Elementary Differential Equations and Boundary Value Problems

TABLE OF CONTENTS 1 Introduction 1.1 Some Basic Mathematical Models 1.2 Solutions of Some Differential Equations 1.3 Classification of Differential Equations 2 First-Order Differential Equations 2.1 Linear Differential Equations; Method of Integrating Factors 2.2 Separable Differential Equations 2.3 Modeling with First-Order Linear Differential Equations 2.4 Differences Between Linear and Nonlinear Differential Equations 2.5 Autonomous Differential Equations and Population Dynamics 2.6 Exact Differential Equations and Integrating Factors 2.7 Numerical Approximations: Euler's Method 2.8 The Existence and Uniqueness Theorem 2.9 First-Order Difference Equations 3 Second-Order Linear Differential Equations 3.1 Homogeneous Differential Equations with Constant Coefficients 3.2 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 Periodic Vibrations 3.9 Central Gravitational Forces and Kepler's Laws 4 Higher-Order Linear Differential Equations 4.1 General Theory of nth Order Linear Differential Equations 4.2 Homogeneous Differential Equations with Constant Coefficients 4.3 The Method of Undetermined Coefficients 4.4 The Method of Variation of Parameters 5 Series Solutions of First-Order and Second-Order Linear Equations 5.1 Review of Power Series 5.2 Series Solution of First Order Equations 5.3 Series Solutions Near an Ordinary Point, Part I 5.4 Series Solutions Near an Ordinary Point, Part II 5.5 Euler Equations; Regular Singular Points 5.6 Series Solutions Near a Regular Singular Point, Part I 5.7 Series Solutions Near a Regular Singular Point, Part II 5.8 Bessel's Equation 6 Systems of First-Order Linear Equations 6.1 Introduction 6.2 Matrices 6.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 6.4 Basic Theory of Systems of First-Order Linear Equations 6.5 Homogeneous Linear Systems with Constant Coefficients 6.6 Complex-Valued Eigenvalues 6.7 Fundamental Matrices 6.8 Repeated Eigenvalues 6.9 Nonhomogeneous Linear Systems 7 The Laplace Transform 7.1 Definition of the Laplace Transform 7.2 Solution of Initial Value Problems 7.3 Step Functions 7.4 Differential Equations with Discontinuous Forcing Functions 7.5 Impulse Functions 7.6 The Convolution Integral 8 Numerical Methods of Solving First Order Equations 8.1 The Euler or Tangent Line Method 8.2 Improvements on the Euler Method 8.3 The Runge-Kutta Method 8.4 Multistep Methods 8.5 Systems of First-Order Equations 8.6 More on Errors; Stability 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 Lyapunov's Second Method 9.7 Periodic Solutions and Limit Cycles 9.8 Chaos and Strange Attractors: The Lorenz Equations 10 Partial Differential Equations and Fourier Series 10.1 Two-Point Boundary Value Problems 10.2 Fourier Series 10.3 The Fourier Convergence Theorem 10.4 Even and Odd Functions 10.5 Separation of Variables; Heat Conduction in a Rod 10.6 Other Heat Conduction Problems 10.7 The Wave Equation: Vibrations of an Elastic String 10.8 Laplace's Equation A APPENDIX 537 B APPENDIX 541 11 Boundary Value Problems and Sturm-Liouville Theory 11.1 The Occurrence of Two-Point Boundary Value Problems 11.2 Sturm-Liouville Boundary Value Problems 11.3 Nonhomogeneous Boundary Value Problems 11.4 Singular Sturm-Liouville Problems 11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 11.6 Series of Orthogonal Functions: Mean Convergence Web Appendix Special Functions: On Legendre Polynomials and Functions ANSWERS TO PROBLEMS INDEX