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#### 1.2 Basic Concepts

We define ordinary differential equations and what it means for a function to be a solution to such an equation.

#### 1.1 Applications Leading to Differential Equations

We discuss population growth, Newton’s law of cooling, glucose absorption, and spread of epidemics as phenomena that can be modeled with differential equations.

#### 2.1 Linear First-Order Differential Equations

We develop a technique for solving first-order linear differential equations.

#### 1.3 Direction Fields for First Order Equations

We explore direction fields (also called slope fields) for some examples of first order differential equations.

#### 2.2 Separable Equations

We define what it means for a first order equation to be separable, and we work out solutions to a few examples of separable equations.

#### 2.3 Existence and Uniqueness of Solutions of Nonlinear Equations

We study an existence and uniqueness theorem for a first-order initial value problem. We do not attempt the proof, as it is beyond the scope of this book.

#### 2.5 Exact Equations

We learn how to recognize whether or not a first-order equation is exact. We also learn how to solve an exact equation.

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#### 2.4A Bernoulli’s Equations

We show how multiplying an equation by an integrating factor can make the equation exact, and we give examples where this is a nice technique for solving a first-order equation.

#### 2.6 Integrating Factors

We show how multiplying an equation by an integrating factor can make the equation exact, and we give examples where this is a nice technique for solving a first-order equation.

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#### 3.2 The Improved Euler Method and Related Methods

We explore some ways to improve upon Euler’s method for approximating the solution of a differential equation.

#### 3.3 Runge-Kutta Method

We study a fourth order method known as Runge-Kutta which is more accurate than any of the other methods studied in this chapter.

#### 4.1 Exponential Growth and Decay

We solve a separable differential equation and describe a few of its many applications.

#### 4.2A Newton’s Law of Cooling

We study Newton’s Law of Cooling as an application of a first order separable differential equation.

#### 4.3 Elementary Mechanics

We study several applications of first order differential equations to elementary mechanics.

#### 4.4 Autonomous Second Order Equations

We define autonomous equations, explain how autonomous second order equations can be reduced to first order equations, and give several applications.

#### 4.5 Applications to Curves

We study a number of ways that families of curves can be defined using differential equations.

#### 5.1 Homogeneous Linear Equations

We develop a technique for solving homogeneous linear differential equations.

#### 5.2 Constant Coefficient Homogeneous Equations

We examine the various possibilities for types of solutions when solving constant coefficient homogeneous equations.

#### 5.3 Nonhomogeneous Linear Equations

We discuss theory related to nonhomogeneous linear equations.

#### 5.4 The Method of Undetermined Coefficients I

We explore the solution of nonhomogeneous linear equations in the case where the forcing function is the product of an exponential function and a polynomial.

#### 5.5 The Method of Undetermined Coefficients II

We explore the solution of nonhomogeneous linear equations with other forcing functions.

#### 5.6 Reduction of Order

We explore a technique for reducing a second order nonhomgeneous linear differential equation to first order when we know a nontrivial solution to the complementary homogeneous equation.

#### 5.7 Variation of Parameters

We study the method of variation of parameters for finding a particular solution to a nonhomogeneous second order linear differential equation.

#### 6.1 Spring Problems I

We study undamped harmonic motion as an application of second order linear differential equations.

#### 6.2 Spring Problems II

We return to our study of harmonic motion as an application of second order linear differential equations, this time considering the cases where damping occurs.

#### 6.3 The RLC Circuit

We study electric circuits as an application of second order linear differential equations.

#### 6.4 Motion Under A Central Force

We study the motion of a object moving under the influence of a central force.

#### 7.1 Review of Power Series

We review the basic properties of power series representation of functions.

#### 7.2 Series Solutions Near an Ordinary Point I

We consider the utilization of power series to determine solutions to certain differential equations.

#### 7.3 Series Solutions Near an Ordinary Point II

We consider the utilization of power series to determine solutions to more general differential equations.

