Electron. J. Diff. Equ., Vol. 2010(2010), No. 174, pp. 1-10.

Necessary and sufficient conditions for the oscillation a third-order differential equation

Pitambar Das, Jitendra Kumar Pati

Abstract:
We show that under certain restrictions the following three conditions are equivalent: The equation
$$ y'''+a(t)y''+b(t)y'+c(t)y=f(t) $$
is oscillatory. The equation
$$ x'''+a(t)x''+b(t)x'+c(t)x=0 $$
is oscillatory. The second-order Riccati equation
$$ z''+3zz'+a(t)z'=z^3+a(t)z^2+b(t)z+c(t) $$
does not admit a non-oscillatory solution that is eventually positive.
Furthermore, we obtain sufficient conditions for the above statements to hold, in terms of the coefficients. These conditions are sharp in the sense that they are both necessary and sufficient when the coefficients $a(t), b(t), c(t)$ are constant.

Submitted April 13, 2010. Published December 6, 2010.
Math Subject Classifications: 34C10, 34C15.
Key Words: Oscillation; non-oscillation; third order differential equations.

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Pitambar Das
Department of Mathematics, Indira Gandhi Institute of Technology
Sarang-759146, Talcher, Orissa, India
email: pdasigit@in.com
Jitendra Kumar Pati
Department of Mathematics, Indira Gandhi Institute of Technology
Sarang-759146, Talcher, Orissa, India
email: jkpati2007@yahoo.com

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