An oscillation criterion for a forced second-order linear differential equation. (English) Zbl 0777.34023

El-Sayed, M. A.

Proc. Am. Math. Soc. 118, No. 3, 813-817 (1993).

The paper deals with oscillation properties of the second order equation \((*)\) \((p(t)y')'+q(t)y=g(t)\), where \(p>0\), \(g\) is an oscillatory function and \(q\) is of arbitrary sign. The used method consists in comparing \((*)\) with a certain oscillatory homogeneous second order equation which is constructed by means of the sequence of zero points of the forcing term \(g\). The main result, when applied to the homogeneous equation, specifies the values of the constants \(a,b\) for which the Mathieu’s equation \(y''+(a+b\cos 2t)y=0\) oscillates. The idea used in the paper is in certain sense similar to the so-called telescopic principle introduced by M. K. Kwong [Ordinary differential equations and operators, Proc. Symp., Dundee 1982. Lect. Notes Math. 1032, 311-327 (1983; Zbl 0534.34017)] but comparison of these two methods is given.

Reviewer: O.Došlý (Brno)

MSC:

34C10

Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

oscillation properties; second order equation; oscillatory homogeneous second order equation; Mathieu’s equation; telescopic principle

Citations:

Zbl 0534.34017