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