“A Study of Some Methods for Solving First-Order Linear and Nonlinear Partial Differential Equations”

Abstract

This research addresses the most important analytical methods for solving first-order partial differential equations. Lagrange’s method is based on transforming a linear partial differential equation into a system of ordinary differential equations through characteristic relations, which facilitates the determination of the general solution.The Cauchy problem, on the other hand, is concerned with finding the solution of a partial differential equation subject to given initial conditions on a curve or a surface. It is widely used in physical applications to establish the existence and uniqueness of solutions.In contrast, Charpit’s method is considered an extension of Lagrange’s method, specifically applied to solving first-order nonlinear partial differential equations. This method transforms the equation into a differential system that assists in obtaining the necessary integrals required to derive the general solution.These methods demonstrate significant importance in simplifying complex equations and transforming them into forms that can be handled mathematically with greater efficiency.