Hill and Matthew's Equations and Some Properties of Their Particular Solutions

In this work we have offered several theorems on oscillating and particular solutions of Hill’s equation which we have not found in the well known monographs [1] and [2]. Namely, we have shown conditions under which Hill’s equation is equation of oscillations. We have also shown that if the coefficient of the equation is periodic function with the period 2 , then Hill’s equation also has periodic particular solution with the period 2 and one equation cannot have two fundamental particular solutions with the same period. By using the first Liouville’s formula we have shown that the poles of the first order of the coefficient (x) are simultaneously zeros of the solution of the equation, which is used for solving many Hill’s equations. Namely, Hiss’s equation is most frequently reduced by substitution   Zdx y e to Riccati’s equation, so it is very important to determine the form of its particular solution.