Answer
The solutions of the given equation are sure to exist on $\mathbb{R}\;=\;(-\infty,+\infty )$
Work Step by Step
The given higher order linear equation is written in the form :
$\frac{d^ny}{dt^n}\;+\;p_{1}(t)\frac{d^{n-1}y}{dt^{n-1}}\;+\;.........\;+\;p_{n-1}\frac{dy}{dt}\;+\;p_{n}(t)y\;=\;g(t)\\\\$
$n=4\;\;\;\;\;\;\;\;\;\;p_{1}(t)=4\;\;\;\;\;\;\;\;\;\;p_{2}(t)=\;p_{3}=0\;\;\;\;\;\;\;\;\;\;p_{4}(t)=3\;\;\;\;\;\;g(t)=t\\\\$
$p_{1},p_{2},p_{3}$ and $p_{4}$ are constant functions and g is an identity function.
therefore all of them are continuous on the interval $I=\mathbb{R}$.
The solutions of the given equation are sure to exist on $\mathbb{R}\;=\;(-\infty,+\infty )$
