Answer
see solution.
Work Step by Step
${y}'''+p_{1}(t){y}''+p_{2}(t){y}'+p_{3}(t)y=0\\\\$
At: $\;\;\;\;\;\;\;y=y_{1}(t)v(t)\\\\$
${y}'={y_{1}}'(t)v(t)+y_{1}(t){v(t)}'\\\\$
${y}''={y_{1}}''(t)v(t)+{y_{1}}'(t){v(t)}'+{y_{1}}'{v(t)}'+y_{1}{v(t)}''=\;$
$\;\;\;\;\;\;{y_{1}}''(t)v(t)+2{y_{1}}'(t){v(t)}'+y_{1}{v(t)}''\\\\$
${y}'''={y_{1}}'''(t)v(t)+{y_{1}}''(t){v(t)}'+2{y_{1}}''{v(t)}'+2{y_{1}}'{v(t)}''+{y_{1}}'{v(t)}''+y_{1}{v(t)}'''=\;$
$\;\;\;\;\;\;\;{y_{1}}'''(t)v(t)+3{y_{1}}''(t){v(t)}'+3{y_{1}}'{v(t)}''+y_{1}{v(t)}'''\\\\$
substitution in equation;
${y_{1}}'''(t)v(t)+3{y_{1}}''(t){v(t)}'+3{y_{1}}'{v(t)}''+y_{1}{v(t)}'''+p_{1}(t){y_{1}}''(t)v(t)+2p_{1}(t){y_{1}}'(t){v(t)}'+p_{1}(t)y_{1}{v(t)}''+p_{2}(t){y_{1}}'(t)v(t)+p_{2}(t)y_{1}(t){v(t)}'+p_{3}(t)y_{1}(t)v(t)\\\\$
$ \Rightarrow \;\;[y_{1}(t){v(t)}'''+(3{y_{1}}'+p_{1}(t)y_{1}(t)]\;{v(t)}''\;+\;[3{y_{1}}''(t)+2p_{1}(t){y_{1}}'(t)+p_{2}(t)y_{1}(t)]\;{v(t)}'\;$
The coefficient of $v$ is equal $0$ because $y_{1}$ is the solution for the equation.
