Elliptic partial differential equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form

where A, B, C, D, E, F, and G are functions of x and y and where u x = ∂ u ∂ x {\displaystyle u_{x}={\frac {\partial u}{\partial x}}}, u x y = ∂ 2 u ∂ x ∂ y {\displaystyle u_{xy}={\frac {\partial ^{2}u}{\partial x\partial y}}} and similarly for u x x, u y, u y y {\displaystyle u_{xx},u_{y},u_{yy}}. A PDE written in this form is elliptic if

with this naming convention inspired by the equation for a planar ellipse. Equations with B 2 − A C = 0 {\displaystyle B^{2}-AC=0} are termed parabolic while those with B 2 − A C > 0 {\displaystyle B^{2}-AC>0} are hyperbolic.

The simplest examples of elliptic PDEs are the Laplace equation, Δ u = u x x + u y y = 0 {\displaystyle \Delta u=u_{xx}+u_{yy}=0}, and the Poisson equation, Δ u = u x x + u y y = f (x, y). {\displaystyle \Delta u=u_{xx}+u_{yy}=f(x,y).} In a sense, any other elliptic PDE in two variables can be considered to be a generalization of one of these equations, as it can always be put into the canonical form

through a change of variables.
startend:quotation