Forced oscillation and asymptotic behavior of solutions of linear differential equations of second order

Abstract

The paper deals with the second order nonhomogeneous linear differential equation $(p(t) y'(t))' + q(t) y(t) = f(t),$ which is oscillatory under the assumption that $p(t)$ and $q(t)$ are positive, continuously differentiable and monotone functions on $[0,\infty)$. Throughout this paper we shall use pairs of quadratic forms, which obtained by different methods than Kusano and Yoshida. This form will lead to a property of qualitative behavior, including amplitudes and slopes, of oscillatory solutions of the above equation. In addition, we will discuss the existence of three types (moderately bounded, small, large) of oscillatory solutions, which are based on results due to Kusano and Yoshida.