On a system of second order differential equations with periodic impulse coefficients

A thorough investigation of the system $$\frac{{d^2 y(x)}}{{dx^2 }} + p(x)y(x) = 0$$ with periodic impulse coefficients $$\begin{gathered} p(x) = \left\{ {\begin{array}{*{20}c} {1, 0 \leqslant x< x_0 (2\pi > x_0 > 0)} \\ { - \eta , x_0 \leqslant x< 2\pi (\eta > 0)} \\ \end{array} } \right. \hfill \\ p(x) = p(x + 2\pi ), ---\infty< x< \infty \hfill \\ \end{gathered} $$ is given, and the method can be applied to one with other periodic impulse coefficients.