# List of topics for Math 2065: "Elementary Differential Equations"

Text: Ordinary Differential Equations, by William A. Adkins and Mark G. Davidson, Springer, 2012.

Chapter 1: First Order Differential Equations
1.1 An introduction to Differential Equations
1.2 Direction Fields
1.3 Separable Differential Equations
1.4 Linear First Order Equations
1.5 Substitutions; Homogeneous and Bernoulli Equations

Chapter 2: The Laplace Transform
2.1 Laplace Transform Method: Introduction
2.2 Definitions, Basic Formulas, and Principles
2.3 Partial Fractions: A Recursive Method for Linear Terms
2.4 Partial Fractions: A Recursive Method for Irreducible Quadratics
2.5 Laplace Inversion
2.6 The Linear Spaces: Special Cases
2.7 The Linear Spaces: The General Case
2.8 Convolution

Chapter 3: Second Order Constant Coefficient Linear Differential Equations
3.1 Notation, Definitions, and some Basic Results
3.2 Linear Independence
3.3 Linear Homogeneous Differential Equations
3.4 The Method of Undetermined Coefficients
3.6 Spring Systems or 3.7 RCL Circuits

Chapter 4: Linear Constant Coefficient Differential Equations
4.1 Notation, Definitions, and Basic Results
4.2 Linear Homogeneous Differential Equations
4.3 Nonhomogeneous Differential Equations

Chapter 5: Second Order Linear Differential Equations
5.1 The Existence and Uniqueness Theorem
5.2 The Homogeneous Case
5.3 The Cauchy-Euler Equations
5.5 Reduction of Order
5.6 Variation of Parameters

Chapter 8: Matrices (no more than two lectures should be spent here)
8.1 Matrix Operations
8.2 Systems of Linear Equations
8.3 Invertible Matrices
8.4 Determinants
8.5 Eigenvectors and Eigenvalues

Chapter 9: Linear Systems of Differential Equations
9.1 Introduction
9.2 Linear Systems of Differential Equations
9.3 The Matrix Exponential Function and its Laplace Transform
9.4 Fulmer's Method
9.5 Constant Coefficient Linear Systems

Optional topics that could be taught at the discretion of the instructor
3.5 The Incomplete Partial Fraction Method
6.3-6.5 Laplace Transform Method for a Heaviside Function, The Dirac Delta Function, etc.

The above topics were chosen by the 2011 Math 2065 Committee: Michael M. Tom, Mark Davidson, Charles Egedy, Jerome W Hoffman, Terrie White