Elliptic partial differential equation

Patent Pending An elliptic partial differential equation is a general partial differential equation of second order of the form Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0\, that satisfies the condition B^2 - AC < 0.\ (Assuming implicitly that u_{xy}=u_{yx}. ) Just as one classifies conic sections and quadratic forms based on the discriminant B^2 - 4AC, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by B^2 - AC, due to the convention (discussion and explanation here).