Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 174, pp. 1-10.
Title: Necessary and sufficient conditions for the oscillation
a third-order differential equation
Authors: Pitambar Das (Indira Gandhi Inst. of Tech., Talcher, India)
Jitendra Kumar Pati (Indira Gandhi Inst. of Tech., Talcher, India)
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 06, 2010.
Math Subject Classifications: 34C10, 34C15.
Key Words: Oscillation; non-oscillation; third order differential equations.