Elementary Differential Equations

Preface Chapter 1 Introduction 1 1.1 Some Basic Mathematical Models; Direction Fields 1.2 Solutions of Some Differential Equations 1.3 Classification of Differential Equations 1.4 Historical Remarks Chapter 2 First Order Differential Equations 2.1 Linear Equations; Method of Integrating Factors 2.2 Separable Equations 2.3 Modeling with First Order Equations 2.4 Differences Between Linear and Nonlinear Equations 2.5 Autonomous Equations and Population Dynamics 2.6 Exact Equations and Integrating Factors 2.7 Numerical Approximations: Euler's Method 2.8 The Existence and Uniqueness Theorem 2.9 First Order Difference Equations Chapter 3 SecondOrder Linear Equations 135 3.1 Homogeneous Equations with Constant Coef?cients 3.2 Fundamental Solutions of Linear Homogeneous Equations; The Wronskian 3.3 Complex Roots of the Characteristic Equation 3.4 Repeated Roots; Reduction of Order 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients 3.6 Variation of Parameters 3.7 Mechanical and Electrical Vibrations 3.8 Forced Vibrations Chapter 4 Higher Order Linear Equations 4.1 General Theory of nth Order Linear Equations 4.2 Homogeneous Equations with Constant Coef?cients 4.3 The Method of Undetermined Coef?cients 4.4 The Method of Variation of Parameters Chapter 5 Series Solutions of Second Order Linear Equations 5.1 Review of Power Series 5.2 Series Solutions Near an Ordinary Point, Part I 5.3 Series Solutions Near an Ordinary Point, Part II 5.4 Euler Equations; Regular Singular Points 5.5 Series Solutions Near a Regular Singular Point, Part I 5.6 Series Solutions Near a Regular Singular Point, Part II 5.7 Bessel's Equation Chapter 6 The Laplace Transform 6.1 Definition of the Laplace Transform 6.2 Solution of Initial Value Problems 6.3 Step Functions 6.4 Differential Equations with Discontinuous Forcing Functions 6.5 Impulse Functions 6.6 The Convolution Integral Chapter 7 Systems of First Order Linear Equations 7.1 Introduction 7.2 Review of Matrices 7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 7.4 Basic Theory of Systems of First Order Linear Equations 7.5 Homogeneous Linear Systems with Constant Coefficients 7.6 Complex Eigenvalues 7.7 Fundamental Matrices 7.8 Repeated Eigenvalues 7.9 Nonhomogeneous Linear Systems Chapter 8 Numerical Methods 8.1 The Euler or Tangent Line Method 8.2 Improvements on the Euler Method 8.3 The Runge-KuttaMethod 8.4 Multistep Methods 8.5 More on Errors; Stability 8.6 Systems of First Order Equations Chapter 9 Nonlinear Differential Equations and Stability 9.1 The Phase Plane: Linear Systems 9.2 Autonomous Systems and Stability 9.3 Locally Linear Systems 9.4 Competing Species 9.5 Predator-Prey Equations 9.6 Liapunov's Second Method 9.7 Periodic Solutions and Limit Cycles 9.8 Chaos and Strange Attractors: The Lorenz Equations Answers to Problems Index