems7.3 Step Functions7.4 Differential Equations with Discontinuous Forcing Functions7.5 Impulse Functions7.6 The Convolution Integral8 Numerical Methods of Solving First Order Equations8.1 The Euler or Tangent Line Method8.2 Improvements on the Euler Method8.3 The Runge-Kutta Method8.4 Multistep Methods8.5 Systems of First-Order Equations8.6 More on Errors; Stability9 Nonlinear Differential Equations and Stability9.1 The Phase Plane: Linear Systems9.2 Autonomous Systems and Stability9.3 Locally Linear Systems9.4 Competing Species9.5 Predator - Prey Equations9.6 Lyapunov's Second Method9.7 Periodic Solutions and Limit Cycles9.8 Chaos and Strange Attractors: The Lorenz Equations10 Partial Differential Equations and Fourier Series10.1 Two-Point Boundary Value Problems10.2 Fourier Series10.3 The Fourier Convergence Theorem10.4 Even and Odd Functions10.5 Separation of Variables; Heat Conduction in a Rod10.6 Other Heat Conduction Problems10.7 The Wave Equation: Vibrations of an Elastic String10.8 Laplace's EquationA APPENDIX 537B APPENDIX 54111 Boundary Value Problems and Sturm-Liouville Theory11.1 The Occurrence of Two-Point Boundary Value Problems11.2 Sturm-Liouville Boundary Value Problems11.3 Nonhomogeneous Boundary Value Problems11.4 Singular Sturm-Liouville Problems11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion11.6 Series of Orthogonal Functions: Mean ConvergenceWeb AppendixSpecial Functions: On Legendre Polynomials and FunctionsANSWERS TO PROBLEMSINDEX