Special trigonometric series in the problem of periodic solutions

This paper describes a method for constructing periodic solutions for special-type nonlinear equations with periodic coefficients. The basis of this method is to represent the desired solution in a nonstandard trigonometric series as a power series in $\sin{t}$. The coefficients of such a series are calculated in a recursive way. Such a representation is permissible not only for continuous periodic solutions, but also for solutions with singularities. In addition, the representation of a singular solution in the form of a non-standard trigonometric series allows localizing its singularities. The equations in question may also have singularities. When finding singular solutions, we use the assumption that in the case of the existence of two such solutions, they are connected by a certain equality. Using this relationship, you can get the equation for these periodic solutions, either as a linear equation of the second order or as an equation of the first order. This allows, for example, to find the boundary curves for the stability zones of the Hill equation with a parameter. The results obtained on the existence of singular periodic solutions supplement the general theorems of [7] obtained by other methods.