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 ${\displaystyle u_{x}={\frac {\partial u}{\partial x}}}$ and similarly for ${\displaystyle u_{xx},u_{y},u_{yy},u_{xy}}$. A PDE written in this form is elliptic if

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