Roughly speaking, a partial differential equation (PDE) is similar to an ordinary differential equation (ODE), except that the dependent variable is a function of not just one, but of several independent variables. Let’s be more precise. Given a function u = u(x1, x2, . . . , xn), a partial differential equation (PDE) in u is an equation which relates any of the partial derivatives of u to each other and/or to any of the variables x1, x2, . . . , xn and u. Before doing some examples, we introduce a bit of notation: Instead of the

∂x2∂y and the like, we will use subscripts whenever possible. We write

ux = ∂u

∂x .