Let z be a function of variables x and y let z_x, z_y, z_xx, z_xy, z_yy, z_xxx,… denote the partial derivatives and higher-order partial derivatives of z. A partial differential equation (PDE for short) is an equation involving x, y, z and its partial derivatives (compare ordinary differential equation, stochastic differential equation). The order of the PDE is the highest total order of any partial derivative that appears. These notions readily generalize to functions of more than two variables.

As the differential operators ∂/∂x and ∂/∂y respectively send functions of y only and x only to 0, a PDE of order n might be expected to have n arbitrary functions, rather than n arbitrary constants, in its general solution. As with ODEs, this is at best a rule of thumb and does not always hold. For example, the first order PDE z_x +  z_y = 0 has general solution z =  f(x–y), and the second order PDE z_xy =  0 has general solution z =  f(x)  +  g(y).

See heat equation, Laplace’s equation, parabolic differential equation, wave equation.