4.1 Basics of differential equations

Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function $y=f(x)$ and its derivative, known as a differential equation . Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.

Techniques for solving differential equations can take many different forms, including direct solution, use of graphs, or computer calculations. We introduce the main ideas in this chapter and describe them in a little more detail later in the course. In this section we study what differential equations are, how to verify their solutions, some methods that are used for solving them, and some examples of common and useful equations.

General differential equations

Consider the equation $y^{\prime }=3x^2,$ which is an example of a differential equation because it includes a derivative. There is a relationship between the variables $x$ and $y\text{:}y$ is an unknown function of $x.$ Furthermore, the left-hand side of the equation is the derivative of $y.$ Therefore we can interpret this equation as follows: Start with some function $y=f(x)$ and take its derivative. The answer must be equal to $3x^2.$ What function has a derivative that is equal to $3x^2?$ One such function is $y=x^3,$ so this function is considered a solution to a differential equation    .

Some examples of differential equations and their solutions appear in [link] .

Note that a solution to a differential equation is not necessarily unique, primarily because the derivative of a constant is zero. For example, $y=x^2+4$ is also a solution to the first differential equation in [link] . We will return to this idea a little bit later in this section. For now, let’s focus on what it means for a function to be a solution to a differential equation.