An analytical solution of Monge differential equation

TL;DR

This paper derives an exact implicit analytical solution to the nonlinear Monge differential equation, clarifying its relation to classical free motion equations and expanding understanding of its integrability.

Contribution

It provides the first explicit analytical implicit solution to the Monge differential equation, linking it to classical mechanics and quadrature methods.

Findings

Exact implicit solution derived

Connection established with free motion equations

Discussion of asymptotic solutions included

Abstract

We present the exact solution to the non linear Monge differential equation lambda(x, t)lambdax(x, t) = lambdat(x, t). It is widely accepted that the Monge equation is equivalent to the ODE d2X/dt2= 0 of free motion for particular conditions. Furthermore, the Monge Type equations are connected with X = F(dX/dt, X; t), which can be integrated with quadratures [1]. Other asymptotic solutions are discussed, see e.g. [2]. The solution was reached with calculations that depend upon dimensional representation, which is given by (x, t) coordinates. We present this analytical solution to the Monge differential equation as an implicit solution.