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

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.