A Form of the Solution to the Mathieu Equation / *V. M. Budanov.* // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2017. № 1. P. 42-47
[Moscow Univ. Mech. Bulletin. Vol. 72, N 2, 2017. P. 7-12].

A special linear transformation is introduced to express the general solution to a second-order differential equation with a periodic coefficient in terms of a particular solution to an auxiliary second-order nonlinear system with a periodically perturbed right-hand side. It is numerically shown that there exist periodic solutions to the auxiliary system outside the instability regions of the solutions to the Mathieu equation. The estimates obtained for the instability regions are in agreement with known results.

*Key words*:
system with periodic coefficients, nonlinear system, Mathieu equation.