Review of methods for solving linear equations with partial derivatives of the first order

Abstract

The review presents the main methods for solving linear inhomogeneous partial differential equations of the first order with two independent variables. The following approaches are considered: transition to new variables through a known integral of the corresponding homogeneous equation, the characteristic system method, and the use of an auxiliary homogeneous equation with three independent variables. Then, the classical and generalized Cauchy problems are formulated and solved. The final examples illustrate the application of these methods to specific equations.