Elementary Differential Equations with Boundary Value Problems

Elementary Differential Equations with Boundary Value Problems integrates the underlying theory, the solution procedures, and the numerical/computational aspects of differential equations in a seamless way. For example, whenever a new type of problem is introduced (such as first-order equations, higher-order equations, systems of differential equations, etc.) the text begins with the basic existence-uniqueness theory. This provides the student the necessary framework to understand and solve differential equations. Theory is presented as simply as possible with an emphasis on how to use it. The Table of Contents is comprehensive and allows flexibility for instructors.1: INTRODUCTION TO DIFFERENTIAL EQUATIONS 1.1 Examples of Differential Equations 1.2 Direction Fields 2: FIRST ORDER DIFFERENTIAL EQUATIONS 2.1 Introduction 2.2 First Order Linear Differential Equations 2.3 Introduction to Mathematical Models 2.4 Population Dynamics and Radioactive Decay 2.5 First Order Nonlinear Differential Equations 2.6 Separable First Order Equations 2.7 Exact Differential Equations 2.8 The Logistic Population Model 2.9 Applications to Mechanics 2.10 Euler's Method 2.11 Review Exercises 3: SECOND AND HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS 3.1 Introduction 3.2 The General Solution of Homogeneous Equations 3.3 Constant Coefficient Homogeneous Equations 3.4 Real Repeated Roots; Reduction of Order 3.5 Complex Roots 3.6 Unforced Mechanical Vibrations 3.7 The General Solution of a Linear Nonhomogeneous Equation 3.8 The Method of Undetermined Coefficients 3.9 The Method of Variation of Parameters 3.10 Forced Mechanical Vibrations, Electrical Networks, and Resonance 3.11 Higher Order Linear Homogeneous Differential Equations 3.12 Higher Order Homogeneous Constant Coefficient Differential Equations 3.13 Higher Order Linear Nonhomogeneous Differential Equations 3.14 Review Exercises 4: FIRST ORDER LINEAR SYSTEMS 4.1 Introduction 4.2 Existence and Uniqueness 4.3 Homogeneous Linear Systems 4.4 Constant Coefficient Homogeneous Systems and the Eigenvalue Problem 4.5 Real Eigenvalues and the Phase Plane 4.6 Complex Eigenvalues 4.7 Repeated Eigenvalues 4.8 Nonhomogeneous Linear Systems 4.9 Numerical Methods for Systems of Differential Equations 4.10 The Exponential Matrix and Diagonalization 4.11 Review Exercises 5: LAPLACE TRANSFORMS 5.1 Introduction 5.2 Laplace Transform Pairs 5.3 The Method of Partial Fractions 5.4 Laplace Transforms of Periodic Functions and System Transfer Functions 5.5 Solving Systems of Differential Equations 5.6 Convolution 5.7 The Delta Function and Impulse Response 6: NONLINEAR SYSTEMS 6.1 Introduction 6.2 Equilibrium Solutions and Direction Fields 6.3 Conservative Systems 6.4 Stability 6.5 Linearization and the Local Picture 6.6 Two-Dimensional Linear Systems 6.7 Predator-Prey Population Models 7: NUMERICAL METHODS 7.1 Euler's Method, Heun's Method, the Modified Euler's Method 7.2 Taylor Series Methods 7.3 Runge-Kutta Methods 8: SERIES SOLUTION OF DIFFERENTIAL EQUATIONS 8.1 Introduction 8.2 Series Solutions near an Ordinary Point 8.3 The Euler Equation 8.4 Solutions Near a Regular Singular Point and the Method of Frobenius 8.5 The Method of Frobenius Continued; Special Cases and a Summary 9: SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES 9.1 Heat Flow in a Thin Bar. Separation of Variables 9.2 Series Solutions 9.3 Calculating the Solution 9.4 Fourier Series 9.5 The Wave Equation 9.6 Laplace's Equation 9.7 Higher-Dimensional Problems; Nonhomogeneous Equations 10: FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS AND THE METHOD OF CHARACTERISTICS 10.1 The Cauchy Problem 10.2 Existence and Uniqueness 10.3 The Method of Characteristics 11: LINEAR TWO-POINT BOUNDARY VALUE PROBLEMS 11.1 Existence and Uniqueness 11.2 Two-Point Boundary Value Problems for Linear Systems 11.3 Sturm-Liouville Boundary Value Problems