Differential equations (DEs) are equations in which the unknown which one is solving for is a function, and the equation that is satisfies contains expressions involving the derivatives of the function, as well as the functions itself.

If the function is just a function of one variable (so that only derivatives in that one variable are involved), then we speak of an ordinary differential equation (or ODE). If the function is a function of several variables, so that partial derivatives are involved, we speak of a partial differential equation (or PDE).

In addition to being classified as ODEs or PDEs, differential equations are also classified by the highest derivative of the unknown function that appears (the order), whether or not they are linear in the unknown (linear vs. non-linear equations), and, in the case of PDEs, by more subtle considerations of the way that the various partial derivatives appear and are related to one another in the equation (so that one speaks of, for example, elliptic, hyperbolic, parabolic, and dispersive equations).