Monge equation

A differential equation of the form

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

G. Monge (see [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.

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 (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. 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

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