Rectifiability of Solutions for a Class of Two-Dimensional Linear Differential Systems

The two-dimensional linear differential system x′=y,y′=-x-h(t)y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x' = y, \quad y' = -x-h(t)y \end{aligned}$$\end{document}is considered, where h∈C1[t0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h \in C^1[t_0,\infty )$$\end{document}. This system is equivalent to the damped linear oscillator x′′+h(t)x′+x=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x'' + h(t) x' + x = 0. \end{aligned}$$\end{document}Necessary and sufficient conditions are established for the length of every nontrivial solution to be finite and infinity.