Implicit Solutions of PDE's

Abstract

Further investigations of implicit solutions to non-linear partial differential equations are pursued. Of particular interest are the equations which are Lorentz invariant. The question of which differential equations of second order for a single unknown $ϕ$ are solved by the imposition of an inhomogeneous quadratic relationship among the independent variables, whose coefficients are functions of $ϕ$ is discussed, and it is shown that if the discriminant of the quadratic vanishes, then an implicit solution of the so-called Universal Field Equation is obtained. The relation to the general solution is discussed.