Answer
The solutions of the given equation are sure to exist on $\mathbb{R}-\{2,-2\}$.
Work Step by Step
The given higher order linear equation is written in the form :
$\frac{d^ny}{dt^n}\;+\;p_{1}(t)\frac{d^{n-1}y}{dt^{n-1}}\;+\;.........\;+\;p_{n-1}\frac{dy}{dt}\;+\;p_{n}(t)y\;=\;g(t)\\\\ $
$n=6\;\;\;\;\;\;\;\;\;\;p_{1}(t)=p_{2}(t)=p_{4}(t)=p_{5}(t)=g(t)=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;p_{3}=\frac{t^2}{t^2-4}\;\;\;\;\;\;\;\;p_{6}(t)=\frac{9}{t^2-4}\\\\$
$p_{1}(t),p_{2}(t),p_{4}(t),p_{5}(t),g(t)$ continuous everywhere on $\mathbb{R}.$
$p_{3}(t) ,p_{4}(t)$ is continuous everywhere on its domain $\mathbb{R}-\{2,-2\}$
The solutions of the given equation are sure to exist on $\mathbb{R}-\{2,-2\}$
