Review 6: Differential Equations

The goal of this module is to give an introduction to differential equations.

In this module you will complete the following activities:

An introduction to differential equations.
How to check whether a function is a solution to a differential equation.
How to classify and identify the different types of differential equations.
Practice the covered concepts through some exercises, called "Now You Try".
Complete a quiz on this topic at the end of the module.

Differential Equations

A differential equation is an equation that relates a function with its derivative.

For example, imagine that you create an ant colony in a box, starting out with 100 ants. Over the next few days, the number of ants in your jar is going to increase as the ants reproduce. Moreover, the more ants there are, the faster you would expect the number of ants to grow.

Let's say the number of ants at time $t$ $t$ i s $A (t),$ $A(t),$ and that the rate at which new ants are born, $\frac{d A}{d t},$ $\frac{dA}{dt},$ is proportional to the number of ants that are already there. That is, we could model the population of ants in your box by the equation

where $k$ $k$ is a constant.

The equation above is an example of a differential equation, since it is an equation involving its derivative $\frac{d A}{d t} .$ $\frac{dA}{dt}.$

Differential equations are very widely used, in many fields. They show up in physics, biology, economics, engineering, medicine, etc. In this module we will talk about the solutions to differential equations, and how to identify and classify the different types of differential equations.