# List of topics for Math 2090: "Elementary Differential Equations and Linear Algebra"

Text: Differential Equations and Linear Algebra, by Stephen W. Goode and Scott A. Annin, Fourth Edition, 2017.

Chapter 1: First-Order Differential Equations
1.2 Basic Ideas and terminology
1.4 Separable Differential Equations
1.6 First Order Linear Differential Equations
1.9 Exact Differential Equations

Chapter 2: Matrices and Systems of Linear Equations
2.1 Matrices: Definitions and Notations
2.2 Matrix Algebra
2.4 Elementary Row Operations and Row-Echelon Matrices
2.5 Gaussian Elimination
2.6 Inverse of a Square Matrix

Chapter 3: Determinants
3.1 The definition of the Determinant
3.2 Properties of the Determinant
3.3 Cofactor Expansions

Chapter 4: Vector Spaces
4.2 Definition of a Vector Space
4.3 Subspaces
4.4 Spanning Sets
4.5 Linear Dependence and Linear Independence
4.6 Bases and Dimension

Chapter 6: Linear Transformations
6.1 Definition of a Linear Transformation
6.3 The Kernel and Range of a Linear Transformation

Chapter 7: Eigenvalues and Eigenvectors
7.1 The Eigenvalue/Eigenvector Problem
7.2 General Results for Eigenvalues and Eigenvectors

Chapter 8: Linear Differential Equations of Order n
8.1 General Theory for Linear Differential Equations
8.2 Constant-Coefficient Homogeneous Linear Differential Equations
8.3 The Method of Undetermined Coefficients. Annihilators
8.7 The Variation of Parameters

Chapter 9: Systems of Differential Equations
9.1 First-Order Linear Systems
9.2 Vector Formulation
9.3 General Results for First-Order Linear Differential Systems
9.4 Vector Differential Equations: Non-defective Coefficient Matrix
9.6 Variation of Parameters for Linear Systems

Chapter 10: The Laplace Transform and Some Elementary Applications
10.1 Definition of Laplace Transform
10.2 Existence of the Laplace Transform and Inverse Transform
10.4 The Transform of Derivatives and Solution of Initial-Value Problems
10.5 The First Shifting Theorem
10.6 The Unit Step Function
10.7 The Second Shifting Theorem

Optional topics that could be taught at the discretion of the instructor include:
1.8 Change of Variables, Homogeneous Equations, Bernoulli's Equation
1.11 Some Higher-Order Differential Equations
7.3 Diagonalization
7.4 An introduction to the Matrix Exponential Function.
8.9 Reduction of Order

Determined by the 2011 Math 2090 Committee: Michael M. Tom, Pat Gilmer, Paul Britt, Charles Egedy,
Boris Rubin; updated by Moscatello on November 2017 for fourth edition.