Stability analysis of Hill systems

The author shows that the (α, β)-plane stability boundaries of the Hill equation, x¨+[α+βf(t)]x=0, f(t)=f(t+T), (which governs the behavior of many physical and engineering systems) can be obtained in a simple fashion by first evaluating a series of constants that depend only on f(t). Definite integral expressions of order k, k=1,2,..., are given that allow one to evaluate the kth constants explicitly for a given f(t). For square, triangular and cisoidal f(t)´s, these constants have been evaluated upto k=6, and the corresponding stability boundaries drawn