Abstract
The general non-linear first-order partial differential equation requires a deeper analysis than its linear and quasi-linear counterparts. Instead of a field of characteristic directions, the non-linear equation delivers a one-parameter family of directions at each point of the space of dependent and independent variables. These directions subtend a local cone-like surface known as the Monge cone. Augmenting the underlying space by adding coordinates representing the first partial derivatives of the unknown field, however, it is possible to recover most of the features of the quasi-linear case so that, ultimately, even the solution of the general non-linear equation can be reduced to the integration of a system of ordinary differential equations.
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