Classes of Hill's equations involving conics, with all bounded solutions or some unbounded solutions

In this note we prove that for any function θ ∈ C 2 ([0, ∞)) bounded from below by a constant θ 0 ∈ R, and for any ∆ 0 > 0, any solution y to the Hill equation is bounded by a constant depending explicitly on the parameters y 0 , z 0 , θ 0 , ∆ 0 . On the contrary, if θ is not bounded from below in [0, ∞) with ∆ 0 < 0, we prove that the solution y is unbounded in [0, ∞) when the parameters y 0 , z 0 , θ(0), θ ′ (0), ∆ 0 satisfy some suitable inequality. Actually, we obtain more general classes of Hill's equations with all bounded solutions or some unbounded solutions. To this end, we show that the pair (y, y ′ ) lies in a conic section -an ellipse when ∆ 0 > 0, and a hyperbola when ∆ 0 < 0 -whose coefficients are solutions to a three-dimensional linear differential system but lie in a constant hyperboloid of R 3 .