#### 7.4 Regular Singular Points: Euler Equations

We consider the utilization of power series to determine solutions to differential equations near a singular point. We also study Euler equations.

#### 7.1 The Method of Frobenius I

We begin our study of the method of Frobenius for finding series solutions of linear second order differential equations.

#### 7.6 The Method of Frobenius II

We continue our study of the method of Frobenius for finding series solutions of linear second order differential equations, extending to the case where the indicial equation has a repeated real root.

#### 7.7 The Method of Frobenius III

We conclude our study of the method of Frobenius for finding series solutions of linear second order differential equations, considering the case where the indicial equation has distinct real roots that differ by an integer.

#### 8.1 Introduction to the Laplace Transform

We begin our study of Laplace transforms with the definition, and we derive the Laplace Transform of some basic functions.

#### 8.2 The Inverse Laplace Transform

Given $F(s)$, the Laplace transform of some function $f(t)$, we study techniques for recovering the function $f(t)$.

#### 8.3 Solution of Initial Value Problems

We demonstrate how Laplace transforms can be used to solve constant coefficient second order initial value problems.

#### 8.4 The Unit Step Function

We introduce the unit step function and some of its applications.

#### 8.5 Constant Coefficient Equations with Piecewise Continuous Forcing Functions

We show how Laplace Transforms may be used to solve initial value problems with piecewise continuous forcing functions.

#### 8.6 Convolution

We define the convolution of two functions, and discuss its application to computing the inverse Laplace transform of a product.

#### 8.7 Constant Coefficient Equations with Impulses

We study the solution of initial value problems where the external force is an impulse.

#### 8.8 A Brief Table of Laplace Transforms

This is a short table of Laplace Transforms.

#### 9.1 Introduction to Linear Higher Order Equations

Given an $n$th order linear differential equation, we discuss necessary and sufficient conditions for a set of $n$ functions to be a fundamental set of solutions.

#### 9.2 Higher Order Constant Coefficients Homogeneous Equations

We discuss the solution of an $n$th order homogeneous linear differential equation.

#### 9.3 Undetermined Coefficients for Higher Order Equations

We discuss the solution of an $n$th order nonhomogeneous linear differential equation, making use of the method of undetermined coefficients to find a particular solution.

#### 9.4 Variation of Parameters for Higher Order Equations

We discuss the solution of an $n$th order nonhomogeneous linear differential equation, making use of variation of parameters to find a particular solution.

#### 10.2 Linear Systems of Differential Equations

We show how linear systems can be written in matrix form, and we make many comparisons to topics we have studied earlier.

#### 10.3 Basic Theory of Homogeneous Linear System

We study the theory of homogeneous linear systems, noting the parallels with the study of linear homogeneous scalar equations.

#### 10.5 Constant Coefficient Homogeneous Systems II

We continue our study of constant coefficient homogeneous systems. In this section we consider the case where $A$ has $n$ real eigenvalues, but does not have $n$ linearly independent eigenvectors.

#### 10.7 Variation of Parameters for Nonhomogeneous Linear Systems

We study constant coefficient nonhomogeneous systems, making use of variation of parameters to find a particular solution.

#### Modeling a spring-mass system

This lab describes an activity with a spring-mass system, designed to explore concepts related to modeling a real world system with wide applicability.

#### Hot Potato! Activity on heating and cooling

An experiment involving Newton’s Law of Cooling.

#### Solving ODEs with Sage

This activity shows how to use Sage to solve differential equations.

#### Experiment with Sage interacts

This activity demonstrates the use of a Sage interact.

#### An Epidemics Model

Activity with a model for an epidemic. Based on: (*** add source ***)

#### Hot Potato! Cooling Activity

An experiment involving Newton’s Law of Cooling.

#### The Simple Pendulum

An experiment involving a simple pendulum.

#### Tank Draining

An experiment involving a draining tank.