On a linear system of differential equations

Abstract

The linear systems of partial differential equations of the first order with the identical main parts
is considered. Аpplying the well-known relation between a normal system of ordinary differential
equations and a linear system of partial differential equations of the first order with the same
main parts, the existence of integral basis of a linear inhomogeneous system of partial differential
equations of the first order adjoining to some solution of the same linear inhomogeneous system
of differential equations with partial derivatives of the first order is proved. A sign at which the
nonlinear system of ordinary differential equations has a neighborhood such that any solution with
initial values from it tends to zero is found. Using the equivalence of a linear system of partial
differential equations of the first order with identical main parts to a linear differential equation
with partial derivatives of the first order, the existence of integral basis of the adjoining to zero
linear homogeneous system of partial differential equations of first order with nonlinear coefficients
is shown.