#### 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.

#### 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.

#### 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.

#### 5.4 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 , the Laplace transform of some function , we study techniques for recovering
the function .

#### 8.3 Solution of Initial Value Problems

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

#### 8.5 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.

#### 9.1 Introduction to Linear Higher Order Equations

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

#### 9.2 Higher Order Constant Coefficients Homogeneous Equations

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

#### 9.3 Undetermined Coefficients for Higher Order Equations

We discuss the solution of an 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 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 has real eigenvalues, but does not have 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.