On linear partial equations of first-order

Abstract

We study a linear differential equation with first-order partial derivatives, where the coefficients
of the equation are given on an unbounded set and have continuous first-order partial derivatives.
Each partial differential equation is closely related to a system of ordinary differential equations,
a system of so-called characteristic equations of a given first-order partial differential equation.
Each first-order partial differential equation under certain conditions has a fundamental system of
integrals or an integral basis. We note that for a general linear partial differential equation of the
first order there can be no nontrivial integral. For a linear first-order partial differential equation,
where the coefficients of the equation are given on an unbounded set and have continuous firstorder
partial derivatives, with the first coefficient equal to one, an integral basis exists. For a
linear first-order partial differential equation, where the coefficients of the equation are given on
an unbounded set and have continuous first-order partial derivatives, with the first coefficient
equal to one, an integral basis exists. For a linear first-order partial differential equation, we define
the asymptotic stability of a linear first-order partial differential equation. A sufficient condition
for the asymptotic stability of a linear partial differential equation of the first order is given. At
present, the theory of partial differential equations finds its application in various fields of natural
science.