Differential equations of Lagrange and Clairaut in the theory of differential equations and their practical applications

Abstract

This article analyzes Clairaut and Lagrange type equations, which are important classes of first-order nonlinear differential equations. The methods for finding the general and singular solutions of these equations are explained from a theoretical perspective. In addition, Problems 287 and 289 taken from Filippov's collection of problems are worked out in detail, and the stages of obtaining the solution using the parametric method are demonstrated. The study also reveals the role of Clairaut and Lagrange equations in the theory of differential equations and their practical significance.