A form of the solution to the Mathieu equation

A linear transformation of a special type is introduced to express the general solution to a second-order differential equation with a periodic coefficient using 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 zones of the Mathieu equation. The obtained estimates for the instability zones are in agreement with known results.