Lyons, T. J.; Reuter, G. E. H. On exponential bounds for solutions of second order differential equations. (English) Zbl 0546.34029 Bull. Lond. Math. Soc. 17, 139-143 (1985). The paper considers the solutions to the equation \(\ddot u+f(t)u=0\), \(t\geq 0\), where f is bounded and nonnegative, say \(0\leq f(t)\leq 1\). The main theorem of this paper proves that \(u(t)\) is \(O(e^{kt})\) where the best possible \(k\) is determined (approximately 0.244). The bound is attained for a function \(f\) which is periodic and piecewise constant with the values 0 and 1. MSC: 34C11 Growth and boundedness of solutions to ordinary differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations Keywords:exponential bounds PDFBibTeX XMLCite \textit{T. J. Lyons} and \textit{G. E. H. Reuter}, Bull. Lond. Math. Soc. 17, 139--143 (1985; Zbl 0546.34029) Full Text: DOI