View source for Monge equation

{{TEX|done}}
A differential equation of the form

$$F(x,y,z,dx,dy,dz)=0.$$

G. Monge (see [[#References|[1]]]) studied these equations in connection with the construction of a geometric theory of first-order partial differential equations. A particular case of a Monge equation is a [[Pfaffian equation|Pfaffian equation]].

For example, consider a first-order partial differential equation for an unknown function $z$ of two independent variables $x,y$:

$$\Phi\left(x,y,z,\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\right)=0,$$

then the directions of the generators of the [[Monge cone|Monge cone]] (characteristic directions) at each point have to satisfy the Monge equation, which can be written in the form

$$M\left(x,y,z,\frac{dy}{dz},\frac{dz}{dx}\right)=0.$$

This is an ordinary differential equation in two unknown functions, that is, it is the simplest case of an [[Underdetermined system|underdetermined system]]. Often an arbitrary underdetermined system of ordinary differential equations, in which the number of equations is less than the number of unknown functions, is called a Monge equation.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Monge,   "Application de l'analyse à la géométrie" , Bachelier  (1850)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.K. Rashevskii,   "Geometric theory of partial differential equations" , Moscow-Leningrad  (1947)  (In Russian)</TD></TR></table>