in case of finding the center of the ellipse or hyperbola for which axis or non parallel to axis we apply partial differential
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Poisson's equation is a partial differential equation of elliptic type. it is used in electrostatics, mechanical engineering and theoretical physics.
Maxwell's equations contain two scalar equations and two vector equations. Gauss' law and Gauss' law for magnetism are the scalar equations. The Maxwell-Faraday equation and Ampere's circuital law are the vector equations.
There is no quadratic equation that is 'linear'. There are linear equations and quadratic equations. Linear equations are equations in which the degree of the variable is 1, and quadratic equations are those equations in which the degree of the variable is 2.
An "inconsistent" set of equations. If they are all linear equations then the matrix of coefficients is singular.