Problems 1.1.1 -

- Chapter 1: First-Order Differential Equations
- Section 1.1: Differential Equations Everywhere
- Section 1.2: Basic Ideas and Terminology
- Section 1.3: The Geometry of First-Order Differential Equations
- Section 1.4: Separable Differential Equations
- Section 1.5: Some Simple Population Models
- Section 1.6: First-Order Linear Differential Equations
- Section 1.7: Modeling Problems Using First-Order Linear Differential Equations
- Section 1.8: Change of Variables
- Section 1.9: Exact Differential Equations
- Section 1.10: Numerical Solution to First-Order Differential Equations
- Section 1.11: Some Higher-Order Differential Equations

- Chapter 2: Matrices and Systems of Linear Equations
- Section 2.1: Matrices: Definitions and Notation
- Section 2.2: Matrix Algebra
- Section 2.3: Terminology for Systems of Linear Equations
- Section 2.4: Row-Echelon Matrices and Elementary Row Operations
- Section 2.5: Gaussian Elimination
- Section 2.6: The Inverse of a Square Matrix
- Section 2.7: Elementary Matrices and the LU Factorization
- Section 2.8: The Invertible Matrix Theorem I

- Chapter 3: Determinants
- Section 3.1: The Definition of the Determinant
- Section 3.2: Properties of Determinants
- Section 3.3: Cofactor Expansions
- Section 3.4: Summary of Determinants

- Chapter 4: Vector Spaces
- Section 4.1: Vectors in R^
*n* - Section 4.2: Definition of a Vector Space
- Section 4.3: Subspaces
- Section 4.4: Spanning Sets
- Section 4.5: Linear Dependence and Linear Independence
- Section 4.6: Bases and Dimension
- Section 4.7: Change of Basis
- Section 4.8: Row Space and Column Space
- Section 4.9: The Rank-Nullity Theorem
- Section 4.10: Invertible Matrix Theorem II

- Section 4.1: Vectors in R^

- Chapter 5: Inner Product Spaces
- Section 5.1: Definition of an Inner Product Space
- Section 5.2: Orthogonal Sets of Vectors and Orthogonal Projections
- Section 5.3: The Gram-Schmidt Process
- Section 5.4: Least Squares Approximation

- Chapter 6: Linear Transformations
- Section 6.1: Definition of a Linear Transformation
- Section 6.2: Transformations of R^2
- Section 6.3: The Kernel and Range of a Linear Transformation
- Section 6.4: Additional Properties of Linear Transformations
- Section 6.5: The Matrix of a Linear Transformation

- Chapter 7: Eigenvalues and Eigenvectors
- Section 7.1: The Eigenvalue/Eigenvector Problem
- Section 7.2: General Results for Eigenvalues and Eigenvectors
- Section 7.3: Diagonalization
- Section 7.4: An Introduction to the Matrix Exponential Function
- Section 7.5: Orthogonal Diagonalization and Quadratic Forms
- Section 7.6: Jordan Canonical Forms

- Chapter 8: Linear Differential Equations of Order
*n*- Section 8.1: General Theory for Linear Differential Equations
- Section 8.2: Constant Coefficient Homogeneous Linear Differential Equations
- Section 8.3: The Method of Undetermined Coefficients: Annihilators
- Section 8.4: Complex-Valued Trial Solutions
- Section 8.5: Oscillations of Mechanical Systems
- Section 8.6: RLC Circuits
- Section 8.7: The Variation of Parameters Method
- Section 8.8: A Differential Equation with Nonconstant Coefficients
- Section 8.9: Reduction of Order

- Chapter 9: Systems of Differential Equations
- Section 9.1: First-Order Linear Systems
- Section 9.2: Vector Formulation
- Section 9.3: General Results for First-Order Linear Differential Systems
- Section 9.4: Vector Differential Equations: Nondefective Coefficient Matrix
- Section 9.5: Vector Differential Equations: Defective Coefficient Matrix
- Section 9.6: Variation-of-Parameters for Linear Systems
- Section 9.7: Some Applications of Linear Systems of Differential Equations
- Section 9.8: Matrix Exponential Function and Systems of Differential Equations
- Section 9.9: The Phase Plane for Linear Autonomous Systems
- Section 9.10: Nonlinear Systems

- Chapter 10: The Laplace Transform and Some Elementary Applications
- Section 10.1: Definition of the Laplace Transform
- Section 10.2: The Existence of the Laplace Transform and the Inverse Transform
- Section 10.3: Periodic Functions and the Laplace Transform
- Section 10.4: The Transform of Derivatives and Solution of Initial-Value Problems
- Section 10.5: The First Shifting Theorem
- Section 10.6: The Unit Step Function
- Section 10.7: The Second Shifting Theorem
- Section 10.8: Impulsive Driving Terms: The Dirac Delta Function
- Section 10.9: The Convolution Integral

- Chapter 11: Series Solutions to Linear Differential Equations
- Section 11.1: A Review of Power Series
- Section 11.2: Series Solutions about an Ordinary Point
- Section 11.3: The Legendre Equation
- Section 11.4: Series Solutions about a Regular Singular Point
- Section 11.5: Frobenius Theory
- Section 11.6: Bessel's Equation of Order
*p*