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On Liapunov stability for \(\ddot x+xf(x)=0\), \(\ddot y+yw(x)=0\). (English) Zbl 0679.34055

Summary: “We consider the problem of the stability of the origin for \[ \ddot x+xf(x)=0,\quad \ddot y+yw(x)=0,\quad (x,y)\in R^ 2,\quad f(0)>0. \] The aim of this paper is to
(a) establish some wide collections of problems related to coexistence of solutions of certain families of Hill’s equations (for example one of them is obtained with \(w=f)\); and
(b) determine all the cases which admit a first integral V quadratic in the velocities, and solve the relative problems in stability of the equilibrium whenever V is a Lyapunov function.”
Reviewer: A.Kanevskij

MSC:

34D20 Stability of solutions to ordinary differential equations
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References:

[1] Magnus , W. and Winkler , S. 1966.Hill’s Equation, 1–119. New York: Interscience. · Zbl 0158.09604
[2] Zampieri G., Proceedings of the Equadiff 87
[3] DOI: 10.1016/0022-0396(88)90005-8 · Zbl 0668.34051 · doi:10.1016/0022-0396(88)90005-8
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