Elementary Differential Equations and Boundary Value Problems

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 Second Order Linear Equations 135
3.1 Homogeneous Equations with Constant Coefficients 
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 Coefficients 
4.3 The Method of Undetermined Coefficients 
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-Kutta Method
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 

Chapter10 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 
Appendix A Derivation of the Heat Conduction Equation 
Appendix B Derivation of the Wave Equation 

Chapter 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 
Answers to Problems 
Index