Elementary Differential Equations with Boundary Value Problems 2

Elementary Differential Equations with Boundary Value Problems Recommended Citation Table of Contents Preface Chapter 1 Introduction Section 1.1 Some Applications Leading to Differential Equations Section 1.2 Basic Concepts Section 1.3 Direction Fields for First Order Equations Chapter 2 First Order Equations Section 2.1 Linear First Order Equations Section 2.2 Separable Equations Section 2.3 Existence and Uniqueness of Solutions of Nonlinear Equations Section 2.4 Transformation of Nonlinear Equations into Separable Equations Section 2.5 Exact Equations Section 2.6 Exact Equations Chapter 3 Numerical Methods Section 3.1 Euler's Method Section 3.2 The Improved Euler Method and Related Methods Section 3.3 The Runge-Kutta Method Chapter 4 Applications of First Order Equations Section 4.1 Growth and Decay Section 4.2 Cooling and Mixing Section 4.3 Elementary Mechanics Section 4.4 Autonomous Second Order Equations Section 4.5 Applications to Curves Chapter 5 Linear Second Order Equations Section 5.1 Homogeneous Linear Equations Section 5.2 Constant Coefficient Homogeneous Equations Section 5.3 Nonhomogeneous Linear Equations Section 5.4 The Method of Undetermined Coefficients I Section 5.5 The Method of Undetermined Coefficients II Section 5.6 Reduction of Order Section 5.7 Variation of Parameters Chapter 6 Applications of Linear Second Order Equations Section 6.1 Spring Problems I Section 6.2 Spring Problems II Section 6.3 The RLC Circuit Section 6.4 Motion Under a Central Force Chapter 7 Series Solutions of Linear Second Equations Section 7.1 Review of Power Series Section 7.2 Series Solutions Near an Ordinary Point I Section 7.3 Series Solutions Near an Ordinary Point II Section 7.4 Regular Singular Points Euler Equations Section 7.5 The Method of Frobenius I Section 7.6 The Method of Frobenius II Section 7.7 The Method of Frobenius III Chapter 8 Laplace Transforms Section 8.1 Introduction to the Laplace Transform Section 8.2 The Inverse Laplace Transform Section 8.3 Solution of Initial Value Problems Section 8.4 The Unit Step Function Section 8.5 Constant Coeefficient Equations with Piecewise Continuous Forcing Functions Section 8.6 Convolution Section 8.7 Constant Coefficient Equations with Impulses 8.8 A Brief Table of Laplace Transforms Chapter 9 Linear Higher Order Equations Section 9.1 Introduction to Linear Higher Order Equations Section 9.2 Higher Order Constant Coefficient Homogeneous Equations Section 9.3 Undetermined Coefficients for Higher Order Equations Section 9.4 Variation of Parameters for Higher Order Equations Chapter 10 Linear Systems of Differential Equations Section 10.1 Introduction to Systems of Differential Equations Section 10.2 Linear Systems of Differential Equations Section 10.3 Basic Theory of Homogeneous Linear System Section 10.4 Constant Coefficient Homogeneous Systems I Section 10.5 Constant Coefficient Homogeneous Systems II Section 10.6 Constant Coefficient Homogeneous Systems III Section 10.7 Variation of Parameters for Nonhomogeneous Linear Systems Chapter 11 Boundary Value Problems and Fourier Expansions Section 11.1 Eigenvalue Problems Section 11.2 Fourier Expansions I Section 11.3 Fourier Expansions II Chapter 12 Fourier Solutions of Partial Differential Section 12.1 The Heat Equation Section 12.2 The Wave Equation Section 12.3 Laplace's Equation in Rectangular Coordinates Section 12.4 Laplace's Equation in Polar Coordinates Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations Section 13.1 Two-Point Boundary Value Problems Section 13.2 Sturm-Liouville Problems A Brief Table of Integrals Answers to Selected Index