On invariant manifolds of linear differential equations. I

TL;DR

This paper extends the integral manifolds method to linear differential equations with variable coefficients, identifying conditions for invariant subspaces that decompose the solution space.

Contribution

It develops necessary and sufficient conditions for invariant linear subspaces in variable coefficient linear differential equations, expanding the integral manifolds method.

Findings

Identifies invariant subspaces for linear differential equations.

Provides necessary and sufficient conditions for invariance.

Decomposes solution space into invariant subspaces.

Abstract

This article is the first in the cycle from two parts. It develops the ideas of integral manifolds method of M. M. Bogolubov in the case of linear differential equations in $R^m$ with variable coefficients. We distinguish linear subspaces $M^n(t)$ and $M^{m-n}(t)$, which have dimensions $n$ and $m-n$ respectively, $m > n$, such that $R^m = M^n(t) \oplus M^{m-n}(t)$, and find necessary and sufficient conditions under which these subspaces are invariant with respect to differential equation under consideration.