This textbook introduces a new type of equation that incorporates not only an unknown function but also how that function changes over time. These equations are called differential equations. In this section, we will provide a clear definition and review the various ways of representing derivatives.
A differential equation (DE) is an equation that involves one or more derivatives of an unknown function. If the function depends on a single variable, it’s called an ordinary differential equation (ODE). If the function depends on more than one variable, it’s called a partial differential equation (PDE).
So, for an equation to qualify as a differential equation, it must include two things: First, it requires a derivative, such as \(f'\) or \(\frac{dy}{dx}\text{.}\) Second, it must have an equal sign.
On the other hand, these are not differential equations, either because they do not contain a derivative or because they do not include an equals sign:
In this text, we will use several common notations for derivatives. These include prime notation, such as \(y'\) or \(y''\text{;}\) Leibniz notation, written as \(\frac{dy}{dx}\text{;}\) and dot notation, typically reserved for derivatives with respect to time.
