Elementary Differential Equations

1 Elementary Differential EquationsBoyce, WISBN-13: 9780470458327 Table of ContentsChapter 1 Introduction Some Basic Mathematical Models; Direction Fields Solutions of Some Differential Equations Classification of Differential Equations Historical Remarks 26 Chapter 2 First Order Differential Equations Linear Equations ; Method of Integrating Factors Separable Equations Modeling with First Order Equations Differences Between Linear and Nonlinear Equations Autonomous Equations and Population Dynamics Exact Equations and Integrating Factors Numerical Approximations: Euler s Method The Existence and Uniqueness Theorem First Order Difference Equations 122 Chapter 3 Second Order Linear Equations Homogeneous Equations with Constant Coefficients Solutions of Linear Homogeneous Equations ; the Wronskian Complex Roots of the Characteristic Equation Repeated Roots; Reduction of Order Nonhomogeneous Equations ; Method of Undetermined Coefficients Variation of Parameters Mechanical and Electrical Vibrations Forced Vibrations 207 Chapter 4 Higher Order Linear Equations General Theory of nth Order Linear Equations Homogeneous Equations with Constant Coefficients The Method of Undetermined Coefficients The Method of Variation of Parameters 241 Chapter 5 Series Solutions of Second Order Linear Equations Review of Power Series Series Solutions Near an Ordinary Point, Part I Series Solutions Near an Ordinary Point, Part II Series Solutions Near an Ordinary Point, Part II Euler Equations .

2 Regular Singular Points Series Solutions Near a Regular Singular Point, Part I Series Solutions Near a Regular Singular Point, Part II Bessel s Equation 296 Chapter 6 The Laplace Transform Definition of the Laplace Transform Solution of Initial Value Problems Step Functions Differential Equations with Discontinuous Forcing Functions Impulse Functions The Convolution Integral 350 Chapter 7 Systems of First Order Linear Equations Introduction Review of Matrices Systems of Linear Algebraic Equations ; Linear Independence, Eigenvalues, Eigenvectors Basic Theory of Systems of First Order Linear Equations Homogeneous Linear Systems with Constant Coefficients Complex Eigenvalues Fundamental Matrices Repeated Eigenvalues Nonhomogeneous Linear Systems 440 Chapter 8 Numerical Methods The Euler or Tangent Line Method Improvements on the Euler Method The Runge Kutta Method Multistep Methods Systems of First Order Equations More on Errors; Stability 482 Chapter 9 Nonlinear Differential Equations and Stability The Phase Plane: Linear Systems Autonomous Systems and Stability Locally Linear Systems Competing Species Predator Prey Equations Liapunov s Second Method Periodic Solutions and Limit Cycles Chaos and Strange Attractors: The Lorenz Equations 577 Answers to Problems 589 Index 637