Global theory of ordinary linear homogeneous differential equations in the real domain. I
In this long paper the author carries out an in-depth investigation and presents a survey of the global properties of linear homogeneous differential equations of the nth order, \(n\geq 2\), with real continuous coefficients defined on open intervals of reals. The set of these equations will be denoted by \({\mathbb{A}}_ n\). Each equation of the set \({\mathbb{A}}_ n\) can be written in the form \[ P_ n(x,y,I)\quad \equiv \quad y^{(n)}+P_{n-1}(x)y^{(n-1)}+...+P_ 0(x)y=0, \] where \(P_ i\in C^ 0(I)\) for \(i=0,1,...,n-1\), the interval \(I\subset {\mathbb{R}}\) is open and \(n\geq 2\) is an integer. The author compares his results with previous local investigations and succeeds in obtaining new results describing global properties of linear homogeneous differential equations of an arbitrary order n, \(n\geq 2\), e.g., he extends Boriuokla's criterion for the global equivalence of the second order linear differential equations into a new criterion of global equivalence of equations of an arbitrary order n from \({\mathbb{A}}_ n\), \(n\geq 3\). In the last section of the paper the author introduces stationary groups, i.e. the groups formed by all global transformations that transform a given ordinary linear differential equation into itself which describe completely the structure of all global transformations in each class of globally equivalent equations.
linear homogeneous differential equations
second order linear differential equations
scientific article; zbMATH DE number 4048102
scientific article