What is a differential equation?

A differential equation is fundamentally defined as an equation that involves derivatives of a function. This signifies that it expresses a relationship that includes the rates of change of a function with respect to one or more variables. Derivatives represent how a quantity changes, and by incorporating them into equations, differential equations model various dynamic systems across physics, engineering, biology, and other fields.

The importance of these equations lies in their ability to describe complex phenomena, including motion, growth, decay, and other processes that change over time or space. These relationships can be ordinary differential equations (ODEs) when they involve functions of a single variable and their derivatives, or partial differential equations (PDEs) when they involve functions of multiple variables.

The other choices do not correctly define a differential equation. For instance, stating that it involves only constants overlooks the core aspect of derivatives that makes it a differential equation. Similarly, defining it as a relationship between angles does not encompass the broader applicability of differential equations. Moreover, suggesting that it can only be solved graphically misrepresents the analytical and numerical methods available for solving these equations. Thus, the essence of a differential equation lies in its definition through derivatives of functions, which is captured by the correct choice.