Chapter 7
Differential Equations
7.1 An Introduction to Differential Equations
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a differential equation and what kinds of information can it tell us?
• How do differential equations arise in the world around us?
• What do we mean by a solution to a differential equation?
Introduction
In previous chapters, we have seen that a function’s derivative tells us the rate at which
the function is changing. More recently, the Fundamental Theorem of Calculus helped us
to determine the total change of a function over an interval when we know the function’s
rate of change. For instance, an object’s velocity tells us the rate of change of that object’s
position. By integrating the velocity over a time interval, we may determine by how much
the position changes over that time interval. In particular, if we know where the object is
at the beginning of that interval, then we have enough information to accurately predict
where it will be at the end of the interval.
In this chapter, we will introduce the concept of differential equations and explore this
idea in more depth. Simply said, a differential equation is an equation that provides a
description of a function’s derivative, which means that it tells us the function’s rate of
change. Using this information, we would like to learn as much as possible about the
function itself. For instance, we would ideally like to have an algebraic description of the
387
Differential Equations
7.1 An Introduction to Differential Equations
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a differential equation and what kinds of information can it tell us?
• How do differential equations arise in the world around us?
• What do we mean by a solution to a differential equation?
Introduction
In previous chapters, we have seen that a function’s derivative tells us the rate at which
the function is changing. More recently, the Fundamental Theorem of Calculus helped us
to determine the total change of a function over an interval when we know the function’s
rate of change. For instance, an object’s velocity tells us the rate of change of that object’s
position. By integrating the velocity over a time interval, we may determine by how much
the position changes over that time interval. In particular, if we know where the object is
at the beginning of that interval, then we have enough information to accurately predict
where it will be at the end of the interval.
In this chapter, we will introduce the concept of differential equations and explore this
idea in more depth. Simply said, a differential equation is an equation that provides a
description of a function’s derivative, which means that it tells us the function’s rate of
change. Using this information, we would like to learn as much as possible about the
function itself. For instance, we would ideally like to have an algebraic description of the
387