A method of constructing integrable linear equations and its application to Hill's equation

Abstract

Starting with a given equation of the form

$$\ddot x + [\lambda + \varepsilon f(t)] x = 0$$

, where λ > 0 and ɛ ≪ l is a small parameter [heref(t) may be periodic, and so Hill's equation is included], we construct an equation of the form y + [λ + ɛf (t) + ɛ² _g (t)]y = 0, integrable by quadratures, close in a certain sense to the original equation. For x₀ = y₀ and x ^′₀ = y ^′₀ , an upper bound is obtained for ¦y—x¦ on an interval of length Δt.