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

Starting with a given equation of the form
$$\ddot{x}+[\lambda+\varepsilon f(t)]x=0,$$
where $\lambda>0$ and $\varepsilon\ll1$ is a small parameter [here $f(t)$ may be periodic, and so Hill's equation is included], we construct an equation of the form $\ddot{y}+[\lambda+\varepsilon f(t)+\varepsilon^2g(t)]y=0$, integrable by quadratures, close in a certain sense to the original equation. For $x_0=y_0$ and $x_0'=y_0'$, an upper bound is obtained for $|y-x|$ on an interval of length $\Delta t$.