Summary
The number of nonoscillatory solutions of a forced second order linear differential equation is studied under the hypothesis that the homogeneous equation is oscillatory. The main technique involves expressing a general solution of the forced equation in terms of two parameters, given a pair of independent solutions of the homogeneous equation (see (2.4) below).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F. V. Atkinson,Asymptotic formulae for linear oscillations, Proceedings Glasgow Mathematical Association III (1957), pp. 105–111.
F. V. Atkinson,On second-order differential inequalities, Proc. Roy. Soc. Edinburgh Sect. A,72, no. 8 (1972–73), pp. 109–127.
J. M.Dolan,On the structure of initial value problems and their influence on the oscillatory behavior of the nonhomogeneous equation (ry′)′+qy=f, Parts I and II, to appear.
J. Graef -P. Spikes,Sufficient conditions for nonoscillation of a second-order nonlinear differential equation, Proc. Amer. Math. Soc.,50 (1975), pp. 289–292.
R. C. Grimmer -W. T. Patula,Nonoscillatory solutions of forced second-order linear equations, J. Math. Anal. Appl.,56 (2) (1976), pp. 452–459.
M. E. Hammett,Oscillation and nonoscillation theorems for nonhomogeneous linear differential equations of second order, Ph. D. thesis, Auburn University, Auburn, Alabama, 1967.
P. Hartman,Ordinary Differential Equations, Wiley, New York, 1964.
P. Hartman,Positive and monotone solutions of linear ordinary differential equations, J. Differential Equations,18 (1975), pp. 431–452.
G. D. Jones,Oscillation properties of y″+p(x)y=f(x), Rend. Mat.,57 (1974), pp. 337–341.
R. M. Kauffman,On the growth of solutions in the oscillatory case, Proc. Amer. Math. Soc.,51 (1975), pp. 49–54.
M. S. Keener,On the solution of certain nonhomogeneous second-order differential equations, Applicable Anal.,1 (1967), pp. 57–63.
Magnus -Winkler,Hill's Equation, Interscience Tracts in Pure and Applied Mathematics (20), Wiley, New York, 1966.
S. M. Rankin,Oscillation theorems for second order nonhomogeneous linear differential equations, J. Math. Anal. Appl.,53 (1976), pp. 550–553.
S. C. Tefteller,Oscillation of second order nonhomogeneous linear differential equation, SIAM J. Appl. Math.,31 (1976), pp. 461–467.
T. Wallgren,Oscillation of solutions of the differential equation y″+p(x)y-f(x), SIAM J. Math. Anal.,7 (1976), pp. 848–857.
A.Wintner,Analytical Foundations of Celestial Mechanics, Princeton, 1941.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Atkinson, F.V., Grimmer, R.C. & Patula, W.T. Nonoscillatory solutions of forced second order linear equations. - II. Annali di Matematica pura ed applicata 126, 299–317 (1980). https://doi.org/10.1007/BF01762513
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01762513