(untitled) (Journal of Mathematics Mechanics and Computer Science, 2018)
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.