Geodesic
A method of constructing integrable linear equations and its application to Hill's equation
Matematičeskie zametki, Tome 8 (1970) no. 6, pp. 783-786 
G. E. Popov. A method of constructing integrable linear equations and its application to Hill's equation. Matematičeskie zametki, Tome 8 (1970) no. 6, pp. 783-786. http://geodesic.mathdoc.fr/item/MZM_1970_8_6_a11/
@article{MZM_1970_8_6_a11,
     author = {G. E. Popov},
     title = {A method of constructing integrable linear equations and its application to {Hill's} equation},
     journal = {Matemati\v{c}eskie zametki},
     pages = {783--786},
     year = {1970},
     volume = {8},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_6_a11/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\Delta t$.