Answer
The solutions of the given equation are sure to exist on $(-\infty\;,\;0) \cup (0,+\infty)$.
Work Step by Step
The given higher order linear equation is written in the form :
$\frac{d^ny}{dt^n}\;+\;p_{1}(t)\frac{d^{n-1}y}{dt^{n-1}}\;+\;.........\;+\;p_{n-1}\frac{dy}{dt}\;+\;p_{n}(t)y\;=\;g(t)\\\\$
$n=3\;\;\;\;\;\;\;\;\;\;p_{1}(t)=\frac{sin(t)}{t}\;\;\;\;\;\;\;\;\;\;p_{2}(t)=0\;\;\;\;\;\;\;\;\;\;p_{3}(t)=\frac{3}{t}\;\;\;\;\;\;g(t)=\frac{cos (t)}{t}\\\\$
$p_{2}$ is a constant function and $p_{1},p_{3},g$ are continuous everywhere except at t=0.
Therefore, all of them are continuous on the domain $(-\infty\;,\;0) \cup (0,+\infty)$
The solutions of the given equation are sure to exist on $(-\infty\;,\;0) \cup (0,+\infty)$.
