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Second-order linear partial differential equations 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.
The simplest nontrivial examples of elliptic PDE's 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 . {\displaystyle \Delta u=u_{xx}+u_{yy}=f.} 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