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

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

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

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

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.

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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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.

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

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

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

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

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

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

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

We develop a technique for solving homogeneous linear differential equations.

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

We discuss theory related to nonhomogeneous linear equations.

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.

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

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.

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

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

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.

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

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

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

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

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

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

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

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.

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.

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

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

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

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

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

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

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

This is a short table of Laplace Transforms.

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.

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

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.

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

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

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

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.

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

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

An experiment involving Newton’s Law of Cooling.

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

This activity demonstrates the use of a Sage interact.

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

An experiment involving Newton’s Law of Cooling.

An experiment involving a simple pendulum.

An experiment involving a draining tank.