Elementary Differential Equations and Boundary Value Problems, 13th Edition

1. Introduction
   1.1  Some Basic Mathematical Models; Direction Fields
   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 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

4. Higher-Order Linear Equations
   4.1  General Theory of nth Order Linear 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 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 I
   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

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

7. Systems of First-Order Linear Equations
   7.1  Introduction
   7.2 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-Valued Eigenvalues
   7.7 Fundamental Matrices
   7.8 Repeated Eigenvalues
   7.9 Nonhomogeneous Linear Systems

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 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 Liapunov’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

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