#### Answer

The solutions of the given equation are sure to exist on $(-\infty,0) \cup (0,1) \cup (1,+\infty)$.

#### 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)\\\\ $
$n=4\;\;\;\;\;\;\;\;\;\;p_{1}(t)=0\;\;\;\;\;\;\;\;\;\;p_{2}=\frac{e^t}{t(t-1)}\;\;\;\;\;\;\;\;p_{3}(t)=0\;\;\;\;\;\;\;\;\;\;p_{4}(t)=\frac{4t}{t-1}\;\;\;\;\;\;g(t)=0\\\\$
$p_{1},p_{3},g$ constants functions that are continuous everywhere on $\mathbb{R}.$
$p_{2}$ is continuous everywhere on its domain $(-\infty,0) \cup (0,1) \cup (1,+\infty)$
$p_{4}$ is continuous everywhere on its domain $(-\infty,1) \cup (1,+\infty)$
The solutions of the given equation are sure to exist on $(-\infty,0) \cup (0,1) \cup (1,+\infty)$