Geodesic
Necessary and sufficient conditions for the oscillation a third-order differential equation
Electronic journal of differential equations, Tome 2010 (2010)
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.
Classification : 34C10, 34C15
Keywords: oscillation, non-oscillation, third order differential equations 
@article{EJDE_2010__2010__a286,
     author = {Das,  Pitambar and Pati,  Jitendra Kumar},
     title = {Necessary and sufficient conditions for the oscillation a third-order differential equation},
     journal = {Electronic journal of differential equations},
     year = {2010},
     volume = {2010},
     zbl = {1208.34039},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a286/}
}
TY  - JOUR
AU  - Das,  Pitambar
AU  - Pati,  Jitendra Kumar
TI  - Necessary and sufficient conditions for the oscillation a third-order differential equation
JO  - Electronic journal of differential equations
PY  - 2010
VL  - 2010
UR  - http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a286/
LA  - en
ID  - EJDE_2010__2010__a286
ER  - 
%0 Journal Article
%A Das,  Pitambar
%A Pati,  Jitendra Kumar
%T Necessary and sufficient conditions for the oscillation a third-order differential equation
%J Electronic journal of differential equations
%D 2010
%V 2010
%U http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a286/
%G en
%F EJDE_2010__2010__a286
Das,  Pitambar; Pati,  Jitendra Kumar. Necessary and sufficient conditions for the oscillation a third-order differential equation. Electronic journal of differential equations, Tome 2010 (2010). http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a286/